Status of Cantor's 'Continuum Hypothesis'

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Re: Status of Cantor's 'Continuum Hypothesis'

Postby Natural ChemE on October 26th, 2015, 8:25 pm 

someguy1 » October 24th, 2015, 8:54 pm wrote:
Natural ChemE » October 24th, 2015, 7:13 pm wrote:someguy1,

I feel that I understand this subject a lot better now. Thank you for your time in explaining it to me!

Oh you are very welcome. Thanks for raising so many stimulating issues. I hope you're not done! We've just cleared out the underbrush, the interesting stuff is ahead.

No matter, if anything occurs to you feel free to ask. You had such a huge rush of wild interconnected ideas ... did I manage to address most of it at some level?

Yeah, I definitely feel like a lot of my confusion has been cleared up as far as it can be.

For example, the problem with my bijection was that it excluded non-terminating decimals, right? To me, numbers are constructions, and non-terminating decimals are just a type of construction that is itself non-terminating.

I see the idea of the set of reals being non-countable as a direct consequence of the implicit construction approach that separates a number's construction from a set's construction. This is, my construction for the set of all reals ultimately constructs all terminating decimals that the construction for a numeric expression of ever would. However, my set of all reals, as defined by my construction of it, would never include a non-terminating decimal in the same sense that no construction of would ever fully reproduce it in decimal form, either.

Due to this perspective, I see the non-countablity of the set of reals as a direct consequence of the implicit, unacknowledged (and thus ambiguous) decision that the construction of a set must occur only after the construction of the members to be added to that set. This makes sense when folks see numbers and sets as fundamentally distinct; a constructionist like me sees both as merely being constructions themselves, making merged constructions possible.

While the full point is above, I would say this in another way. This is, I see cardinality as sort of a loop process in modern programming, wherein the order of cardinality is directly equivalent to the depth of nested loops that must go arbitrarily far. For example, any set is countable if and only if it can be constructed in code as
    Code: Select all
    List<TypeOfSetElements> countableSet = new List<TypeOfSetElements>();
    for (int i=0; i -> infinity; ++i)
    {
        countableSet.Add( FiniteProcedureForForNewElement(i) );
    }
. Then the issue with my construction of the set of reals is that it does not contain a FiniteProcedureForForNewElement(i); rather, some elements (such as the non-terminating decimals you pointed out) must be generated by infinite procedures InfiniteProcedureForForNewElement(i). As such, the full construction method for the set of reals would have to contain at least one more nested infinite loop to construct the reals.

Assuming such a construction procedure to be like
    Code: Select all
    public TypeOfSetElements InfiniteProcedureForForNewElement(int i)
    {
        TypeOfSetElements toReturn = new TypeOfSetElements();
        for (int j=0; j -> infinity; ++j)
        {
            // Omitted here is some procedure for
            // constructing a real number.
            // This omitted procedure may itself contain
            // nested infinite loops.
            // If this omitted procedure does contain an
            // infinite loop, then the Continuum Hypothesis,
            // which states
            //     "There is no set whose cardinality is strictly
            //      between that of the integers and the real
            //      numbers."
            // is wrong because other sets may be constructed
            // without a nested infinite loop.  However, if this
            // construction doesn't contain an infinite loop,
            // then there's no cardinality between that of
            // integers and real numbers, proving the
            // Continuum Hypothesis to be correct.
        }
        return toReturn;
    }
    .

However, I'm entirely comfortable with the fact that my construction of all real numbers must itself run for infinity to construct any construction that must itself be constructed by running to infinity, e.g. non-terminating decimals. This directly leads me to see the differing cardinalities of the sets of integers and reals to be an immediate consequence of the common decision to construct non-terminating decimals in a nested loop. However, since I can (and have) demonstrated a construction that fully performs the same by merely flattening the loop, this notion of cardanlity is nothing to me but a direct, immediate, and obvious consequence of the classical distinction between numbers and sets of those numbers.

And this is why I'm satisfied to leave it here. I feel that the fundamental issue was that classical mathematicians see there to be a reason to separate the construction of numbers from the construction of a set of those numbers such that cardinalities make sense. And because I'm disinterested in false distinctions between these constructions, I'm entirely content to consider this resolved.

Anyway, the bottom line is that I really do appreciate you explaining this subject to me. I now understand that the distinction between numbers and sets of them to be the fundamental assumption that I was missing, and all the stuff that I've been reading now makes sense within that context.
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Re: Status of Cantor's 'Continuum Hypothesis'

Postby someguy1 on October 27th, 2015, 2:41 pm 

Good morning!

I'm mindful that you said you are satisfied. Therefore I should say, "Nice chatting with you," and leave it at that. This is the Science chat forum, after all.

However, you've chosen to post a number of profound misunderstandings of set theory and the real numbers. I have to respond to at least clear these up.

I agree this can be a little tedious. Going forward, if you'd like me to stop correcting your errors, you're going to have to stop making them, at least in print. If you're satisfied, leave it at that. Don't bait me with falsehoods I have to counter. I haven't the willpower to resist; and I can't abide the idea that someone might read what you've written and take my silence for assent.

I'll make a couple of general points before jumping into the specifics of your post.

* Can you do me a favor and tell me what you've been reading that you mentioned at the end of your post? If I knew what you were reading I could have a much better sense of where your ideas are coming from and how I can most effectively put them into proper mathematical context.

* In modern math, everything is a set. Numbers are sets. The elements of sets are other sets. Everything is a set. Sets are logically prior to numbers. The number 3 is a set. The number pi is a set. The imaginary unit i is a set. Did I mention that everything is a set?

* You cannot generate the set of real numbers with an algorithm, as the term is currently understood. An algorithm is a Turing machine (TM) and no TM can generate all the real numbers. There are only countably many TMs and uncountably many real numbers.

* Constructivism can not be logically refuted. Nor can solipsism, the belief that I am conscious and everything else is a figment of my imagination. I'm a brain in a vat and you're not real. You can't disprove this to me; but it's not a sensible thing to believe. Nor is mathematical constructivism. The construcive real line has more holes than points. The real numbers fail to be topologically complete. The intermediate value theorem is false. Why adhere to such an idea? You can't do science with it. You can't do physics, classical or quantum. You could not go back and rebuild the chemical engineering curriculum based on the constructive real numbers. Why are you claiming to believe something that nobody believes?

https://en.wikipedia.org/wiki/Construct ... athematics)
https://en.wikipedia.org/wiki/Solipsism

* Even in Computer Science, the first thing they tell people after explaining computability, is to prove the existence of a noncomputable problem. https://en.wikipedia.org/wiki/Halting_problem Even CS understands there are noncomputable things in the world.

You're like a guy who knows everything there is to know about fish, and when he sees a cow he says, "That's a fish." When you explain that it's a cow and not a fish, he says, "Don't be silly. Everything is a fish." Don't be that guy. Some things are computable. Some things aren't. Even the computer science professors know that and teach it to the undergrads.


Natural ChemE » October 26th, 2015, 6:25 pm wrote:For example, the problem with my bijection was that it excluded non-terminating decimals, right? To me, numbers are constructions, and non-terminating decimals are just a type of construction that is itself non-terminating.


Ok, this is the beginning of your (to my mind strange) ideas about the real numbers and set theory. I am not even sure what you're saying, but I have to start here to try to untangle this.

We all agree that the set of terminating decimals is countable, and in fact you gave a reasonable account of a way of enumerating them. It's always good to agree on something.

Your second sentence is problemetic. "To me, numbers are constructions, and non-terminating decimals are just a type of construction that is itself non-terminating."

- "Numbers are constructions." If by "construction" you mean construction by an algorithm, then most of the sets used in mathematics are not constructible. For example if I have a finite set like S = {1, 2, 3} I can use the Power Set axiom to form the set of all eight subsets S. In addition, I could write an algorithm to generate all the subsets. It would be a programming exercise for a beginner.

What if we have the set N of natural numbers? Then the Power Set axiom states that the power set of N exists. But there is no possible algorithm to crank out all the subsets of N. There are uncountably many of them and they are not computable. They include all the computable sets and a lot of noncomputable sets too.

You need to distinguish clearly between constructions given by an algorithm; and statements of existence in set theory. The latter are also called "constructions." For example we can "construct" the real numbers within set theory. It's not an algorithmic construction; it's a set theory construction.

Very important to be clear about that. Most of the real numbers can not be constructed algorithmically but they can be "constructed" from the axioms of set theory.

- "non-terminating decimals are just a type of construction that is itself non-terminating." You're confusing nonterminating decimals with noncomputability. The number 1/3 = .333... has a very simple program: "Repeat: Print 3". Likewise sqrt(2) and pi have finite-length programs that generate their digits. We are interested in the noncomputable reals, the ones that don't have any algorithm. Minor point.


Natural ChemE » October 26th, 2015, 6:25 pm wrote:
I see the idea of the set of reals being non-countable as a direct consequence of the implicit construction approach that separates a number's construction from a set's construction.


That's not how it works. In set theory (and by extension, modern mathematics) sets are logically prior to numbers. First we have the axioms of set theory; then we have the universe of sets; and within that universe are models of everything we think of as a number: naturals, integers, rationals, reals, complext numbers, quaternions, transfinite ordinals and cardinals, p-adics, and lots more. All kinds of numbers live in the world of sets. The sets come first. Every number IS in fact some set.

I do know what you're thinking of, forming sets as collections of things. First we have things, then we have collections of things. But the "things" we are collecting are themselves sets. Modern set theory is "pure" set theory. Everything is a set. There are variants of set theory with "urlements" or atoms that are not sets. But in ZFC, everything is a set. The number 46 is a set, the number pi is a set.



