Clarifying Infinity

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Clarifying Infinity

Postby scientificphilosophe on April 11th, 2017, 12:47 am 

I have been challenged on another thread about the nature of infinity.
I have argued that infinity cannot be exceeded or added to.
If something is infinite it is without end in all possible directions and therefore covers everything everywhere and potentially more if the 'theoretical' can go beyond the actual.

I was challenged by reference to Hilberts Hotel

Maybe we should visit Hilbert's Hotel, to better examine this notion....

http://world.mathigon.org/Infinity


This was proposed as a mathematical proof, but to my thinking anything which argues that you can add to infinity is fundamentally flawed.

You can only have infinity or less.

If I am wrong, what is my error?
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Re: Clarifying Infinity

Postby someguy1 on April 11th, 2017, 1:02 am 

You can start by answering my question about the counting numbers 1, 2, 3, 4, ... I think we'd all agree it's an infinite set. (Unless we're finitists, in which case it's an infinite collection but not a set. Or ultrafinitists, in which case there's no such thing as the collection of all the counting numbers).

Can't we add 0? In which case, didn't we just add to an infinite set? Next I'll ask about adding the negative numbers, the rationals, the reals (cardinality jump, but nobody knows how big a jump), the complex numbers, the quaternions, etc. We have an upward hierarchy of common mathematical systems of numbers. Agree? Disagree? Alternate point of view?

Hilbert's hotel is not a mathematical argument. It's a popularization of a mathematical argument.
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Re: Clarifying Infinity

Postby scientificphilosophe on April 11th, 2017, 7:31 am 

Hi someguy1

I think you pointed out the example of infinitely sub-dividing the 'space' between 0 and 1. This is a good example of illustrating that infinities apply in a direction or a dimension and that in other directions or dimensions the sequence is finite/truncated/limited.

In the case of the sequence 1,2,3 4 etc it is infinite in the positive direction and truncated in the opposite direction.
When you add zero you are not adding to the infinite but adding to the finite, because you are adding it in the limited direction. You are not touching the infinite.

The problem with Hilbert's Hotel is that it suggests you can add to the infinite.
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Re: Clarifying Infinity

Postby someguy1 on April 11th, 2017, 10:40 am 

Anyone who believes they have a theory of the entire universe should be willing to educate themselves on the modern mathematical conception of infinity. You don't have to agree with these ideas, but you do need to be aware of them.

Here are some web resources, and of course you can Google others.

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

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

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

https://nrich.maths.org/2756

If you have specific questions I'd be glad to answer them, but I prefer not to get into a dispute about technical material that's already accepted in standard math.

scientificphilosophe » April 11th, 2017, 5:31 am wrote:
In the case of the sequence 1,2,3 4 etc it is infinite in the positive direction and truncated in the opposite direction.
When you add zero you are not adding to the infinite but adding to the finite, because you are adding it in the limited direction. You are not touching the infinite.


We can add a number at the high end too. We just go: 1, 2, 3, 4, ..., 0. All we've done here is put the 0 at the end. We can formalize this by defining a new "less than" relation on the counting numbers which is the same as the usual <, except that everything is less than 0. It's just a rearrangement, putting the same set of elements into a different order. In math this is called the order type or ordinal ω + 1, where that's the Greek letter lower-case omega.

Again, I'm only here to report what mathematicians do. There's no right or wrong to it. If you say, "I don't like it," I won't argue the point. But you can't ignore the fact that these ideas are widely accepted in modern math.

scientificphilosophe » April 11th, 2017, 5:31 am wrote:The problem with Hilbert's Hotel is that it suggests you can add to the infinite.


Hilbert's hotel is not a mathematical argument. It's a popularized account of a mathematical argument. It's like using a bowling ball on a rubber sheet to illustrate how Einstein's gravity works. It's not physics, it's a popularization.

But there's no difficulty adding an element to an infinite set. If you show me the integers (infinite in both directions in your terminology) I'll toss in a rational. If you show me the reals, I'll toss in the square root of -1. If you show me the complex numbers I'll toss in a quaternion. You can always add an element to an infinite set.

