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Kurt Gödel and the romance of logic

prospectmagazine.co.uk

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Re: Kurt Gödel and the romance of logic

#21

Post by AlexCoventry »

whatshisface wrote: post #19
skh wrote: post #13

Earlier quoted context omitted.

An important nitpick. His Incompleteness Theorem deals with recursively enumerable axiomatic systems. The second order Peano Axioms are categorical. That is, they have only one model up to isomorphism. It’s easy to come up with a complete axiomatic system for the standard model of the natural numbers. Just take as your axiomatic system the collection of all true statements. This ins’t a useful system since there is n…

How would you be able to take every true statement as an axiom? Without proving anything I don't see how you could identify any statements as true.

That's the joke.

Re: Kurt Gödel and the romance of logic

#22

Post by kkylin »

Anyone interested in this article may also be interested in this discussion from a while back: https://news.ycombinator.com/item?id=18115696

Also of interest is Godel's AMS Gibbs Lecture, which unfortunately I have not been able to locate on-line. It can be found in Volume 3 of Godel's Collected Works, alongside his paper / lecture notes on closed time loops in general relativity.

Re: Kurt Gödel and the romance of logic

#23

Post by antidesitter »

whatshisface wrote: post #19
skh wrote: post #13

Earlier quoted context omitted.

An important nitpick. His Incompleteness Theorem deals with recursively enumerable axiomatic systems. The second order Peano Axioms are categorical. That is, they have only one model up to isomorphism. It’s easy to come up with a complete axiomatic system for the standard model of the natural numbers. Just take as your axiomatic system the collection of all true statements. This ins’t a useful system since there is n…

How would you be able to take every true statement as an axiom? Without proving anything I don't see how you could identify any statements as true.

The point is that the set of all true statements about arithmetic [1] exists “out there” (and thus as a “theory” in a very general sense) even if we can’t identify it.

[1] https://en.wikipedia.org/wiki/True_arithmetic

Re: Kurt Gödel and the romance of logic

#24

Post by antidesitter »

infinity0 wrote: post #5

> rescued the idea that there are truths that humans can never prove This is a gross misinterpretation of Gödel's actual theorem that helps perpetuate irrational superstitious attitudes against science, mathematics, and logic. What Gödel showed was that proofs are relative to some underlying axiomatic model and that for any particular axiomatic model there are always truths unprovable by it. That doesn't mean "there…

> That doesn't mean "there are truths that humans can never prove"

Well, if there’s an analogue of a Gödel sentence for humans...

Re: Kurt Gödel and the romance of logic

#25

Post by neokantian »

The biggest qualm that I have with the example for the incompleteness theorem, is the use of two-valued, Boolean/Aristotelian logic, in which "not true" automatically becomes "false".

If you allow the use of a three-valued logic, for example, with (true, undetermined, false), the Gödel statement, “I am not provable" amounts to saying "My provability is false or undetermined".

The same problem occurs in Russell's paradox, "Does the set of all sets that do NOT contain themselves, contain itself?"

The use of the NOT-operator is degenerated in a cyclic group Z2. It is the only situation in which the NOT-operator is not set-valued.

In my opinion, if the "isProvable()" predicate allowed for multi-valued logic, Gödel's incompleteness theorem would look much less paradoxical. In other words, the paradoxical outcome could simply be the result of Boolean shoehorning.

Re: Kurt Gödel and the romance of logic

#26

Post by [deleted] »

neokantian wrote: post #25

The biggest qualm that I have with the example for the incompleteness theorem, is the use of two-valued, Boolean/Aristotelian logic, in which "not true" automatically becomes "false". If you allow the use of a three-valued logic, for example, with (true, undetermined, false), the Gödel statement, “I am not provable" amounts to saying "My provability is false or undetermined". The same problem occurs in Russell's para…

[deleted]

Re: Kurt Gödel and the romance of logic

#27

Post by ginnungagap »

whatshisface wrote: post #19
skh wrote: post #13

Earlier quoted context omitted.