Natural ChemE » October 26th, 2015, 6:25 pm wrote:
This is, my construction for the set of all reals ultimately constructs all terminating decimals that the construction for a numeric expression of ever would.


You should be thinking of Turing machines or algorithms. Then you could crank out the digits of pi. (Not in finite time, of course; this is still an abstract execution of an algorithm not constrained by physical law).


Natural ChemE » October 26th, 2015, 6:25 pm wrote:
However, my set of all reals, as defined by my construction of it, would never include a non-terminating decimal in the same sense that no construction of would ever fully reproduce it in decimal form, either.


Fine, so you can only construct the computable numbers. You immediately conclude from that that the noncomputable numbers don't exist. That's a totally unjustified leap. How do you justify it? "I know all about fish so everything's a fish." Really, that is exactly what you're doing here. "I understand computability so everything must be computable." Even computer science acknowledges (and proves!) the existence of noncomputability.





Natural ChemE » October 26th, 2015, 6:25 pm wrote:
Due to this perspective, I see the non-countablity of the set of reals as a direct consequence of the implicit, unacknowledged (and thus ambiguous) decision that the construction of a set must occur only after the construction of the members to be added to that set.


That's a strawman argument because nobody else holds that belief. If you like, think of the universe of sets as a static universe. Look, there's 1, there's 2, and there's 3. And there's {1, 2, 3}. They all got created at the same time if you like.

It's not "implicit" and "unacknowledged." Rather it's something you just made up. I suppose we agree that in everyday language we put the groceries in the bag so that the groceries are logically prior to the bag of groceries. In set theory the groceries and the bag of groceries all coexist, if you want to think of it that way. You are making a distinction that is not present in mathematics.

[It's true that in set theory one can stratify sets into levels for purposes of proving various things. This does not affect my point].

Natural ChemE » October 26th, 2015, 6:25 pm wrote:This makes sense when folks see numbers and sets as fundamentally distinct


Yes but you're the only folk doing that. Mathematicians know that numbers are sets. Numbers are particular sets of interest to us; but they are sets. Of course the "Platonic" numbers aren't sets. Numbers are sets in set theory. If we're doing modern math, numbers are sets. There is no distinction in the ontological status between a number and a set. Both are sets. A number is a particular set that's recruited for the purpose of standing in for a number.


Natural ChemE » October 26th, 2015, 6:25 pm wrote:a constructionist like me sees both as merely being constructions themselves, making merged constructions possible.


Do you intend that to be synonymous with constructivist? https://en.wikipedia.org/wiki/Construct ... athematics)

You've said your training is in chemical engineering. I daresay that if you went back to your undergrad curriculum and tried to re-learn everything you know but using only constructive mathematics, you couldn't do it. You can't do physics, classical or quantum, without the real numbers and uncountable spaces. Hilbert space is not a constructive object. Hilbert space is an infinite-dimensional space in which you can take limits. How are you going to do modern physics without noncomputable concepts?

Constructivism is like solipsism. I can't prove it's wrong. It's just a pointless and somewhat nihilistic thing to believe. You have to deny most of mainstream science, math, and computer science in order to claim there are no noncomputable reals.

I don't take constructivism as a serious intellectual position.

Natural ChemE » October 26th, 2015, 6:25 pm wrote:
https://en.wikipedia.org/wiki/Construct ... athematics)
I would say this in another way. This is, I see cardinality as sort of a loop process in modern programming,


I can't make sense out of that. Nor is it true in any way that I can see. Algorithms can only generate countable sets.

Natural ChemE » October 26th, 2015, 6:25 pm wrote: Loops in programming only produce countable sets.
For example, any set is countable if and only if it can be constructed in code ...


I appreciate your zeal but code fragments are not going to advance this discussion. Programs can only produce computable sets. You're being the fish guy again. You know programming so everything's a program. But even the computer scientists teach the undergrads that there are noncomputable objects. They just don't study them in computer science, they leave them to the mathematicians.

Natural ChemE » October 26th, 2015, 6:25 pm wrote:However, I'm entirely comfortable with the fact that my construction of all real numbers must itself run for infinity to construct any construction that must itself be constructed by running to infinity, e.g. non-terminating decimals.


You are comfortable but you are wrong in a very deep sense. No algorithm or program can produce a noncomputable object. And even Computer Science easily proves that there exist noncomputable objects. You are simply wrong here.


Natural ChemE » October 26th, 2015, 6:25 pm wrote:
This directly leads me to see the differing cardinalities of the sets of integers and reals to be an immediate consequence of the common decision to construct non-terminating decimals in a nested loop.


Nonsense. You can't algorithmically construct the real numbers at all. This is such a key point that as many times as you claim it, I have to take the time to point out that you're wrong. Even the Computer Scientists prove that there are noncomputable objects.


And (minor point) you keep confusing nonterminating decimals with noncomputable numbers. 1/3 = .333... is computable, as well as pi and sqrt(2). They all have algorithms that generate their decimal digits.


Natural ChemE » October 26th, 2015, 6:25 pm wrote:However, since I can (and have) demonstrated a construction that fully performs the same by merely flattening the loop, this notion of cardanlity is nothing to be but a direct, immediate, and obvious consequence of the classical distinction between numbers and sets of those numbers.


What classical distinction? You're the only one claiming there's a distinction. In modern math, numbers are sets. There is no distinction, classical or otherwise. And you can not algorithmically construct the reals, period. Even a quantum computer can't do that. (Off-topic, but quantum computers can not compute anything that a classical computer can't. They can only offer time speedups in special cases.)

Natural ChemE » October 26th, 2015, 6:25 pm wrote:
And this is why I'm satisfied to leave it here.


I'd be satisfied too, however I simply can not allow my silence to be taken as agreement with anything you've written.

Natural ChemE » October 26th, 2015, 6:25 pm wrote:
I feel that the fundamental issue was that classical mathematicians see there to be a reason to separate the construction of numbers from the construction of a set of those numbers


There is no such separation in mathematics. I've said so several times. Numbers are sets. The elements of sets are sets. Everything is a set. In modern math, everything is a set. Even in Category theory, where some things are not sets, one adds new axioms to set theory to ensure that in the end, everything is a set.

https://en.wikipedia.org/wiki/Category_theory

Cardinalities make perfect sense in set theory. There's no distinction between numbers and sets. None. We define cardinalities from the concept of bijective function. I'm pretty sure you must know that.

Natural ChemE » October 26th, 2015, 6:25 pm wrote:
And because I'm disinterested in false distinctions between these constructions, I'm entirely content to consider this resolved.


It's fine that you have other things to do besides continue this thread. I just don't want you to go through life with these bizarre ideas about sets and real numbers. Because your earlier posts show that you have an interest in understanding these things and their relationship to the nature of mind and the universe.

If you say, "I prefer not to continue this discussion," I am perfectly happy to let the matter drop. But then why enumerate a laundry list of misinformation perfectly designed to bait me into responding? Like I say, I haven't the willpower to resist. And I can't allow my silence to be taken for agreement.

Natural ChemE » October 26th, 2015, 6:25 pm wrote:
Anyway, the bottom line is that I really do appreciate you explaining this subject to me.


I'm glad I've been able to help. It would be better for you to just say "Thanks," rather than, "Thanks for explaining that everything's a fish," when I've just tried to show you a cow. I'm sure you can see how frustrating that must be to me.

I truly am glad I've been able to move the conversation forward. But you must understand that the real numbers can not possibly, even in principle, be generated by an algorithm. And that even the computer scientists can prove that problems exist that can not be solved by algorithms. That's the famous Halting problem, one of many problems that can not possibly, even in principle, and even allowing infinite time and space for the computation, be solved by an algorithm.


Natural ChemE » October 26th, 2015, 6:25 pm wrote:
I now understand that the distinction between numbers and sets of them to be the fundamental assumption that I was missing, and all the stuff that I've been reading now makes sense within that context.


There is no such fundamental assumption. And to the extent that you believe there is, you are allowing yourself to reach false conclusions.

But please tell me, what are you reading? That would help me to understand where you are getting all these ideas.
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Re: Status of Cantor's 'Continuum Hypothesis'

Postby someguy1 on October 27th, 2015, 4:11 pm 

ps ... I was thinking about your "wheels within wheels" construction of the real numbers and perhaps you are intuiting the construction (set-theoretic construction, not algorithmic construction) of the ordinal numbers.

I'd been hoping to talk about the ordinals in the context of CH but we never got that far. If you count all the finite numbers 1, 2, 3, ... and put them into a set N and call it w (lower-case omega) you can keep on going w + 1, w + 2, ... all the way to w + w, w + w + w, etc. If you allow this process to go on forever (metaphorical "forever", no time implied) you get the transfinite ordinal numbers, one of the most beautiful elementary objects of math.

When people talk about infinity they always talk about cardinalities. But the cardinals are in fact defined as ordinals so the ordinals are logically prior. And you can prove in ZF (no Choice needed) that an uncountable ordinal exists.

An uncountable ordinal is surely one of the most bizarre objects in math. It's an ordinal, meaning that every nonempty subset has a least element, just like the positive integers. But it's uncountable, just like the reals. Totally bizarre but provable to exist in ZF.

https://en.wikipedia.org/wiki/Ordinal_number
https://en.wikipedia.org/wiki/First_uncountable_ordinal

Anyway I'm thinking that perhaps this is the intuition you're getting at. You run an algorithm countable times then you keep on going by taking unions. I think your InfiniteProcedureForForNewElement is an intuition in the direction of the ordinals. Perhaps.
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Re: Status of Cantor's 'Continuum Hypothesis'

Postby Natural ChemE on October 27th, 2015, 10:40 pm 

someguy1 » October 27th, 2015, 1:41 pm wrote:Going forward, if you'd like me to stop correcting your errors, you're going to have to stop making them, at least in print.