That's not a matter of opinion, it's a matter of accepted math. You are entitled to reject it, but you can't claim it doesn't exist.

Not every person needs to know about the mathematical theory of infinity. But people with an interest in putting forth their own ideas about the universe should at least have a glancing knowledge of mathematical infinity.
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Re: Clarifying Infinity

Postby scientificphilosophe on April 13th, 2017, 8:02 am 

As you say, this is what mathematicians do to avoid some awkward principles about infinity that break mathematical rules... such as infinity+1 is still infinity, not infinity +1.

You are also correct in saying that there is no right or wrong to this, as long as there are other logical interpretations that don't break the facts.
There are.
We start from the basics - infinity must be unlimited in at least one direction and in that direction it therefore can't be added to.
We then recognise that infinity may be limited 'along' or within some parameters by making them finite.
Those aspects/directions which are limited must be finite.

If infinity exists then it must have a direction towards an open/unlimited sequence. That is the direction of infinity.
You can have a 'forwards' form of infinity from a start point, or a 'negative' direction from an end point. By limiting infinity in other parameters it may be squeezed into a channel with a flow in one or mire directions.

It doesn't matter whether you add different 'dimensions' (new directions) in which to seek a new level of infinity, or close down more directions from an infinite set of directions, the basics don't change. Infinity can only exist without an end in the direction of travel.

In all of your examples you are adding to a sequence in directions that are not yet infinite.
I say again - Hilbert Hotel was presumably not pursued because is fundamentally incorrect in trying to add levels in the direction that is already infinite.
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Re: Clarifying Infinity

Postby vivian maxine on April 13th, 2017, 11:57 am 

someguy, are we adding numbers to that already infinite set? Or are we merely adding words for naming those far out numbers that we have not yet approached and had to make use of? Seems to me, that if the set is infinite, we cannot add to it. We can only start applying names farther out than we currently do.
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Re: Clarifying Infinity

Postby Lomax on April 13th, 2017, 12:12 pm 

scientificphilosophe » April 13th, 2017, 1:02 pm wrote:We start from the basics - infinity must be unlimited in at least one direction and in that direction it therefore can't be added to.
We then recognise that infinity may be limited 'along' or within some parameters by making them finite.
Those aspects/directions which are limited must be finite.

If infinity exists then it must have a direction towards an open/unlimited sequence. That is the direction of infinity.

Infinity does not need to have a "direction" at all. You are talking specifically about countable sets. (A sequence is different to a set, by the way.)

It's fine to have an opinion on whether there is more than one infinity but someguy1 is right: until you understand the difference between cardinality and ordinality you will never be able to persuade the mathematically informed.
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Re: Clarifying Infinity

Postby Lomax on April 13th, 2017, 12:15 pm 

Anyway, the infinity of infinities is provable, and valid even within ZFC.

Cantor (or was it Frege? I forget...) noticed that the (set of even integers) and the (set of all integers) have a one-to-one correspondence:

2, 4, 6, 8, 10, 12...
1, 2, 3, 4, 5, 6...

If I may borrow from a more eloquent man:

Bertrand Russell wrote:It was formerly believed that this was a contradiction; even Leibnitz, although he was a partisan of the actual infinite, denied infinite number because of this supposed contradiction. But to demonstrate that there is a contradiction we must suppose that all numbers obey mathematical induction. To explain mathematical induction, let us call by the name “hereditary property” of a number a property which belongs to n +1 whenever it belongs to n. Such is, for example, the property of being greater than 100. If a number is greater than 100, the next number after it is greater than 100. Let us call by the name “inductive property” of a number a hereditary property which is possessed by the number zero. Such a property must belong to 1, since it is hereditary and belongs to 0; in the same way, it must belong to 2, since it belongs to 1; and so on. Consequently the numbers of daily life possess every inductive property. Now, amongst the inductive properties of numbers is found the following. If any collection has the number n, no part of this collection can have the same number n. Consequently, if all numbers possess all inductive properties, there is a contradiction with the result that there are collections which have the same number as a part of themselves. This contradiction, however, ceases to subsist as soon as we admit that there are numbers which do not possess all inductive properties.


