
Forest_Dump wrote:it seems to me that some of this work being done, and by apparently wellrespected authorities, has no clear application..

You know what's stupid to the point of being offensive? It appears that hyksos's characterization of the historical understanding of Gödel's work is correct. Gödel was actually that limited; the man literally starved himself to death, and he had no head for things outside of his limited domain of professional expertise.
Sorry Hyksos.. no derailment of your topic intended.
Regards,
Dave :^)
Natural ChemE wrote:I think that it's just reductionism. Just like physicists are always looking for the most fundamental particles and forces to explain the universe, computationalists are looking for the most fundamental components of logic to explain math/logic/computation/thought/knowledge/whateveryoucallit.
Wolfram spent a lot of his life researching this sorta thing, and to great effect; he ended up making Mathematica, WolframAlpha, and some other cool stuff that's really helped Science progress. So, I guess that he kept looking further, trying to find that ultimate, most fundamental understanding that everyone should base their work on.
While I don't know the particulars of how he got into it, later in his life Wolfram became obsessed with the observation that cellular automata are Turing complete. To be fair, it's a really cool observation. Since any Turingcomplete system can emulate any other, this means that a very simplistic system can fully reproduce everything that our modern computers can  including 3D video games, quantum physics simulations, and even artificial intelligence.
If I had to guess, I'd think that Wolfram is trying to take his work to its ultimate conclusion. He's trying to find that great, eternal philosophy to encompass all others. And if he's successful, he'll spin it into another program that, like Mathematica, will help revolutionize Science.
Dave_Oblad wrote:One such application might be to explain how/why we Exist.
That Math doesn't just simply describe our Universe.. that the Universe actually Exists as a subset of Math. IMHO.. lol.

Braininvat » January 13th, 2017, 9:47 am wrote:Scott, understanding "prior art" in any STEM field is essential. We encourage members to be fully familiar with groundbreakers like Von Neuman, Turing, Godel, Shannon, Church, Hilbert, et al when entering a chat like this.

IF some are sincerely 'authoritative' in some field, they should be skilled enough to prove their own credibility by relating to the less 'authoritative' at their level.
Braininvat » January 16th, 2017, 9:17 pm wrote: We don't study and cite because we are enraptured with authority or celebrity, or feel that someone "owned" an idea, but rather to be informed, and sometimes inspired, by groundbreaking work that was key in a particular field and has stood the rigorous tests of decades of experiment and observation.