Hah yeah, that's how forums normally work, right?

Seriously dude, if you can prove me wrong, I'll be so grossly interested that I might just pay you consulting fees. Also, I realize that I have a "mod" tag next to my name, so please allow me to make this perfectly clear: to the maximum extent that the other moderators allow (and I'd ask other mods to respect my wishes on this), you can be as ruthless and blunt in your assessments of my arguments as you please. I truly do appreciate your attempts at criticism, though I'll appreciate them yet more if they can be supported. Right now we disagree, but you may yet show me that I'm wrong.

That said, please allow me a bit to take care of some business, then I'm gonna read through your responses!
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Re: Status of Cantor's 'Continuum Hypothesis'

Postby Natural ChemE on October 28th, 2015, 1:42 am 

someguy1,

Okay, after reading through your points, there's a ton of stuff that I'd disagree with - I'd probably write a huge response debunking your stuff point-by-point, and you'd respond in similar fashion, sinking tons of time. I can't really afford that much time, and I suspect you couldn't, either.

I think that our problem is that we disagree on a rather fundamental level. As such, I'd recommend that we focus on the most fundamental issues involved before discussing derived points. As we build up from a mutually agreeable fundamental basis, we'll enjoy having that solid basis for clear reference. I'd hope that this would allow us to stay on the same wave length.

Would you agree to this approach, perhaps with modifications?

If so, I'd propose starting at computational theory (though our true starting point will be discussing what our "starting point" should be =P). I personally perceive mathematics to be a specific mode of operation for computers (usually human brains, but increasingly other devices), so I think it's useful to acknowledge what, at this very fundamental level, math can and can't be. Once we agree upon this sort of basis, we can start adding in axioms, etc., with the ultimate goal of reproducing a general notion of the concepts that we're disagreeing over, e.g. infinities, computability, and cardinaliy.

PS - If you were interested in my response to any particular point, I'd be happy to provide it. Just, in the interest of ultimately syncing up our discussion, I'd like to primarily focus on arriving at a common framework for us to refer to in our points.
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Re: Status of Cantor's 'Continuum Hypothesis'

Postby Natural ChemE on October 28th, 2015, 2:29 am 

someguy1,

If I could add one question as we start, I'd like to ask you about if you agree with my take on logic-must-be-constructable. I feel that this question is fundamental to our discussion and perhaps a source of disagreement.

I argue:
  1. Logic is a description of procedures conductable by computers, including both human brains and electronic computers.
    • Computers conduct logic by enacting the procedures it dictates. For example, evaluating to requires the following steps (plus others that I'm not showing for brevity):
      1. Identifying the function of addition.
      2. Identifying the arguments to this function ( and then another ).
      3. Conducting the function of addition on the arguments and .
    • Computers can only use logic that the can conduct. Therefore, logical constructs that cannot be conducted on any computer are effectively non-existent.
      • This is, I don't care what you mean by "infinity" or how you construct it - but if it's a valid concept that somehow exists somewhere in your mind, then there must be a way you thought of it. Thinking of it was constructing it. Any arguments over what different people call "infinity" can be resolved by exactly specifying exactly how you constructed your thought of "infinity" such that others can reproduce your steps. Academics often describe such brain-construction-procedures in terms of logical operations. I'd prefer this route.
    Is this agreeable? Because, to me, this is an extremely fundamental point. And if you don't agree, I want to know how you're thinking thoughts that your brain isn't able to put together.
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    Re: Status of Cantor's 'Continuum Hypothesis'

    Postby Natural ChemE on October 28th, 2015, 3:50 am 

    someguy1,

    In violating my own proposal (so please see this as tangential), if you'll allow me this tangent, I'd like to restart my bijection in a more classical thought experiment (which violates constructivism, but I suspect that this isn't an issue for you). I figure that this construction of it will avoid concerns over irrationals, etc.

    So, hypothetically, say that there's a mathematician in a room. The mathematician is immortal, ageless, never needs to sleep, eat, etc., etc. Every day, he makes a blog post with two numbers: one integer number and one real number. These blog posts don't necessarily correspond even when able. For example, the mathematician is free to post as an integer and as the real, even though . He's also careful to never post a duplicate member to a set in which it already exists.

    I'd say that:
    1. Over infinity, he constructs the set of all integers and all reals.
      • We can define a bijection as, "Any member of either set corresponds to the member of the other set in the same blog post."
        • For example, to map in the set of all reals to in the set of all integers by looking up the blog post on the day that he posted .
      As far as I can tell, there's no fundamental reason why we can't say that this bijection works - it seems like a simple enough process to me. I figure that, at some point in deriving the rules for modern mathematics, there's an asserted assumption that prohibits this construction. This is, either:
      1. this bijection works; or
      2. this bijection fails.
      But, what's the missing assumption that prohibits this bijection?

      As I understand it, the Löwenheim–Skolem theorem already notes this for first-order languages in model theory. And since modern CPU's are first-order evaluators
        Argument:
          CPU instructions are defined by the hardware and never modified, making them the first-order functions that operate only on data. I'm sure that some folks will argue that setting the function pointer register is a higher-order operation, and they certainly can argue this; however, it's equally valid to say that the CPU's intrinsic function merely operates according to the dumb value in that register, which makes it first-order. For folks trying to picture a CPU as first-order, just see there to be some function CPU() that repeatedly operates on the collection of the states accessible to the CPU, including the function pointer.
      that can implement higher-order logic (e.g. Lisp or any other reflective programming platform), then all logic systems have no fundamental concept of cardinality until some element in higher-order logic systems introduces it.

      Therefore, I would say that cardinality exists only in higher-order logic (per the Löwenheim–Skolem theorem) and even then only if one of the functions-that-operates-on-other-functions introduced to the first-order basis of that higher-order language effects a meaningful, consistent concept of cardinality.

      Further, as proven by Paul Cohen, the continuum hypothesis, ZF, and the axiom of choice are all mutually independent, right? So whatever folks are claiming is causing this reduction isn't within ZFC.

      Anyway, that's just the thought-before-bed tonight. I think that, following our derivation of math, we'll ultimately come to this conclusion. Just curious what the source of confusion will prove to be. Metamathematics for the win, right? =P
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      Re: Status of Cantor's 'Continuum Hypothesis'

      Postby Natural ChemE on October 28th, 2015, 6:23 am 

      someguy1,

      Okay, one more tangential post. I'll probably split off my tangents here into different threads to avoid derailing this subject through excessive branching. =P

      So, Cantor's diagonal argument,
        Image,
      goes something like (please take my handshake math with a grain of salt; too tired to check it):
      1. Construct a set of binary numeric strings of length .
        • For any set with members of length , we can always construct a member that does not exist. To do this, start:
          • Consider a new, empty string.
          • Set the first member of that new string to the compliment of the first member of the first string in the set.
          • Set the second member of that new string to the compliment of the second member of the second string in the set.
          • [...]
          • Set the th member of the new string to the compliment of the th member of the th string in the set.
          • Because this new string differs from any set member in its th member, this new strong necessarily does not exist in the set.
        • We can actually go further, allowing members to be non-contained strings of length to be constructed through arbitrary bijections between set-member and string-member .
        • After this, the arguments get sketchy. I can see differing ways to go forward, but since none of them are correct, I'm not sure which to assert - I can be accused of a straw man no matter which I pick. However, I'll arbitrarily pick one and invite critics to select their own route forward.
          • Consider an infinite set of length of strings of length . This level of infinity, i.e. this value of , is the cardinality of the set of integers.
            • No matter the value of , there will always exist members of length that are not members of the set, per the construction specified in Steps (2) and (3).
              • Therefore, any set of countably-infinite-long strings (i.e. strings of length ) cannot be enumerated with countably infinitely many (i.e. ) indices.
                • In fact, a countably infinite set must lack members, then such a set must be only
                  of the full set's size. And since is infinite, then the set of integers is infinitely smaller than any set composed of members of countably many bits (0-or-1 values).
                  • Real numbers have infinitely long representations, so let's assume that they're of countably infinite length.
                    • Therefore, the set of all reals is infinitely larger than the set of all integers.

                    The above argument considers countably many set members with countably many digits. Due to the factor where , this argument would hold any time the precision of the members is finitely mappable to the set size.

                    So I suppose that we could then note that, for each bit of precision, we'd need infinitely many additional set members. I think that this is the obvious, intuitive mapping that, for any number integer , there must exist an infinity of reals in . I think that we all get that natural feeling, and Cantor's diagonal argument is one way to arrive at it as a conclusion.

                    However, despite how natural this approach feels, I can't see why we couldn't say that the size of the set of reals is infinitely larger than the precision of any individual real, recovering the countability of the set of reals? This type of construction is also very natural.