Anyway, here's where it gets interesting: suppose an infinite sequence (with a given first term) of binary numbers. We can write this out as:

S1: 0,0,0,0,0,0,0,0,0,0...

Suppose another. And another. Make a few. Write them out together. For instance:

S1: 0,0,0,0,0,0,0...
S2: 1,1,1,1,1,1,1...
S3: 1,0,1,0,1,0,1...
S4: 0,1,0,1,0,1,0...
S5: 1,1,0,0,1,1,0...
S6: 0,0,1,1,0,0,1...
S7: 1,1,1,0,0,0,1...

And so on. Now take the first term from the first row, the second term from the second row, and so on:

S1: 0,0,0,0,0,0,0...
S2: 1,1,1,1,1,1,1...
S3: 1,0,1,0,1,0,1...
S4: 0,1,0,1,0,1,0...
S5: 1,1,0,0,1,1,0...
S6: 0,0,1,1,0,0,1...
S7: 1,1,1,0,0,0,1...

Make the highlighted terms into a new sequence (in this example: 0,1,1,1,1,0,1...), then make each one of them its opposite (so: 1,0,0,0,0,1,0...). Call this new set S0.

S0 is therefore different to every set we already had. (To explain why: suppose some set Sn. The nth term of Sn will be different to the nth term of S0, because that is how we deliberately constructed S0). Add S0 to your list of possible sequences and repeat the process, and you will always yield a new infinite set that you didn't already have on your list*.

This is called Cantor's "diagonal argument". What it shows us is that, not only is infinity as infinite as it seems, it is more so; there are an infinity of infinities.

Lomax

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* don't actually do this, it would literally take forever. Just get a monkey and a typewriter.
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Re: Clarifying Infinity

Postby someguy1 on April 13th, 2017, 12:33 pm 

I'm not sure how to respond to this. I offered to discuss mathematical infinity. You have a personal theory, which is fine; but your theory makes up terms without defining them and doesn't make much sense to me. This forum has a Personal Theories section, which is where personal theories go. I see no discussion of mathematical infinity here.

I'll offer some specific comments here, but if you desire to learn about mathematical infinity you should start with the links I gave earlier. Else there's no place to go with this.


scientificphilosophe » April 13th, 2017, 6:02 am wrote:As you say, this is what mathematicians do to avoid some awkward principles about infinity that break mathematical rules... such as infinity+1 is still infinity, not infinity +1.


I don't recall saying anything like that. In any event, in the transfinite ordinals, adding to an infinite ordinal sometimes gives you a new one. You should add this to your reading list.

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

scientificphilosophe » April 13th, 2017, 6:02 am wrote:You are also correct in saying that there is no right or wrong to this, as long as there are other logical interpretations that don't break the facts.
There are.


I have no problem with this in theory. I don't think you've presented much of an alternative.


scientificphilosophe » April 13th, 2017, 6:02 am wrote:We start from the basics - infinity must be unlimited in at least one direction and in that direction it therefore can't be added to.


I have three separate problems here.

* Basics of what? Certainly not the basics of the mathematical theory of infinity. Rather, the basics of your own personal theory. In which case this thread goes in Personal Theories.

* You are confusing boundedness with cardinality. There are infinitely many real numbers between 0 and 1, but they form a bounded set. You can draw a finite circle around them. In fact a circle contains finite area, goes in every possible direction, yet contains infinitely many points.

