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Comment by SemanticStrengh | Hacker News Reader

I feel like godel incompleteness is a cheap trick that mislead to think there are statements in ZFC that can't be proven except that those proven statement are contrived useless meta-statements.

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I'm going to assume you mean that the true unprovable sentences that Gödel constructed are contrived. I don't think I can convince you otherwise, except to say that others thought like you and eventually Paris and Harrington came up with [1]. You can make even more "concrete" statements, see the work of Chaitin, but all this stems from Hilbert's very concrete 10th problem [2].

I must say, be careful with "reasoning". For example, once a philosophy PhD minimized mathematics eloquently by saying all mathematical results are "tautological", hence uninteresting. Hence, why bother studying it, and in your case, why bother thinking Gödel did anything special.

Why bother about anything at all... there's the rub.

[1] https://en.wikipedia.org/wiki/Paris–Harrington_theorem

[2] https://en.wikipedia.org/wiki/Hilbert's_tenth_problem

> Since Peano arithmetic cannot prove its own consistency by Gödel's second incompleteness theorem, this shows that Peano arithmetic cannot prove the strengthened finite Ramsey theorem.

This seems quite circular, where the (claims mine) contrived statements enabling godel 2 incompletude theorem, make it unable to prove a non-contrived statement?

Thanks for the reference though, I'll admit that is a topic I'm not familiar enough with to fully challenge it.

> I must say, be careful with "reasoning". For example, once a philosophy PhD minimized mathematics eloquently by saying all mathematical results are "tautological", hence uninteresting. Hence, why bother studying it, and in your case, why bother thinking Gödel did anything special.

Yeah that's a classic, it's a good thought experiment but that only prove that one must not be careful with the pruning that can enable high level reasoning but more with being careful of fallacious or misleading reasonings. Yes in theory, mathematical chain proofs are only tautologies derived from ZFC/higher order logic, so yes mathematics doesn't say anything new. However in practice, the task of unfolding reasoning chains and being able to refer to past lemnas as abtractions/objects, enable the reader to optimize for cognition, and hence the more proofs advances, the more we can discover, retain, understand and refer, useful tautological yet transitive knowledge.

This is not true. For example, the statement "there exists a bijection between the real numbers and the powerset of the integers" is not contrived or meta or useless, but is independent of ZFC.

independently of ZFC, can your statement be mathematically proven in a reasonable alternative axiomatic system?

Also, I am aware there are non-contrived statements that are can't be proven in ZFC such as https://en.wikipedia.org/wiki/List_of_statements_independent.... I'm only claiming that the incompleteness theorems only find contrived ones, and that for the non-contrived ones humanity has found, it is NOT that they are mathematically unproveable, it is that ZFC is not enough.

> independently of ZFC, can your statement be mathematically proven in a reasonable alternative axiomatic system?

There are plenty of people who find ZFC + ¬CH a reasonable axiomatic system.

EDIT: I misread samth's statement (https://news.ycombinator.com/item?id=31424581), which I think is proveable in ZFC; I thought their statement was ¬CH.

It seems its not enough ? https://mathoverflow.net/questions/273861/independence-over-... (Although its CH not the negation of it) what additional axioms be needed there?

I misread the initial statement as "there exist sets intermediate in size between the real numbers and the integers", which may have been what samth meant (https://news.ycombinator.com/item?id=31424581). That is ¬CH, so ZFC + ¬CH can obviously prove it. If you meant that ZFC + ¬CH is not enough to prove interesting mathematics, then it surely is, since ZFC itself is already enough to prove interesting mathematics. If you meant that ZFC + ¬CH (or ZFC + CH) is not enough to decide all "natural" statements … well, I agree with that, but it seemed that you were arguing against, not for, that position (https://news.ycombinator.com/item?id=31423948).

I am making the distinction between natural statements and contrived statements. I claim godel only find contrived statements. However there exist non-contrived, natural statements that are unprovable in ZFC. My question is, what set of additional axioms are needed to have a maximal coverage/provability of those independent statements, while introducing the least possible axioms. And yes indeed ZFC + ¬CH seems like a good first step.

There are a large number of barriers to formalizing your question, such as (i) the definition of natural, (ii) the definition of "maximal" when comparing countably infinite sets, and (iii) the fact that asking about the provability of X in vanilla ZFC is still arguably interesting in the system ZFC+X (see: reverse mathematics), so it's not clear what adding it as an axiom does towards the "largeness" of the set of provable natural statements.

About formalising (i) I guess it's about accepting everything that isn't a meta-token, I don't remember godel incompleteness theorems sufficently to be precise about it, anyway the useful list would be the wikipedia list of independent statements since most of them are not contrived (although very abstract). So yes my question can be specialized as to how to achieve maximum coverage of the wikipedia page. Otherwise, great comment and it's been a while since I hadn't heard the term reverse mathematics!

My statement was slightly misphrased, it was intended to be just a statement of CH.

There are a variety of stronger axiomatic systems (typically ZFC + some other axioms). Some of them prove the continuum hypothesis (which is what I stated), others prove the negation. There's no way to decide which one is "right", though, just like there's no way to tell if the Axiom of Choice is "right".

Your second paragraph is confused. Given an axiom system like ZFC, there are (a) statements that can be proved true or false using it, (b) statements that can't be proved that are true in a particular model, (c) statements that can't be proved that are false in that model. It's set (b) that the incompleteness theorem tells us must exist. The theorem doesn't "find" statements, it proves that (b) exists by constructing a particular one, which is necessarily meta because it applies to every formal system.

However, we do not have access to a model which tells us the answers for things like the CH. So all you can do is decide on the axiom scheme you like, and then some things are provable. You can always add more axioms, like CH, but you can add their negation instead, if you want. So there's no sense in which there's really a right answer for CH but we haven't found it yet.

Well, Gödel set out to prove incompleteness. The first statement that comes to mind is “there is an unprovable sentence” but that is too general. What Gödel did was find out that “this statement is unprovable” is “definable” in Peano’s system (once it has been encoded), and that’s it.

Nothing especially contrived there. Less so for a professional logician.

Halting problem and difficulties in compilation and static analysis are not contrived.

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