                    How does this argument do anything but provide a pseudo-justification for the concept of fairness in construction? This is, why do the infinities of set size and member entropy have to be on the same order? If we drop this assumption, Cantor's diagonal argument breaks.
                      Note:
                        More generally, Cantor's argument holds so long as the size of the set of reals divided by the square of the entropy of a real is infinitesimal. In his basic argument, he asserts that the size of the set of reals is equal to the entropy of a real, such that this inequality ends up holding. His argument would even hold with a small relaxation, so long as the entropy of the reals is not reduced down to the square root of the size of the set of reals. It's at that point his argument would break.
                    So, again, I get lots of reasons that the set of reals feels larger than the set of integers, but we can trivially construct them with the same cardinality, including with easily-described bijections. It just seems to be that our usual system of cardinalities is a direct consequence of the choice of construction method such as the choice to make the set size finitely scalable with the entropy of reals.

                    PS - Also, if we do assume that the size of the set of reals must scale finitely with the entropy of the reals, then doesn't the factor prove the continuum hypothesis? Or I guess it'd be easier to say that the set of reals is times larger. This is, it's first-order with respect to , therefore there's no cardinality strictly between the set of integers and the set of reals. So as long as folks accept the assumptions necessary to arrive at the current notion of cardinality, then the continuum hypothesis must be true.

                    ...unless you're willing to allow sets that differ by non-integer powers of , e.g. a set of reals constructed with entropy rather than ? In this case we can reject the continuum hypothesis, since even is infinitely smaller than and infinitely larger than for .

                    Bah, just seems like a bunch of arbitrary choices to me. And since differing choices lead to differing results, it's logically inconsistent to refuse to specify constructions.

                    In general, I'd say that cardinality is indeterminate until these assumptions are stated. And it bugs me that folks are acting like the choice of construction is moot to the conclusion when this position is demonstrably false.
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                    Re: Status of Cantor's 'Continuum Hypothesis'

                    Postby Natural ChemE on October 28th, 2015, 7:19 am 

                    someguy1,

                    Alright, it's about dawn, so I'll quit being silly here and get back to work. I can split my tangents later, as to stick with the derive-math-from-computational-theory approach I'd suggested, assuming you're up for it.
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                    Re: Status of Cantor's 'Continuum Hypothesis'

                    Postby someguy1 on October 28th, 2015, 1:28 pm 

                    You said that you were "satisfied" and considered the matter "resolved" and now you stay up all night writing a couple thousand more words? Most of it Cantor denialism? You're going to be disappointed this morning. I'm going to make a few brief remarks and depart from this thread.

                    The bottom line is that I don't argue with Cantor denialists. I've been on the Internet a long time and I've seen hundreds of these threads. Maybe thousands. I simply no longer engage. It's a personal choice.

                    I will respond briefly to each of your new posts.

                    Natural ChemE » October 27th, 2015, 11:42 pm wrote:
                    If so, I'd propose starting at computational theory (though our true starting point will be discussing what our "starting point" should be =P). I personally perceive mathematics to be a specific mode of operation for computers.


                    I thought we established earlier that whatever math is, it's the output of the brain/mind just as is the latest Star Wars movie. If you aren't going to take math on its own terms I honestly have no contribution to make. I can't teach someone to drive who is only here to argue about the existence of steering wheels.

                    But if you want to start at computational theory, what do you think of the Halting problem, which demonstrates that there is an easily-stated problem that can not possibly be solved by any computation? That's another point I've raised several times and that you've chosen to ignore.

                    If you're a finist, that's fine. You might be interested in studying the work of the late Ed Nelson and the still-living Doron Zeilberger (both easily Google-able with lots of accessible material online). They are the two finitists I know who are legit mathematicians. Zeilberger in particular is a computationalist like you. He writes brilliant articles accessible to nonspecialists. You would enjoy him.

                    Finitism is just not something I can argue with you about.

                    Natural ChemE » October 28th, 2015, 12:29 am wrote:
                    If I could add one question as we start, I'd like to ask you about if you agree with my take on logic-must-be-constructable. I feel that this question is fundamental to our discussion and perhaps a source of disagreement.


                    You've used the word constructible so many different ways, without ever defining it, that the question has no meaning. I've challenged you many times on this point and you never respond. But let me ask you this. Are Star Wars movies, being the product of the human mind, constructible? What does that mean? Are you harrumphing about Wookies again?

                    Natural ChemE » October 28th, 2015, 1:50 am wrote:[*]Over infinity, he constructs the set of all integers and all reals.


                    Cantor denialism? The uncountability of the reals is a 140 year old universally accepted theorem of mathematics.

                    It's one thing to make the common mistake of enumerating the finite strings. I was glad to clarify that for you.

                    It's quite another thing to stay up all night writing Cantor denialist screeds. It's beneath you. You are entitled to your opinion but this is a conversation I gave up a long time ago. I daresay that if I took a standard result in chemical engineering and wrote a couple of thousand words trying to deny it, you'd feel the same. I think instead of arguing about Cantor's results we should argue about whether you can use a long plastic straw to transport volatile chemicals over long distances. I hope you see my point but Cantor denialists never do see the point.

                    Natural ChemE » October 28th, 2015, 12:29 am wrote: And if you don't agree, I want to know how you're thinking thoughts that your brain isn't able to put together.


                    If the works of Escher and the true historical example of non-Euclidean geometry don't move you, then I have nothing else to say. Do you understand what a profound shock to the intellectual world was the discovery of non-Euclidean geometry? And then 70 years later it turns out to be true about the real world? Did I not make my point? Did you not understand the point? Perhaps I'm too steeped in this example and assume people are getting the point when perhaps they're not. Did you want me to talk about it more?



                    Sorry, I did not read the rest of this post. You are on the side of the Internet cranks here, do you understand that? I get that you spent a lot of time on this post. It doesn't matter. If I send you a 200 page paper showing how to transport volatile chemicals over long distances using little red cocktail straws scrounged from the local bar, you don't need to read the whole paper before tossing it in the trash. I don't need to read the details of yet another online "refutation" of Cantor.

                    Natural ChemE » October 28th, 2015, 5:19 am wrote:
                    I'll quit being silly here


                    If only. Then we could discuss some math.

                    Natural ChemE » October 28th, 2015, 5:19 am wrote:I can split my tangents later, as to stick with the derive-math-from-computational-theory approach I'd suggested, assuming you're up for it.


                    I'm dismayed that you've devolved into Cantor crankery. I can no longer participate. And even if you backed off the Cantor angle, I'm not a finitist. If you are, that's fine. You should read Zeilberger. Here's a link to his "Opinions," which are a collection of highly readable and entertaining articles about finitism and computationalism. I think you'll find a kindred spirit.

                    http://www.math.rutgers.edu/~zeilberg/OPINIONS.html

                    I've appreciated our conversation. I don't mean for this to end badly. It's just that honestly, after taking the time to work through your initial multi-post outpouring last month, I took a long time to craft a thoughtful reply that got to the essence of your concerns. I admit that I am genuinely dismayed that your argument comes down to trying to enumerate the reals. I feel punked. Do you understand that in every model of ZF, including the constructible universe, the non-enumerability of the reals is a theorem? Do you understand that L is a proper class, a mathematical object so big that it's too big to be a set? Do you understand that the construction of L involves transfinite recursion over the class of ordinals? I tried to explain that L is hardly the finitist paradise you think it is, but to no avail.

                    I'm also disappointed that you've never engaged with any of the specific points I've made. Not the "product of the mind" argument, which shows that the output of our minds need not conform to logic or physical law, yet is nonetheless meaningful. Not the Halting problem, which proves there are easily-stated problems that can never be solved by a computer even given infinite time and space. Not the example of non-Euclidean geometry, which shows that a mathematical idea may be considered impossible for 2000 years; then be shown possible but regarded as a curiosity; and finally shown to be the TRUE nature of reality. Not the fact that Cantor's results are true in every conceivable model of set theory because they follow logically from the axioms.

                    You have chosen not to engage directly with a single one of these points. Your idea is to enumerate the reals, something that cannot be done in any consistent theory of math.

                    I simply choose not to engage finitists, who deny the bulk of modern math with an argument that's at least interesting. I love reading Zeilberger, but I wouldn't try to argue with him. Let alone Cantor deniers, whose arguments can never be logically consistent since they contradict the axioms of set theory. [Unless, of course, the axioms of set theory are themselves inconsistent. Is that perhaps something you're arguing? Ed Nelson, who died just a year ago, was convinced the Peano axioms are inconsistent and spent the bulk of his career trying to prove that. He was by far the most reputable finitist. https://en.wikipedia.org/wiki/Edward_Nelson].

                    I know you are a serious and thoughtful person and I regret that my words probably come off as disputatious. I simply cannot tell you how profoundly disappointed I am that your entire argument comes down to trying to enumerate the reals.

                    Peace.
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                    Re: Status of Cantor's 'Continuum Hypothesis'

                    Postby Natural ChemE on October 28th, 2015, 6:19 pm 

                    someguy1,

                    Your disappointment is understated. I fail everything like this. All my life, I've denied all assertions - be they Cantor's or Einstein's - unless I could derive them myself. As a consequence, I know and believe in very little; if I can't derive it, I can't use it.

                    In this case the assertion is that there's a fundamental reason that all possible systems of math absolutely must prohibit bijections between the set of all integers and the set of all reals. And since I can't derive why that assertion must be true, it merely seems to be an axiom for a strict subset of possible maths.

                    But, no worries, I'm not gonna go selling a perpetual motion machine based on my perspective here. Merely, I personally can't use the concept of cardinalities in general math because I'm unable to derive it without asserting reductions. Therefore, I personally can't work with cardinalities outside of these constrained contexts, nor will I be able to understand other works that rely on these concepts except within these constrained contexts. For me to understand these concepts in the general context, I would need to come to understand why all alternatives are fundamentally impossible

                    I dunno why this is so bad though. I get the literature just fine so long as I read it with the understanding that the authors are working under the constraints that I personally require for my own derivations of the underlying principles. If anything, I simply feel more free because I can also work outside of those contexts. This is a pretty cool trick - whenever someone proves something impossible under certain constraints, you can often prove it possible by relaxing those constraints. Seems liberating to me, though it's possible that this liberation would be of no value if it turns out that the assertion is fundamentally true and thus the space that I see as open is of zero size.