* You can certainly add to a "directional" infinity as you call it. Consider the counting numbers 0, 1, 2, 3, ... Now just rearrange them in this new order: 1, 2, 3, ..., 0. The second order represents a different ordinal number than the first. You can see that because there's an order-preserving bijection between 0, 1, 2, ... and 1, 2, 3, ...; but in the second case we have 0 left over at the end. We DID just add an element at the unbounded end of an infinite set. Ordinals are really cool, it's too bad people don't hear about them much.


scientificphilosophe » April 13th, 2017, 6:02 am wrote:We then recognise that infinity may be limited 'along' or within some parameters by making them finite.
Those aspects/directions which are limited must be finite.


I don't know what that means. I think you're confusing boundedness with cardinality, or with order, or with something.

scientificphilosophe » April 13th, 2017, 6:02 am wrote:If infinity exists then it must have a direction towards an open/unlimited sequence.


Evidence? Examples? What about the infinite set of points between 0 and 1? What about a circle in the plane?

scientificphilosophe » April 13th, 2017, 6:02 am wrote:That is the direction of infinity.


You'll need to offer a formal definition else you're making word salad. "Direction of infinity?" That's not defined by you at all.

If you have vague ideas in your head, you can read Wiki and ask questions online to help clarify your ideas. There's little point in doubling down on your own confusions.


scientificphilosophe » April 13th, 2017, 6:02 am wrote:You can have a 'forwards' form of infinity from a start point, or a 'negative' direction from an end point. By limiting infinity in other parameters it may be squeezed into a channel with a flow in one or mire directions.


Squeezed into a channel with a flow? Does this word salad come with croutons?

scientificphilosophe » April 13th, 2017, 6:02 am wrote:It doesn't matter whether you add different 'dimensions' (new directions) in which to seek a new level of infinity, or close down more directions from an infinite set of directions, the basics don't change. Infinity can only exist without an end in the direction of travel.


You're just making stuff up. No law against it, but you're not giving me anything sensible to work with.

scientificphilosophe » April 13th, 2017, 6:02 am wrote:In all of your examples you are adding to a sequence in directions that are not yet infinite.


If 0, 1, 2, 3, 4, ... isn't infinite, I don't know what is. I already showed how to add something to the unbounded end.

scientificphilosophe » April 13th, 2017, 6:02 am wrote:I say again - Hilbert Hotel was presumably not pursued because is fundamentally incorrect in trying to add levels in the direction that is already infinite.


When the bowling ball distorts the rubber sheet, what pulls the bowling ball down? Meta-gravity? No, it's that the rubber sheet model is a fable, a simplified popularization. Same with the Hilbert hotel.
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Re: Clarifying Infinity

Postby someguy1 on April 13th, 2017, 12:35 pm 

Lomax » April 13th, 2017, 10:15 am wrote:Anyway, the infinity of infinities is provable, and valid even within ZFC.

Cantor (or was it Frege? I forget...) noticed that the (set of even integers) and the (set of all integers) have a one-to-one correspondence:

2, 4, 6, 8, 10, 12...
1, 2, 3, 4, 5, 6...


Galileo, although what's now called Galileo's paradox was already known in the early middle ages.

https://en.wikipedia.org/wiki/Galileo%27s_paradox

Today we take the property of a set being in bijection with a proper subset of itself as one of the definitions of an infinite set.
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Re: Clarifying Infinity

Postby someguy1 on April 13th, 2017, 12:39 pm 

vivian maxine » April 13th, 2017, 9:57 am wrote:someguy, are we adding numbers to that already infinite set? Or are we merely adding words for naming those far out numbers that we have not yet approached and had to make use of? Seems to me, that if the set is infinite, we cannot add to it. We can only start applying names farther out than we currently do.


Interesting question. Suppose I have the integers, ..., -3, -2, -1, 0, 1, 2, 3, ... Surely this is an infinite set. It's infinite in "both directions" as even the OP would agree. A more precise phrase is that it's unbounded in both directions.

Now suppose we want to add in the rational numbers like 1/2, 3/47, -97/12, etc. Would you say we are just adding words?" Or are we finding one infinite set properly contained within another?

After all, aren't there infinitely many even numbers? And don't they live inside the set of all whole numbers, which live inside the set of all rational numbers, which live inside the set of all real numbers?

