Hacker News Board Hacker News, browsed like a classic message board
Live data from Hacker News

Kurt Gödel and the romance of logic

prospectmagazine.co.uk

Post reply on HN
Print view View original

11–20 of 52 posts

Re: Kurt Gödel and the romance of logic

#11

Post by joe_the_user »

madmax96 wrote: post #9
joe_the_user wrote: post #8

Earlier quoted context omitted.

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…

What’s the problem with the disjunction?

Well, the only thing that Godel showed was that if you have consistent assignment of a truth value to every well-formed statement in a given logic language (at least as complex the Peano postulates), you will get some statements which are assigned "true" or "false" but whose truth cannot be deduced in a given axiom system for the language. However, the above statement very much involves a statement about "truth" in much transcendent sense - see the discussion of Godel's Platonism and religious beliefs in the article.

Re: Kurt Gödel and the romance of logic

#12

Post by skh »

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…

You are using the word “truth” but it is more correct to use provable/non-provable. What Godel showed is that -limiting the discussion to the natural numbers for simplicity - there are statements that are true in the standard model of the natural numbers that are not provable in the first order Peano Axiomatic system for the natural numbers. What this means is that such a statement will be false in some non-standard model of the natural numbers.

It’s important that we are talking about models of the first order Peano Axioms. One can always find a system of axioms in which all true statements are provable. To do this just take the collection of all true statements in the standard model. Now every true statement is a theorem. It’s easy to have a complete set of axioms. What can’t happen is a recursively enumerable set of axioms that is complete and consistent.

Recursively enumerability is needed so that one can have an effective means of determining if a statement is an axiom. Think computable when I say effective.

When talking about truth we need to be careful because this is tied to a model of an axiomatic system. By the Completeness Theorem a statement that is true in all models of the system is provable.

Re: Kurt Gödel and the romance of logic

#13

Post by skh »

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…

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 no procedure for determining if a statement is an axiom or not.

Re: Kurt Gödel and the romance of logic

#14

Post by skh »

joe_the_user wrote: post #4

Actually, Gödel's theory is fairly accessible compared to, say, the General Theory of Relativity. All you have to know that the process of proving a statement is a fairly mechanical, step-by-step process. At that point, "statements that can be proven true" and "statements that be proven false" are two fairly defined sets. At that point, "there are some true statements that cannot be proven true" is moderately clear.…

The quality of being “true” is dependent on the model one is using. One can not talk about “truth” without being in a model. (Assuming we are talking about standard mathematical logic.). A statement in a first order system is provable if and only if it is true in all models for that system.

Re: Kurt Gödel and the romance of logic

#15

Post by KMag »

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.

But the very interesting part is that you might keep extending and extending two axiomatic systems until every true statement is provable in one or the other system. However, you're guaranteed that in that case, the two systems will be inconsistent with each other such that they cannot be combined into a consistent whole.

Re: Kurt Gödel and the romance of logic

#16

Post by jeromebaek »

joe_the_user wrote: post #4

Actually, Gödel's theory is fairly accessible compared to, say, the General Theory of Relativity. All you have to know that the process of proving a statement is a fairly mechanical, step-by-step process. At that point, "statements that can be proven true" and "statements that be proven false" are two fairly defined sets. At that point, "there are some true statements that cannot be proven true" is moderately clear.…

It's true, a glimmer of the proof can be easily grasped. The analogy between the Godel sentence and the Liar's Paradox, "this sentence is false", is not perfect but almost isomorphic. I think this is rather the most curious feature of the theorem, that a 5-year old could understand it, at least a glimmer of it. Now there is a big gap between grasping it somewhat and being able to consider its full ramifications.

Re: Kurt Gödel and the romance of logic

#17

Post by joe_the_user »

skh wrote: post #14
joe_the_user wrote: post #4

Actually, Gödel's theory is fairly accessible compared to, say, the General Theory of Relativity. All you have to know that the process of proving a statement is a fairly mechanical, step-by-step process. At that point, "statements that can be proven true" and "statements that be proven false" are two fairly defined sets. At that point, "there are some true statements that cannot be proven true" is moderately clear.…

The quality of being “true” is dependent on the model one is using. One can not talk about “truth” without being in a model. (Assuming we are talking about standard mathematical logic.). A statement in a first order system is provable if and only if it is true in all models for that system.

That's it, plus a model is an assignment of a truth value to every well-formed statement of logic language.

Re: Kurt Gödel and the romance of logic

#18

Post by JoeSmithson »

oska wrote: post #6
craigr1972 wrote: post #2

" He announced that he had studied the US constitution in detail, and—no doubt, forensically examining its propositions one at a time and perhaps testing it against thought experiments against wild possible futures in which the president was allowed to get out of control—he had discovered how the US could legally be turned into a dictatorship."

Suggestions as to what this found flaw might have been has been discussed on Quora a couple of times [1][2] [1] https://www.quora.com/How-did-G%C3%B6del-believe-the-US-coul... [2] https://www.quora.com/What-was-the-flaw-Kurt-G%C3%B6del-disc...

The first answer is from someone claiming a BS from the University of Autodidacts, who finishes with an unexplained dig at the Democratic Party.

Re: Kurt Gödel and the romance of logic

#19

Post by whatshisface »

skh wrote: post #13
truantbuick wrote: post #10

Earlier quoted context omitted.

> 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…

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.

Re: Kurt Gödel and the romance of logic

#20

Post by AlexCoventry »

jeromebaek wrote: post #16
joe_the_user wrote: post #4

Actually, Gödel's theory is fairly accessible compared to, say, the General Theory of Relativity. All you have to know that the process of proving a statement is a fairly mechanical, step-by-step process. At that point, "statements that can be proven true" and "statements that be proven false" are two fairly defined sets. At that point, "there are some true statements that cannot be proven true" is moderately clear.…

It's true, a glimmer of the proof can be easily grasped. The analogy between the Godel sentence and the Liar's Paradox, "this sentence is false", is not perfect but almost isomorphic. I think this is rather the most curious feature of the theorem, that a 5-year old could understand it, at least a glimmer of it. Now there is a big gap between grasping it somewhat and being able to consider its full ramifications.

It's definitely easier than General Relativity. I read Goedel's original paper when I was 19, whereas there are aspects of GR I'm still getting my head around.
Post reply on HN