                    Also, you're entirely free to attack Chemical Engineering! The very best way to learn a discipline is to derive it while attacking those derivations with all you've got. I've been pretty dang successful doing this sorta thing, and I highly recommend it.
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                    Re: Status of Cantor's 'Continuum Hypothesis'

                    Postby someguy1 on October 28th, 2015, 7:27 pm 

                    I didn't read your latest yet but I just wrote this up. Perhaps it will help.


                    Proof using simple finitary, constructive logic that there is no bijection between the naturals and the reals, in any mathematical ontology you like

                    It occurs to me that I know how to elevate this conversation and move it forward. Clearly we are stimulating each other's thoughts. This could be productive if we can just get ourselves on the same page, right?

                    I am going to prove to you that the set of natural numbers has strictly smaller cardinality than the set of real numbers .

                    And I am going to do so using a proof that is valid in every possible mathematical ontology: ultrafinitist, finitist, constructivist, and mainstream (infinitist, I guess we could call it).

                    Once we do that -- once you see that there is no possible way, even in your constructive world, to biject the naturals and the reals, perhaps you'll come around to actually talking about math. Or not. But at least I'm going to give this my best shot.

                    First, let me briefly outline the mathematical ontologies for clarity and reference.

                    * Mainstream math. We accept large infinities, uncountable sets, noncomputable sets, nonconstructive proofs. This is where I live.

                    * Constructivism. We accept infinite sets as long as they can be constructed. I imagine among the constructivists there must be discussions regarding exactly what types of construction are legal. Doesn't interest me but it's got a Wiki entry so it must interest someone. I think this is where you live perhaps?

                    * Finitism. The physical universe is finite, after all, so we might as well disallow infinite sets. We do allow arbitrarily large finite sets, though: 1, 2, 3, 4, 5, 6, ... all exist. What does NOT exist is the COMPLETED set of all of them. This idea has an interesting set-theoretic model, known as the hereditarily finite sets.

                    https://en.wikipedia.org/wiki/Finitism
                    https://en.wikipedia.org/wiki/Hereditarily_finite_set

                    * Ultrafinitism. If one is going to require that math must conform to the known laws of physics, this is the only sensible choice. We know there are only hydrogen atoms in the universe. And one time I worked out to my own satisfaction that there are, give or take a handful, around quarks in the universe. Those are very small numbers in the mathematical scheme of things.

                    So what sense can it make to talk about, say, ? Such a number makes no possible sense in the physical world. [Exponents precede from right to left, so this is a really big number].

                    An ultrafinitist is somone who does not believe in the existence of sufficiently large finite sets. Now that's an extreme position; but if math is constrained by physics, as you seem to believe, you have to admit the ultrafinitists have a point. https://en.wikipedia.org/wiki/Ultrafinitism

                    Now I am going to present to you a proof of the uncountability of the real numbers that holds no matter which of these mathematical ontologies you believe. I trust this will settle the matter so that we can begin to talk about something interesting. Cantor denialism isn't interesting because it's so easily refuted.

                    In truth I already showed you this proof earlier, or rather linked to it and strongly suggested that you look at it. Perhaps you didn't. So here it is again. It's based on Cantor's theorem, which is a proof of uncountability that has NOTHING TO DO WITH DECIMALS or any type of infinitary reasoning at all. It's so simple I wish they'd teach this proof instead of the diagonal argument, which confuses everyone.

                    https://en.wikipedia.org/wiki/Cantor%27s_theorem

                    First, a preliminary theorem to relate the powerset of the naturals to the real numbers.

                    Theorem: There is a bijection between the set of real numbers and the powerset of the natural numbers. Recall that the powersset of a set is the set of all subsets of the original set. So if then

                    .

                    You see that if the cardinality of a (finite) set is , then the cardinality of its powerset is . This fact actually applies to infinite sets as well (with a suitable definition of exponentiation) and in fact this leads directly to the statement of the Continuum Hypothesis, so if we ever get there, this is good to keep in mind.

                    If we think of the real numbers (between zero and 1) as infinitely long decimal expressions, we can also think of them as infinitely long binary expressions, in other words bitstrings like 010101001010001111000... where the bits go on forever. It's easier to do this argument for reals between 0 and 1 so I don't have to worry about the integers to the left of the binary point, and in fact I can just leave off the binary point for all the difference it makes. I'm really just talking about bitstrings.

                    Now suppose I have a bitstring like 1010101010... where 1's and 0's alternate forever. If I number the bit positions starting from the left as 1, 2, 3, 4, ... you can see that the bit positions containing a '1' pick out a particular subset of the natural numbers, namely the odd numbers 1, 3, 5, 7, ... In other words every bitstring specifies a subset of the naturals; and conversely, every subset of the naturals specifies a bitstring.

                    There is a bijection between the set of subsets of the naturals, and the reals.

                    And please note. I didn't say anything about mathematical ontology. You accept the full powerset? Then you have a bijection between the full powerset and the full collection of bitstrings. You only like constructible subsets of the naturals? You have a bijection to the constructible bitstrings. You only like finite or ultrafinite sets of naturals? Those biject to the finite or ultrafinite bitstrings.

                    Whatever your mathematical ontology, this bijection works. That's important. So from now on when we talk about real numbers, we don't care about whether they're represented as decimals, bitstrings, or subsets of . They're all the same thing.

                    I hope you can see that ignoring the integers to the left of the binary point does not alter this argument in any material way. If I need to work out the details I suppose I could but I don't think anyone ever bothers. Proof by handwaving.

                    Theorem (Cantor, 1891). There is no bijection from a set to its powerset.

                    Proof:

                    Let be a set, its powerset.

                    Let be any function. We will show that the assumption that is a bijection leads to a contradiction.

                    If you think about any function from to , it's clear that for any given element , either or . For example if we may have mapped to or we may have mapped to .

                    Now consider the set .

                    is a subset of ; so if is a bijection, then must map some element to .

                    I ask you: Is ? If it is, then hence .

                    But if , then hence !

                    In other words is an element of if and only if is NOT an element of .

                    This is absurd. Hence THERE IS NO SUCH that is mapped by to .

                    Put another way: Given any function whatsoever from a set to its powerset, we can easily construct a member of the powerset that can not possibly be hit by that function. There can never be a surjection from a set to its powerset. QED.

                    By the way, the map that sends to is an injection, so we know that a set is less than or equal in cardinality to its powerset. And we've just proved that it's not equal. So the cardinality of any set is strictly less than that of its powerset.

                    Putting this together with the earlier bijection between and , we conclude that there is no possible bijection from to .

                    Remark 1: Note that this proof uses no infinitary reasoning, makes no appeals to infinite anything. It's so simple and straightforward that they should just ban the diagonal proof on the grounds that it confuses people on the Internet.

                    Remark 2: Note that this proof is totally obvious in the finite case. A set with 3 elements has an 8-element powerset. A set with 10 elements has a 1024-element powerset. It is extremely instructive to work through the details of the proof in the finite case and I recommend that readers do so.

                    Remark 3. Now we can do the same thing with and and on and on, obtaining an endless hierarchy of strictly larger and larger cardinalities. And you see we did this with simple finitary logical reasoning; and in each case we constructed a specific member of the powerset that can't possibly be hit by any function from the underlying set. I assume that you approve of this proof, it's finitary and constructive. That's the thing about Cantor's beautiful and revolutionary work. It's perfectly logical from the ground up.

                    Remark 4: I have actually saved you a boatload of time and energy. All the work you've ever done and all the work you ever would have done to try to enumerate the reals? You can go do something else more productive instead. There is no such enumeration and you don't need any kind of infinitary reasoning to prove it. It's straight logic.

                    Remark 5: We can define the sequence of transfinite cardinals . I have not done the preliminary work to explain this, that's another lengthy post, but it's not difficult.

                    Which one of those is the cardinality of the reals?? CH is the claim that the cardinality of the reals is . In the absence of CH, the cardinality of the reals could be almost anything, subject to some technical restrictions. Without CH, for all we know, the cardinality of the reals is . There's no way to know. That's what CH is about.

                    If you have any questions about anything I've written, please please please ask.

                    Once we're past this, you need spend no more time in your life trying to enumerate the reals or argue that cardinality is some artifact of mathematicians misunderstanding set theory. And then you and I can talk some math and some philosophy.
                    Last edited by someguy1 on October 28th, 2015, 8:01 pm, edited 6 times in total.
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                    Re: Status of Cantor's 'Continuum Hypothesis'

                    Postby someguy1 on October 28th, 2015, 7:41 pm 

                    Natural ChemE » October 28th, 2015, 4:19 pm wrote:In this case the assertion is that there's a fundamental reason that all possible systems of math absolutely must prohibit bijections between the set of all integers and the set of all reals. And since I can't derive why that assertion must be true, it merely seems to be an axiom for a strict subset of possible maths.


                    I hope you will take the time to review and understand Cantor's beautiful proof above. We needn't feel bad that we didn't think of this ourselves. Cantor was a genius. His short proof I gave above should be much more widely known. It doesn't induce the confusion in students that the diagonal argument does. By the way neither of these were Cantor's original proof. His first proof from 1872 was topological in nature.

                    ps -- Different point. I still detect that you think math must be constrained by physics. The universe is finite therefore all these crazy infinities in math must be wrong.