Whether you consider this math or word games is a matter of personal viewpoint. You wouldn't be the first to suggest that math is really only word games. Philosophically there is something to that point of view. After all, we use finite strings of symbols to talk about infinity.

Do any of these examples help clarify your question? It's a good question, but mathematically it's clear that infinite sets can live inside other infinite sets.
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Re: Clarifying Infinity

Postby scientificphilosophe on May 7th, 2017, 9:51 pm 

Lomax » April 13th, 2017, 5:12 pm

Infinity does not need to have a "direction" at all. You are talking specifically about countable sets. (A sequence is different to a set, by the way.)

It's fine to have an opinion on whether there is more than one infinity but someguy1 is right: until you understand the difference between cardinality and ordinality you will never be able to persuade the mathematically informed.



Hi Lomax

I can't claim to be a mathematician - as you've probably guessed - but I don't think my comments confuse cardinality and ordinality. Neither do I say that infinity has to have a direction, but where one is imposed then the direction should be recognised.

The full set of positive and negative integers will not be limited, but if you begin a sequence from zero and then just proceed with the positive sequence, you will have established a direction. If you move along the sequence away from 0 then it will be unlimited, but if you move towards zero it will be finite as it will not extend beyond 0.

If we look at Time running up to the present moment then it is finite in the 'forwards direction' because we cannot go further than the current moment, but if we look back into the past then it is potentially unlimited/eternal in that direction.

If you sub-divide the gap between 0 and 1 you are effectively establishing a new dimension for the numbers because you are no longer dealing with integers, and in the 'direction' of that new dimension you can find another infinity.

At headline level I agree with the infinity of infinities, but I also recognise that people or circumstances can limit an infinite sequence - thereby establishing a direction
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Re: Clarifying Infinity

Postby scientificphilosophe on May 7th, 2017, 10:12 pm 

Someguy

Please read my last reply to Lomax. That will set a context.

Your examples do set limits because you start from zero and do not move in the negative direction.

You talk about re-arranging sequences but I don't see how that adds anything if you are just dealing with the same numbers, and not going negative. I think you also have to define what this re-arrangement of numbers represents if it is to reflect anything real. I have given examples with Time etc. - what are yours?

Sub-dividing adds a different dimension to the sequences you were originally working with.
I don't have a problem with an infinite set of sub-divisions - but it isn't the same set-up that we started with - Integers. You are very ford of changing the parameters of your examples. Deal with the basics.

If you start from zero you have applied a restriction/limit/direction.

I honestly don't see the relevance of your circle other than another opportunity for sub-division. Was that it?

A circle/ring (2 dimensions) is only infinite in its potential for sub-division - a direction - in other respects it is finite - as you said yourself. I think you missed my point when you try to say that a circle covers all directions.
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Re: Clarifying Infinity

Postby someguy1 on May 7th, 2017, 10:51 pm 

I'm replying here to your reply to Lomax, which you asked me to read.


scientificphilosophe » May 7th, 2017, 7:51 pm wrote:I can't claim to be a mathematician - as you've probably guessed - but I don't think my comments confuse cardinality and ordinality.


Indeed they do, as we shall see.

scientificphilosophe » May 7th, 2017, 7:51 pm wrote: Neither do I say that infinity has to have a direction, but where one is imposed then the direction should be recognised.


The fundamental entity is the set. Sets are disorderly collections of things. No order, no structure, no arithmetic, etc. All those other things are imposed on top of sets.

So if we have an infinite set, it doesn't have an order unless we give it one. And if we give a set an order, we can instead give it a different order. The same set may have may different orders put on it.

Think of it as a classroom full of unruly kids running around. That's a set. Now we tell them to line up in order of height. That's one way to order a set. Then we ask them to line up in alphabetical order of last name. That's the same set with a different order. Then we ask them to line up by reverse-alphabetical order of middle name. That's yet another example of the same set with a different order. The raw set consisting of 47 screaming kids is a cardinal concept. "How many." When we put an order on the kids, it's a different ordered set.

scientificphilosophe » May 7th, 2017, 7:51 pm wrote:The full set of positive and negative integers will not be limited, but if you begin a sequence from zero and then just proceed with the positive sequence, you will have established a direction.