But while their domains are apparently limited, each of those domains of math and computation are a subset of a larger inclusive domain of all subjects collectively. If part of the 'larger' domain has some system of reasoning that could encompass those truths by the same assumptions of 'consistency',
even if that metalogic is itself 'complete', those theorems in it can only be trusted on the faith of the validity of that metalogic.
Yet, Godel already covered this or his theorem using his reasoning would be a mere subdomain of a yet larger domain to which he couldn't speak (or decide) on whether there is or is not some consistentbased logic that could completely exhaust all math truths.
You cannot find a perfectly complete system of logic that encompasses all truths in reality where it is dependent on consistency. AND, you cannot expect to BEGIN with some consistentessential system of logic that can selfjustify its own system.
perfectly complete system of logic that encompasses all truths in reality
hyksos » January 21st, 2017, 8:11 am wrote:But while their domains are apparently limited, each of those domains of math and computation are a subset of a larger inclusive domain of all subjects collectively. If part of the 'larger' domain has some system of reasoning that could encompass those truths by the same assumptions of 'consistency',
There are two very different senses in which a domain is made larger. One of them is that you start tacking on extra axioms onto the existing ones. This has actually been done in practice. The axiom of Choice is known to be independent of the other axioms of ZFC. So you can just use the "deductive system" of ZF, which has no Choice Axiom in it. Or you can include Choice Axiom, according to taste. This domainextending is what is usually referred to if I make a statement like "Complex Analysis is a subset of ZFC" {call this Sense I}
The second way is to define a universe of symbols and their syntax, and start talking about the system from the outside. This is what you are calling "metalogic". A deductive system appearing to be very narrow, might seem boring to us citizens of 2017, but in the 1920s this stuff was revolutionary in scope. {Call This Sense II}
even if that metalogic is itself 'complete', those theorems in it can only be trusted on the faith of the validity of that metalogic.
Luckily, incompleteness is not the same thing as invalidity.
Yet, Godel already covered this or his theorem using his reasoning would be a mere subdomain of a yet larger domain to which he couldn't speak (or decide) on whether there is or is not some consistentbased logic that could completely exhaust all math truths.
This is the difference between the 1st Incompleteness Theorem and the 2nd Incompleteness theorem. Godel explicitly addressed the issue of larger domains in the 2nd theorem. And he did so in the {Sense I} way. The adding of extra axioms (provided they are genuinely independent of the existing ones) will always place us up into a larger system containing the previous ones. In this way we scale a hierarchy of embedded systems, each larger than the previous and containing the previous inside it.
The question for the 2nd theorem was : if we tack on enough extra axioms, will we tie up all the loose ends and happen upon a system that is complete and consistent? The theorem says No. Any system rich enough to depict arithmetic must either be incomplete or inconsistent. All of them are subject to this restraint.
You cannot find a perfectly complete system of logic that encompasses all truths in reality where it is dependent on consistency. AND, you cannot expect to BEGIN with some consistentessential system of logic that can selfjustify its own system.
The other example was metalingual reference. Ironically metalingual reference appears in Godel's 2nd theorem.
The third example would be the question as to whether the class of computable functions is larger than ZFC.
perfectly complete system of logic that encompasses all truths in reality
This is where the conversation goes off the rails for me. When you use phrases like "encompass all truths in reality", I believe you are no longer talking about math. Your meaning is unclear. What do you mean by "encompass"? And what do you mean by "reality" here? Do you mean the physical universe?
I have already mentioned several issues of LOGIC which exceed the scope of Godel's theorem. The first example was an agent who acts in a world under uncertainty, and whose beliefs changes whenever new data comes in. That agent's logic is going to be very different from a static system that is assumed to contain a priori all of its consequences hanging motionless in a platonic superchest. (if you will)
The underlined "uncertainty" implies incompleteness and potential inconsistency.
I thought you understood I was referencing ALL reality, not simply math. ?? Yes, the theorems hold for the physical laws of science too. As long as they are consistent logics, Godel's system holds true for all such logic.
"Godel showed that the Continuum Hypothesis is independent of ZFC."
Gödel's completeness theorem establishes an equivalence in firstorder logic between the formal provability of a formula and its truth in all possible models.
hyksos » January 24th, 2017, 2:52 am wrote:So I was banging around the wikipedia article on Foundations of Mathematics.
https://en.wikipedia.org/wiki/Foundations_of_mathematics
hyksos » January 24th, 2017, 2:52 am wrote:So again, (for the third time) do not try learn Proof Theory from wikipedia.
hyksos » January 21st, 2017, 7:11 am wrote:"Complex Analysis is a subset of ZFC" {call this Sense I}
hyksos » January 24th, 2017, 2:35 am wrote:Listen. Bertrand Russell and Alfred North Whitehead discovered that all branches of mathematics, which were once considered separate disciplines, were all united under the same umbrella. That umbrella is something called ZermeloFraenkelChoice Set Theory.
Which I promised myself I'd stay out of. But this latest ... who are you to say who can post what in a discussion forum?
Please, remember why you started this thread. It was to explain, not fight, make fun of people's clothing choices, or to say that a simple idea like Gödel's completeness theorem requires grad level math.
hyksos » January 24th, 2017, 12:23 pm wrote:What I will say, more clearly now, is that I am picking up (using my spideysense) that Scott Mayers got his training in higher mathematics from reading the internet.
hyksos » January 24th, 2017, 12:23 pm wrote:I don't who the f&%k you are, "someguy1" but if you want to sit in this thread, and be Scott Mayer's personal mathematics tutor  have at it. He's all yours. Don't let me stop you. Tutor him to your own delight.
I have the same sense about you. Your OP and your most recent post that I responded to were highly inaccurate.
Nicely avoiding the direct challenges I gave to you. Your remarks about complex analysis and Russell and Whitehead were so far off the mark I had to call them out.
Frankly your remarks about Russell and Whitehead show that you didn't even bother to read the Internet, since the Wikipedia entry on PM explicitly refutes your claim that they said anything about ZFC.
hyksos » January 24th, 2017, 12:37 pm wrote:I have the same sense about you. Your OP and your most recent post that I responded to were highly inaccurate.
I am universitytrained in higher mathematics. Some of the books from those years are still on my shelf. I could photograph them in front of the screen.
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