                    I thought we handled this. Escher, the Wookies. Crazy things come out of the mind that are nevertheless interesting and meaningful. Math is one of those things. And in the case of non-Euclidean geometry, one of the craziest things ever to come out of math turned out to be the following century's basic physics.

                    So we MUST take math on its own terms. Yes the universe is finite and for sake of discussion I don't care if my mind is a computer. But just as I take Escher and Wookies on their own terms, I take the crazy infinities of math on their own terms. And who's to say they won't be the foundation of next century's physics.

                    I really thought you agreed with that. Who cares if it's true. Wookies aren't true but people are lining up to spend money on them. Non-real things are very important!! You should think about that.
                    Last edited by someguy1 on October 28th, 2015, 8:37 pm, edited 1 time in total.
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                    Re: Status of Cantor's 'Continuum Hypothesis'

                    Postby Natural ChemE on October 28th, 2015, 8:27 pm 

                    someguy1,

                    I suppose that geniuses are as inscrutable as the wizards of yore.

                    I'll check out that new theory later tonight. Just gotta get some groceries and write this silly book. Thank goodness Windows 10 runs Office 2013 so much better than Windows 8 did; the embedded links and such don't crash out huge documents anymore! Though I guess 2016 came out and gotta do that?

                    Bah gotta pull myself away from math stuff because it's too much fun. This book's due Friday, plus tons of other stuff. Life's crazy, but I'm gonna read that proof and catch up soon. ...ideally not too soon, because I really should sleep one day, but I suppose we all have our addictions.
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                    Re: Status of Cantor's 'Continuum Hypothesis'

                    Postby someguy1 on October 28th, 2015, 8:41 pm 

                    Natural ChemE » October 28th, 2015, 6:27 pm wrote:someguy1,

                    I suppose that geniuses are as inscrutable as the wizards of yore.

                    I'll check out that new theory later tonight. Just gotta get some groceries and write this silly book. Thank goodness Windows 10 runs Office 2013 so much better than Windows 8 did; the embedded links and such don't crash out huge documents anymore! Though I guess 2016 came out and gotta do that?

                    Bah gotta pull myself away from math stuff because it's too much fun. This book's due Friday, plus tons of other stuff. Life's crazy, but I'm gonna read that proof and catch up soon. ...ideally not too soon, because I really should sleep one day, but I suppose we all have our addictions.


                    Take your time. Hope your work's going well. The meat of my previous post is this short proof. All the rest is commentary. Here's the stripped down version.

                    Just read through this proof and then forget about it and let it percolate in your brain. From the mind of Cantor to our minds ... his computer brain was a very good one, don't you think?

                    Theorem (Cantor, 1891). There is no bijection from a set to its powerset.

                    Proof:

                    Let be a set, its powerset.

                    Let be any function. We will show that the assumption that is a bijection leads to a contradiction.

                    If you think about any function from to , it's clear that for any given element , either or . For example if we may have mapped to or we may have mapped to .

                    Now consider the set .

                    is a subset of ; so if is a bijection, then must map some element to .

                    I ask you: Is ? If it is, then hence .

                    But if , then hence !

                    In other words is an element of if and only if is NOT an element of .

                    This is absurd. Hence THERE IS NO SUCH that is mapped by to .

                    Put another way: Given any function whatsoever from a set to its powerset, we can easily construct a member of the powerset that can not possibly be hit by that function. There can never be a surjection from a set to its powerset. QED.

                    As a special case, note that the reals can be bijected to the powerset of the naturals; therefore there is no bijection from the naturals to the reals. Bam. Done.

                    That's it. Once you get this you have the infinite ladder of powersets in your mind forever. And yes, your finite computer brain produced it. Your finite computational brain just gave you a clear logical construction of an endless hierarchy of larger and larger transfinite cardinalities.

                    What do you think that means?
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                    Re: Status of Cantor's 'Continuum Hypothesis'

                    Postby Natural ChemE on October 29th, 2015, 1:18 am 

                    someguy1,

                    Yup, looks like we agree. The whole powerset thing is related to arguments for binomials, as I'd noted last night as a handshake problem.

                    Apparently that axiom I was originally missing - that non-fundamental assumption I'd been referring to - is simply the axiom of power set. This axiom specifies a construction method from which cardinalities follow. My "general math" comments were referring to the more general set of math before we reduce the possibility space by introducing the axiom of power set.

                    In some sense, I feel like this closes the issue. I agree with Cantor's conclusion so long as we stipulate that his results apply within the context of the axiom of power set, which appears to be the general consensus. For example, Cantor's arguments don't apply to general set theory.

                    Wanna solve the continuum hypothesis? I suspect that the general note in this post is just about there; we merely need to check if there are allowed construction methods besides the axiom of power set (which is a convention thing, so I'd ask if you know if any other construction methods are allowed in normal theory).
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                    Re: Status of Cantor's 'Continuum Hypothesis'

                    Postby Natural ChemE on October 29th, 2015, 1:31 am 

                    someguy1,

                    Wikipedia noted that constructivists (hello!) have tried to generalize the axiom of power set in constructive set theory. However, since it's first-order, they're not really able to do much with infinities due to the reasoning like that in the Löwenheim–Skolem theorem, so I think that their "weaker" versions are pretty much doomed to stay in sync with the axiom of power set (though not quite sure about that one off-hand).

                    I have to say that it's weird working with first-order logic. There's so much you can't do. Shedding these limitations is a big reason that I love the whole meta-programming thing. Reflective programming is higher-order, which opens up whole new worlds.

                    I know you don't like descriptions in code, but seriously, it's all the same thing - just different semantics (and often far easier demonstration/debugging). Following from this, I'd note that reflective programming is pretty much reflective math by another name; higher-order math is vastly more beautiful than the old stuff.

                    Also, ya know how I started out this thread with a rant about dualism and computation in brains and such? My entire point was that the full set of possible axioms is defined by physics within computers like our brains. These cute little math systems that we talk about are a very narrow subset. The full glory of math isn't represented in modern formalisms.
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                    Re: Status of Cantor's 'Continuum Hypothesis'

                    Postby neuro on October 29th, 2015, 5:52 am 

                    Sorry to interrupt, I'd just propose a layman's observation.
                    It seems that Cantor was so foolish in the first place that he thought one can apply the concept of cardinality - which in principle refers to our capability of "counting", and therefore apply a tag (a finite number) to something - to infinity.

                    The funny thing is that this way cardinality loses its connection with counting: there are infinities that are uncountable, there are infinities of different cardinality... But, come on! as far as we can conceive, infinity is infinity and that's it. Full stop.

                    On the other hand, this is so reasonable to become trivial.
                    Take a line, take it as long as you wish. Take it INFINITELY long.
                    If it is a line (only one dimension), it will not be able to fill a surface, not even that of a stamp.
                    Then take a surface. Take it large. As large as you wish. Take it INFINITELY large.
                    If it is a surface (only two dimensions), it will not be able to fill a space, not even the volume of a cell.

                    No surprise, we know it well: infinity, large as it may be, cannot do it, is not enough, does not reach there.
                    Natural numbers can't do it, NaturalChemE!

                    There is always something farther. Always. Anyway.
                    And it is not simply a problem of lines that cannot enclose a surface, of infinite surfaces that can wrap an object but cannot fill and pervade it.
                    The question - the reason why Cantor is right and higher infinities MUST exist and be dealt with - is that no infinite space can encompass more than an instant, or include movement. The question is that no physical reality in space or time can embrace an emotion, an idea, an interpretation, a desire, a passion, a dream.

                    Excuse me. I just felt like saying this, as a homage to the beauty of maths.
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                    Re: Status of Cantor's 'Continuum Hypothesis'

                    Postby Natural ChemE on October 29th, 2015, 9:01 am 

                    neuro,

                    Hah, I do love the romanticism! I guess it's weird to think that such intuitive-feeling principles only apply within a subset of math, huh?

                    About romanticism for math, it's funny to see where folks find it. Personally I love the higher-order stuff; it's fluid, plastic, mutable.. beauty there. Though by contrast, I think some folks find beauty in the idea that math is some enduring, fundamental truth of the universe.
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                    Re: Status of Cantor's 'Continuum Hypothesis'

                    Postby someguy1 on October 29th, 2015, 3:26 pm 

                    Natural ChemE » October 28th, 2015, 11:31 pm wrote:Also, ya know how I started out this thread with a rant about dualism and computation in brains and such? My entire point was that the full set of possible axioms is defined by physics within computers like our brains. These cute little math systems that we talk about are a very narrow subset. The full glory of math isn't represented in modern formalisms.


                    To the extent I was able to discern your intent, you were arguing the exact opposite. That math is constrained by physics hence mathematical infinities were "inconsistent," a word you repeated many times without ever giving a definition. Now you are agreeing that your finite, computational mind is giving you a perfectly clear demonstration of an endless sequences of strictly larger and larger transfinite cardinalities.

                    So what do you think it MEANS that our finite, computational brains are capable of demonstrating the logical existence of all these huge cardinalities?

                    * Do you think all these cardinalities are fictions, like a Star Wars movie or an Escher drawing?

                    * Do you think they are true of mathematics but impossible in the real world?