You've taken a set (cardinal) and imposed an order on it (ordinal). We could order the natural numbers in the usual way 1, 2, 3, 4, ... Or we could order them in reverse: ..., 4, 3, 2, 1. Or evens before odds. Or primes before composites. There are lots and lots of ways to order the natural numbers. They are all the same underlying set but they are all distinct ordered sets.

scientificphilosophe » May 7th, 2017, 7:51 pm wrote: If you move along the sequence away from 0 then it will be unlimited, but if you move towards zero it will be finite as it will not extend beyond 0.


In that particular order.

scientificphilosophe » May 7th, 2017, 7:51 pm wrote:If we look at Time running up to the present moment then it is finite in the 'forwards direction' because we cannot go further than the current moment, but if we look back into the past then it is potentially unlimited/eternal in that direction.


I don't know anything about time. That's physics. One of the biggest sources of confusion is to confuse math with physics. We are just talking about sets, their cardinality, and the many different ways we can impose order on them.

Models of time are really something else entirely. You can't reason about the nature of time by contemplating the usual order on the integers. Mathematically, one order is just as good as another and none of them are necessarily the way the universe works.

In fact when people start talking about the nature of time, and using as evidence the standard order on the natural numbers, I'm immediately reminded of the religious sophistry of William Lane Craig and his Kalam cosmological argument. It sets my alarm bells flashing.

scientificphilosophe » May 7th, 2017, 7:51 pm wrote:If you sub-divide the gap between 0 and 1 you are effectively establishing a new dimension for the numbers because you are no longer dealing with integers, and in the 'direction' of that new dimension you can find another infinity.


That statement I don't understand. Can you clarify?

scientificphilosophe » May 7th, 2017, 7:51 pm wrote:At headline level I agree with the infinity of infinities, but I also recognise that people or circumstances can limit an infinite sequence - thereby establishing a direction


"Establishing a direction" is precisely what we mean by imposing an order on a set.

I'll leave your other post for a little later, since I think this post of yours raised a lot of important issues.

The moment you talk about "direction" in a set, you are talking about order properties.
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Re: Clarifying Infinity

Postby scientificphilosophe on May 11th, 2017, 10:23 pm 

Someguy - If you are so isolated from the world that you don't understand the basic concept of Time that we experience then I can't explain it to you.

Equally, both cardinality and ordinality are present in each mathematical debate - that doesn't mean they are confused.

To say that you can simply re-order numbers in a set with impunity is possibly valid if you are using those numbers as a code or a set of names which are equal - but to deny the intrinsic value of each integer seems ridiculous. Those numbers are not equal, and their inherent value determines their position in the sequence.

If you do deny the natural order then mathematics has no rationale.

By limiting your set to positive integers only you are placing an order on them because you are limiting that set in one direction only. That limitation sets a boundary and therefore a direction.

On your point about fractions - eg. the space between 0 and 1 - they are not integers, but they may have their value place within the natural order. By not being integers they are not part of the integer set. If you wish to position them within the integer set then must represent something else - a different dimension of numbers. I don't think that's a hard concept.

If you operate in a different dimension then you have a new direction to explore/travel in.

That is my point - what's yours?
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Re: Clarifying Infinity

Postby someguy1 on May 12th, 2017, 12:35 am 

scientificphilosophe » May 11th, 2017, 8:23 pm wrote:
If you operate in a different dimension then you have a new direction to explore/travel in.

That is my point - what's yours?


To be perfectly honest I no longer understand what you're talking about. You initially had some questions about Hilbert's hotel and I see that I haven't been very helpful. Your most recent post seems rather far afield from anything I can understand. You might read up on ordinal numbers to see how we can put different orders on the same set.