An important nitpick. His Incompleteness Theorem deals with recursively enumerable axiomatic systems. The second order Peano Axioms are categorical. That is, they have only one model up to isomorphism. It’s easy to come up with a complete axiomatic system for the standard model of the natural numbers. Just take as your axiomatic system the collection of all true statements. This ins’t a useful system since there is n…

How would you be able to take every true statement as an axiom? Without proving anything I don't see how you could identify any statements as true.

Whenever you have a structure M of some language L you can take the so called complete theory of the structure, denoted Th(M), which is just the set of all L-sentences true in M. In particular if L is the language of PA and M are the standard natural number with the standard operations Th(M) is a theory called true arithmetic. This theory is complete (that's because Th(M) is always complete) and clearly enough to talk about the integers, but it escapes Gödel's theorem since it's axioms are not recursively axiomatizable.

You seem to be confusing "true" and "provable", as far as first order logic is concerned those are equivalent (by another theorem of Gödel, the completeness theorem), but the first in defined in terms of models and the second is purely syntactic

Re: Kurt Gödel and the romance of logic

#28

Post by theaeolist »

neokantian wrote: post #25

The biggest qualm that I have with the example for the incompleteness theorem, is the use of two-valued, Boolean/Aristotelian logic, in which "not true" automatically becomes "false". If you allow the use of a three-valued logic, for example, with (true, undetermined, false), the Gödel statement, “I am not provable" amounts to saying "My provability is false or undetermined". The same problem occurs in Russell's para…

The incompleteness result does not even mention "true" and "false". It says that "there are statements of the language of F which can neither be proved nor disproved in F." Whether those statements are true/false is left unsaid.

Re: Kurt Gödel and the romance of logic

#29

Post by sorokod »

truantbuick wrote: post #10
infinity0 wrote: post #5

> rescued the idea that there are truths that humans can never prove This is a gross misinterpretation of Gödel's actual theorem that helps perpetuate irrational superstitious attitudes against science, mathematics, and logic. What Gödel showed was that proofs are relative to some underlying axiomatic model and that for any particular axiomatic model there are always truths unprovable by it. That doesn't mean "there…

> That doesn't mean "there are truths that humans can never prove", all it means is that we have to extend our axiomatic systems in order to prove some truths. If you believe that the only consequence to Gödel's theorem is we need to "extend our axiomatic system", I do think you've missed the point. For one thing, I think Gödel's theorem and Gödel's proof are unfortunately conflated. Gödel's proof is lovely and elega…

"Gödel's proof is lovely and elegant, and can be understood with minimal knowledge of logic"

I suspect you haven't gone through the actual rigorous proof which is highly technical and requires more than a "minimal knowledge of logic"

Re: Kurt Gödel and the romance of logic

#30

Post by james_s_tayler »

joe_the_user wrote: post #8
infinity0 wrote: post #5

> rescued the idea that there are truths that humans can never prove This is a gross misinterpretation of Gödel's actual theorem that helps perpetuate irrational superstitious attitudes against science, mathematics, and logic. What Gödel showed was that proofs are relative to some underlying axiomatic model and that for any particular axiomatic model there are always truths unprovable by it. That doesn't mean "there…

This is a gross misinterpretation of Gödel's actual theorem that helps perpetuate irrational superstitious attitudes against science, mathematics, and logic. It is, it really is. Yet is also an interpretation that Gödel himself would indulge in. Consider his most famous quote: "Either mathematics is too big for the human mind, or the human mind is more than a machine." (I remember reading this statement in the Time-L…

But what's say there was some conjecture that if proved to be the case proved a unifying theory of everything in Physics that relied on that conjecture, but that conjecture couldn't be proven under the given set of axioms... then in that case wouldn't it be that there were some truths about the universe that aren't possible to prove?

I don't think it's irrational or anti-science to say or think that. It just seems like it might be a possibility, which sure, why the hell not?

"There's this thing we can't prove under the given axioms"..."Ok, well just extend the axioms"..."Right but there is still this other thing we can't prove under that set of axioms."..."Ok, so rinse and repeat?"

How, if you can't prove it, do you figure out if it is an axiom in the first place?

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