                    * Do you think perhaps that these higher cardinalities are perhaps the foundation of the physics of the 22nd century or beyond, in the same way non-Euclidean geometry was once thought to be impossible and is now considered to be physically true?
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                    Re: Status of Cantor's 'Continuum Hypothesis'

                    Postby someguy1 on October 29th, 2015, 4:11 pm 

                    neuro » October 29th, 2015, 3:52 am wrote:Take a line, take it as long as you wish. Take it INFINITELY long.
                    If it is a line (only one dimension), it will not be able to fill a surface, not even that of a stamp.


                    There is a continuous space-filling curve.

                    https://en.wikipedia.org/wiki/Space-filling_curve
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                    Re: Status of Cantor's 'Continuum Hypothesis'

                    Postby Natural ChemE on October 30th, 2015, 1:31 am 

                    someguy1,

                    I tried writing a response before, but it feels hard to get this across in a few words after we've had a lot of miscommunications.

                    Okay, so for one-step-at-a-time, lemme ask: I'd say that there's a difference between:
                    1. acknowledging that we can construct a power set that we can then pull the set of integers and the set of reals out from; and
                      • claiming that any comparison between the set of integers and the set of reals must view the two sets as though there were fundamentally related by such a power set.
                      Would this be agreeable? 'cause I'd like to use it as the basis for my explanation, but if it's controversial I'd like to work that out first.

                      Also, are you of the opinion that, in the most rigorous of senses, in all cases? Would it make sense to you if I said that not all procedures for producing are the same? This is, it's possible for
                      due to differences "after infinity".

                      Just to be clear, I'm not making up crazy stuff - this is all easy to prove. I'm just curious where you're at. I can demonstrate stuff that doesn't make sense. For example, I can demonstrate by constructing two infinitely-repeating series such that their difference is non-zero, e.g. hyperreals.

                      Sorry, last question in this post (book's gonna keep me from going crazy tonight): do you buy into the Law of Continuity?

                      PS - Just to be clear, I'm aware of stuff like in systems that prohibit infinitesimals. I assume that we're both well beyond the intro-level math most people learn in grade school.

                      ...also I'm aware that most people regard this stuff as harder than my words might suggest. Frankly it pisses me off that people make such a big deal out of this junk. It's simple and should be regarded as such. I simply have no interest in discussing grade school simplifications when I have the option of a more intelligent conversation.
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                      Re: Status of Cantor's 'Continuum Hypothesis'

                      Postby someguy1 on October 30th, 2015, 11:58 am 

                      Natural ChemE » October 29th, 2015, 11:31 pm wrote:
                      I tried writing a response before, but it feels hard to get this across in a few words after we've had a lot of miscommunications.


                      They've been interesting miscommunications. And we've arrived at Cantor's beautiful seven-liner that there can be no bijection between a set and its powerset. This is a perfectly constructive, finitary proof. As a corollary, can never be bijected to . As a second corollary, there is an endless tower of ever larger transfinite cardinalities.



                      Natural ChemE » October 29th, 2015, 11:31 pm wrote:
                      Okay, so for one-step-at-a-time, lemme ask: I'd say that there's a difference between:
                      1. acknowledging that we can construct a power set that we can then pull the set of integers and the set of reals out from; and




                        Semantic note. The powerset axiom says that the powerset exists. If is a set, there is a set having the property that if , then . There is no recipe or procedure or algorithm to generate the elements of , except in special cases. For example you could write a program to crank out the elements of the powerset of a finite set. But even given infinite time and space, you could not write a program to generate the elements of the powerset of an infinite set.

                        We sometimes use the word "construct" to mean an existence proof in set theory. But this is a misleading overloading of the word. Not important today, but good to keep in mind.

                        Natural ChemE » October 29th, 2015, 11:31 pm wrote:
                      2. claiming that any comparison between the set of integers and the set of reals must view the two sets as though there were fundamentally related by such a power set.
                      3. Would this be agreeable? 'cause I'd like to use it as the basis for my explanation, but if it's controversial I'd like to work that out first.


                        There's an obvious bijection between the reals and the power set of the naturals. A bitstring specifies a subset; a subset specifies a bitstring. I showed this a couple of posts back. The bitstring 10101010101010... of alternating 1's and 0's picks out the set of odd numbers {1, 3, 5, 7, ...}. Put a binary point in front of the bitstring and you have a real number. It's not possible to deny the bijection.

                        Of course the powerset of the naturals lacks all the structure of the reals such as order, topology, arithmetic operations, etc. So they're not the same thing. But there's a bijection between them. Whatever they each may be, they have the same cardinality.

                        Perhaps you can tell me where you're going with this. Related by a powerset? Yes, the reals are bijectively equivalent to the powerset of the naturals. I don't see how any reasonable person could deny that.


                        Natural ChemE » October 29th, 2015, 11:31 pm wrote:
                        Also, are you of the opinion that, in the most rigorous of senses, in all cases?


                        How many cases are there? I only know of one.

                        Natural ChemE » October 29th, 2015, 11:31 pm wrote:
                        Would it make sense to you if I said that not all procedures for producing are the same?


                        The expression .999... is defined in math as the sum of the geometric series 9/10 + 9/100 + ...; and the sum of a geometric series is defined as the limit of the sequence of partial sums; and in this case, that limit is 1.

                        That's the only definition there is. It produces the same answer every time; and in fact it's a theorem of real analysis that the limit of a sequence of real numbers is unique. [This is not true in general; there are mathematical spaces in which a sequence can have more than one limit. But in the real numbers, limits are unique if they exist at all].


                        Natural ChemE » October 29th, 2015, 11:31 pm wrote:This is, it's possible for a repeating-series of 's to not equal another repeating-series of 's?


                        A thing is equal to itself. That's the law of identity, a principle of logic that precedes set theory. https://en.wikipedia.org/wiki/Law_of_identity

                        Without the law of identity, we can't even get set theory off the ground.

                        Put it this way. If I asked you to accept that 4 isn't equal to 4, wouldn't that pretty much preclude any further rational discussion? In everyday speech I can say, "I'm not myself today." But I could NEVER start a mathematical conversation by saying, "Let 4 not be 4." That can't happen.

                        Of course you could define a DIFFERENT meaning of .999... if you wanted to, as long as you reminded us which definition you're using. There's the standard way and you can make up some other way. But if you use the same definition each time, there's only one possible answer.


                        Natural ChemE » October 29th, 2015, 11:31 pm wrote:If I said that this difference is visible "after infinity", would that be meaningful to you?


                        I would regard that as ill-informed, lacking in understanding of what the notation .999... means; and lacking in understanding of how infinity is treated in mathematics. There is no meaning to the phrase, "after infinity." What's after infinity? Nevermind, I don't want to know.


                        Natural ChemE » October 29th, 2015, 11:31 pm wrote:Just to be clear, I'm not making up crazy stuff


                        Where are you getting it from then? You mentioned earlier that you are reading something. I asked you what you are reading. I'm very curious. If you seek to understand modern math, you are making your own task much harder.

                        Natural ChemE » October 29th, 2015, 11:31 pm wrote:- this is all easy to prove.


                        You can't prove that a thing is not equal to itself. Aristotle would spin in his grave. Do you suppose he spins the other way in the southern hemisphere?


                        Natural ChemE » October 29th, 2015, 11:31 pm wrote:I'm just curious where you're at.


                        Wondering where you're going with all this. Dismayed that .999... crankery seems to be on the agenda. Say it ain't so.


                        Natural ChemE » October 29th, 2015, 11:31 pm wrote:I can demonstrate stuff that doesn't make sense.


                        Didn't the example of the terminating decimals open your mind to the possibility that you might simply be mistaken?
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                        Re: Status of Cantor's 'Continuum Hypothesis'

                        Postby someguy1 on October 31st, 2015, 4:33 pm 

                        It occurs to me that I can try to go beyond the mathematical aspect to comment on the metaphysics.

                        I found this old post of yours. I think I see what's on your mind.
                        viewtopic.php?f=19&t=23334&p=227959#p227959

                        The underlying idea you're getting at is the infinitesimal. An infinitesimal is a quantity that is greater than zero but less than 1/n for every positive integer n. https://en.wikipedia.otrarg/wiki/Infinitesimal

                        You will see right away that there are no infinitesimals in the real numbers. If you pick any real number whatsoever, call it x, then there exists some positive integer n such that 0 < 1/n < x.

                        It's common in online discussions of this topic for people to mention two alternative models of the real numbers that do incorporate infinitisimals. These are Nonstandard Analysis (NSA) and the hypperreals.

                        https://en.wikipedia.org/wiki/Non-standard_analysis
                        https://en.wikipedia.org/wiki/Hyperreal_number

                        Without going into detail -- not least because I don't know anything about these systems -- they are both subject to the transfer principle, which says that any first-order sentence true in the standard reals is true in the extended models. https://en.wikipedia.org/wiki/Transfer_principle

                        It follows that .999... = 1 is a theorem in both NSA and the hyperreals. [This is my understanding].

                        Philosophically, infinitesimals have a long and honorable history. I would not deny anyone's right to think about them or imagine them as much as you like. My understanding is that physicists basically think in terms of infinitesimals no matter how much the math professors try to correct them!

                        But no contemporary theory of mathematics, standard or nonstandard, is able to squeeze anything between .999... and 1. There's just no known logically or mathematically sensible theory in which .999... fails to be 1.

                        I hope this resonates with your thoughts.
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                        Re: Status of Cantor's 'Continuum Hypothesis'

                        Postby Natural ChemE on November 1st, 2015, 4:23 am 

                        someguy1,

                        I'm an engineer. As far as I care, math is just a way to think about stuff. It's useful for writing physics, solving problems, etc. I look into stuff like What's a number? because generalizing/redefining stuff like numbers can be stupidly useful. I even have my simulations explicitly deconstruct numbers at a conceptual level to get really cool computations.