If you can frame some specific questions I might be able to answer them.

Let me go back to your first post.

scientificphilosophe » April 10th, 2017, 10:47 pm wrote:This was proposed as a mathematical proof, but to my thinking anything which argues that you can add to infinity is fundamentally flawed.

You can only have infinity or less.

If I am wrong, what is my error?


Would you agree that the set of even positive integers is infinite? This is the set {2, 4, 6, 8, 10, 12, 14, ...}.

Now what if I add in the odd numbers? I've added to an infinite set, right?
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Re: Clarifying Infinity

Postby scientificphilosophe on May 12th, 2017, 6:46 am 

If your set is of even numbers only then odd numbers can't feature in that set.
Even numbers do continue to infinity.
If you add odd numbers it is a different set - it is no longer a set of even numbers but a set of all integers.

When you define a set by excluding a particular category of number, that is part of the definition of your set.
You can say that it is infinite within those parameters and you can't add to the set within those parameters..
When you change the definition of your set it is a new set.

They are apples and pairs.
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Re: Clarifying Infinity

Postby Lomax on May 12th, 2017, 9:42 am 

scientificphilosophe » May 12th, 2017, 3:23 am wrote:To say that you can simply re-order numbers in a set with impunity is possibly valid if you are using those numbers as a code or a set of names which are equal - but to deny the intrinsic value of each integer seems ridiculous. Those numbers are not equal, and their inherent value determines their position in the sequence.

This belief is your confusion between cardinality and ordinality.
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Re: Clarifying Infinity

Postby Heavy_Water on May 12th, 2017, 1:45 pm 

I agree that Infinity cannot be added to.

That's why, uh, it's called Infinity.

LOL

Tis a pity the homo sapien mind is totally incapable of truly understanding the term Infinity.

Just as we cannot fully grasp the term "Nothingness."

The latter fact is why the godists invent, well, their gods.

And their Afterlifes.

Their Heavens.

To close, I once heard a nice and imagery-pleasing description of infinity. I really like it, and if you sit and take a moment and try to imagine it, you SORT OF get an inkling of the concept of Infinity. Though, as I said, we can't ever really get it. Since we of course have no experience with it during our short lives.


Anyway, here it is. Sorry that I cannot recall the author. I wish I had invented it! But, alas, no...........


"Imagine that the moon is not made of rock, but instead of dense and solid black iron. Then let us see a butterfly flutters away from the Earth, and navigate through space, traversing the entire 240,000 miles to this Iron Moon.

The Butterfly then flies around the moon once, As he does, his inside wing juuuust barely brushes the Iron Moon.

The Butterfly then begins his journey back to Earth. When he touches down on Earth, he immediately begins another trip to the Iron Moon. And again his wing brushes is as he orbits it. And he again returns to Earth.

He continues this process. Over and Over.

When the Iron Moon has been reduced to the size of a marble, from the countless brushes of the Butterfly's wing.......

INFINITY HAS JUST BEGUN."



Cheers.
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Re: Clarifying Infinity

Postby someguy1 on May 12th, 2017, 2:26 pm 

scientificphilosophe » May 12th, 2017, 4:46 am wrote:
They are apples and pairs.


That's a pretty good pun, even if unintentional. Although in this case we're not using the pairing axiom, but rather the axiom of unions.

https://en.wikipedia.org/wiki/Union_(set_theory)

And speaking of pears ... what's yellow and equivalent to the axiom of choice?

Zorn's lemon.
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Re: Clarifying Infinity

Postby someguy1 on May 12th, 2017, 7:48 pm 

scientificphilosophe » May 12th, 2017, 4:46 am wrote:If your set is of even numbers only then odd numbers can't feature in that set.
Even numbers do continue to infinity.
If you add odd numbers it is a different set - it is no longer a set of even numbers but a set of all integers.


But by that logic you can never add a number to a finite set either, because it would become a different set. So a finite set is no different than an infinite set in this regard. Right?
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