                        As you linked, I've explained , but that was in the context of basic math. As Wikipedia explains, the more rigorous equation is
                          .
                        This is exactly as from my prior explanation:
                        Natural ChemE » February 15th, 2013, 2:59 pm wrote:Written correctly, it’s , which doesn’t address the fundamental issue.
                        The "fundamental issue" was something I'd referred to earlier in that same thread:
                        Natural ChemE » December 30th, 2012, 8:08 pm wrote:It’s probably that that seems weird.
                        In other words, elementary math ignores infinitesimals.

                        Earlier I'd said that I was satisfied enough with this thread. The big thing that I realized was that a lot of research continues to be based in elementary math.

                        Regardless of what some philosophers like to tell themselves, we're still gonna make artificial intelligence. Ditto for using advanced math; for an engineer, the idea of being limited to elementary math is pointless and silly.
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                        Re: Status of Cantor's 'Continuum Hypothesis'

                        Postby someguy1 on November 2nd, 2015, 1:02 am 

                        Natural ChemE » November 1st, 2015, 2:23 am wrote:
                        It’s probably that that seems weird. In other words, elementary math ignores infinitesimals.


                        That notation is utter nonsense. You just made it up.

                        And @NatChemE my friend ... did you simply not bother to read the discussion of infinitesimals that I posted? This is not the first time you ignored something I wrote that specifically addressed one of your concerns. Did you find my remarks on infinitesimals far too trivial to even deserve acknowledgement? I thought they were rather on point and anticipated your view.

                        Natural ChemE » November 1st, 2015, 2:23 am wrote:
                        Regardless of what some philosophers like to tell themselves, we're still gonna make artificial intelligence. Ditto for using advanced math; for an engineer, the idea of being limited to elementary math is pointless and silly.


                        My take is that you prefer to dismiss what you can't understand. Computers can't represent real numbers. You don't know what a real number is and you seem to not want to know. You studied engineering math but not math. I've tried my best to bridge the gap for you. Perhaps something I've written will prove helpful to you someday.

                        That you think I'm making some kind of argument regarding artificial intelligence is telling. You still don't understand the output argument. The Wookies. John Searle did not send me here to torment you. That you think this is about Searle is also telling. I only came here to try to put some modern math into context for you.

                        I'd still like to know what you're reading that you referred to earlier. It might give me a clue as to where you're coming from. A lot of engineers don't seem to grok the real numbers. And the real numbers are not taught rigorously till the classes intended for the math majors. But what I don't understand is the stubbornness to not want to learn, but rather to dismiss what has not yet been learned. That I do not get.

                        IEEE-754 is not the real numbers. Let's leave it at that. https://en.wikipedia.org/wiki/IEEE_floating_point

                        Anyway thanks for the chat. I've enjoyed composing these posts whether they did any good. Perhaps some future reader will encounter the proof of Cantor's theorem and will be astonished and delighted by it as I am. They would experience their feelings regardless of the nature of their mind. If we're computers, then we're computers who can appreciate Cantor's theorem. At least I am. You perhaps run a different program. That could be. What do you think?
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                        Re: Status of Cantor's 'Continuum Hypothesis'

                        Postby someguy1 on November 2nd, 2015, 2:35 am 

                        I also wanted to add that even though we may miscommunicate on some levels, I believe that we are communicating very well on others. I believe we are both interested in the same things, in understanding how physics and computation relate to mind. And where mathematics fits into the picture.

                        I've really enjoyed the heck out of this chat. I've found your point of view as stimulating to my thoughts as it is often wrong. Your ideas are the opposite of "not even wrong." For me, they are wonderfully and inspiringly wrong. I've been forced to explore and clarify some of my own thoughts.

                        So I just wanted to say I've really enjoyed this and I hope you feel the occasional frustration was worth it too.

                        Ok!
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                        Re: Status of Cantor's 'Continuum Hypothesis'

                        Postby Natural ChemE on November 2nd, 2015, 9:16 am 

                        someguy1,

                        I've enjoyed our chat too. And I get all of your points, and agree with them within the limited context that you've meant them in.

                        The big thing that I learned in this thread was just how limited that context really is. Before this thread, I'd have assumed that the ambiguity in was a simple, elementary topic; it's not 'til you responded did I really start to fully grasp how I was wrong about that. I had no idea that elementary math was anything more to mathematicians than spherical cows are to physicists.

                        Now that I understand the context in which they're made, I get your points as well as other online content surrounding topics like the continuum hypothesis. And since this all makes sense now, there's just nothing left for me to ask.
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                        Re: Status of Cantor's 'Continuum Hypothesis'

                        Postby Natural ChemE on November 2nd, 2015, 9:31 am 

                        This might help explain why I kept complaining that stuff was ambiguous or specific to limited contexts:
                        Weak ontology, Wikipedia wrote:In computer science, a weak ontology is an ontology that is not sufficiently rigorous to allow software to infer new facts without intervention by humans (the end users of the software system).

                        By this standard – which evolved as artificial intelligence methods became more sophisticated, and computers were used to model high human impact decisions – most databases use weak ontologies.

                        A weak ontology is adequate for many purposes, including education, where one teaches a set of distinctions and trying to induce the power to make those distinctions in the student. Stronger ontologies only tend to evolve as the weaker ones prove deficient. This phenomenon of ontology becoming stronger over time parallels observations in folk taxonomy about taxonomy: as a society practices more labour specialization, it tends to become intolerant of confusions and mixed metaphors, and sorts them into formal professions or practices. Ultimately, these are expected to reason about them in common, with mathematics, especially statistics and logic, as the common ground.

                        I've been doing AI stuff for so long that I employ generalizations of math even in simple Engineering calculations. The idea that there's a single weak ontology that we should all focus on just seems so absurdly limited that I forgot classical mathematicians actually believe in such silliness.

                        Don't get me wrong, I don't think you have any concern about AI or Searle; I do hope that you were joking in that response. However I do care about such things. For me, strong ontology (what I was trying to describe as "constructivism") is necessary.

                        Strong ontology's a lot like constructivism, and our axioms are literally the set of reproducible physical phenomena. All computations are necessarily able to be done in this regime or are else fundamentally impossible to reproducibly perform. This follows from the observations (A) that human thinking is a physical procedure and (B) that logic must be constructed based on reproducible steps.
                          Note:
                            I keep using the "reproducible" qualifier because, technically, you could say that a human might "predict" or "just know" something without being able to explain why. However, in the absence of reproducibility, such predictions are indistinguishable from coincidence.
                        All weak ontologies - e.g. elementary math - are strict subsets of strong ontology. You can state any problem in a weak ontology in a stronger ontology by:
                        1. Select a stronger ontology that you wish to state the problem in.
                        2. Derive a series of reductions that can be applied to the stronger ontology to arrive at the weaker ontology.
                        3. Amend those reductions to the problem statement.
                        4. The original problem, plus the amended reductions, are now meaningful in the stronger ontology.
                        5. Optionally, you may wish to reduce the new problem statement since parts of it may be redundant.
                        The strongest ontology is physics. Any logical problem in any domain or any set of domains is necessarily reducible to a physical phenomena or else is fundamentally ambiguous throughout the set of all possible ontologies. The only type of exception is ignorance on how to well-state the mapping, e.g. classically it's hard to get electronic computers to reproduce human brains because we didn't know how human brains did what they did.

                        Then questions within weak ontologies, e.g. the continuum hypothesis, can either:
                        1. be solved by stating them in strong ontological form;
                        2. can't be stated in strong ontological form (ambiguous).
                        So when I see stuff like the continuum hypothesis, my impulse is to translate it into a stronger ontology and solve it within that context. Having done so (demonstrated previously), I concluded that the correctness of the continuum hypothesis is ambiguous because it depends upon the non-availability of a construction method that scales more weakly than power set construction. This is, in general it's possible to construct a set with a cardinality strictly between the set of integers and reals, but only if you're allowed to use a construction that scales more weakly than power set construction (since power set construction is how the reals scale against the integers in elementary math). And whether or not you're allowed to use a construction method that scales more weakly than power set construction is simply a matter of what axioms the system allows; obviously it's entirely possible in general, but perhaps not possible in particular formal math systems that don't provide access to such construction methods.

                        But, again, I do get that this isn't how classical folks saw stuff.
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                        Re: Status of Cantor's 'Continuum Hypothesis'

                        Postby someguy1 on November 2nd, 2015, 3:12 pm 

                        Natural ChemE » November 2nd, 2015, 7:31 am wrote:The strongest ontology is physics. Any logical problem in any domain or any set of domains is necessarily reducible to a physical phenomena


                        If you lived in 1840 you'd be presenting the exact same argument in opposition to non-Euclidean geometry. You still don't get that mathematics isn't subject to physical law, let alone historically contingent contemporary theories of physics; any more than the latest Star Wars movie or an Escher staircase, each of which are inconsistent with physical reality yet nonetheless meaningful.


                        Natural ChemE » November 2nd, 2015, 7:31 am wrote:I've been doing AI stuff for so long


                        As far as I can tell you've never claimed to work in AI, not in this thread or in any other. Can you say what you've been doing in AI?


                        Natural ChemE » November 2nd, 2015, 7:31 am wrote:that I employ generalizations of math even in simple Engineering calculations.


                        Perhaps you'd favor us with an example of such a generalization of math.
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