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Comment by zzless | Hacker News Reader

With all due respect, no, it isn't. His drivel against set theory shows that he didn't even read the basic axiomatic set theory texts. In one of his papers, he is ranting against the axiom of infinity saying that 'there exists an infinite set' is not a precise mathematical statement. However, the axiom of infinity does not say any such thing! It precisely states the existence of some object than can be thought of as infinite but does not assign any semantics to it. Ironically, if he looked deeper, he would realize that the most interesting set theoretic proofs (independence results) are really the results in basic arithmetic (although covered in a lot of abstractions) and thus no less 'constructive' than his rational trigonometry.

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Almost every critique of the axiom of infinity is philosophical. I don't think you can just say "the axiom is sound, so what's your point". And you don't even get to claim that because of Godel's incompleteness theorem.

The axioms were not handed to us from above. They were a product of a thought process anchored to intuition about the real world. The outcomes of that process can be argued about. This includes the belief that the outcomes are wrong even if we can't point to any obvious paradox.

"Sound" means free of contradiction with respect to the axioms assumed.

If you can derive a contradiction using his methods of computation I would study that with interest.

By "sound" I do not mean provably sound. I mean I have not seen a proof of unsoundness yet.

To clarify:

“Sound” != proof of soundness in the same way that the Riemann Hypothesis being true is not the same as RH being proven.

Not a bad analogy. Damn good.

> "Sound" means free of contradiction with respect to the axioms assumed.

Gödel wept.

If with an axiomatic system there are undecidable propositions, that is not the same with the axiomatic system being contradictory, i.e. where you can prove that a proposition is both true and false.

An undecidable proposition is neither true nor false, it is not both true and false.

A system with undecidable propositions may be perfectly fine, while a contradictory system is useless.

Thus what the previous poster has said has nothing to do with what Gödel had proved.

Ensuring that the system of axioms that you use is non-contradictory has remained as useful today as by the time of Euclid and basing your reasoning on clearly stated non-contradictory axioms has also remained equally important, even if we are now aware that there may be undecidable things (which are normally irrelevant in practice anyway).

The results of Gödel may be interpreted as a demonstration that the use of ternary logic is unavoidable in mathematics, like it already was in real life, where it cannot always be determined whether a claim is true or false.

Indeed. Soundness and completeness are different things.

There are two well accepted definitions of soundness. One of them is the inability to prove true == false, that is, one cannot prove a contradiction from within that axiomatic system.

They aren't completely different, because trying to achieve wine generally harms the other.

True, I guess you understood what I meant, that they are different things.

Indeed, as you allude, you cannot have both in an expressive enough system.

> It precisely states the existence of some object than can be thought of as infinite but does not assign any semantics to it

Can you elaborate on this? I think many understand that the "existence of some object" implies there is some semantic difference even if there isn't a practical one.

I really enjoyed Wildberger's take back in high school and college. It can be far more intuitive to avoid unnecessary invocation of calculation and abstraction when possible.

I think the main thrust of his argument was that if we're going to give in to notions of infinity, irrationals, etc. it should be when they're truly needed. Most students are being given the opposite (as early as possible and with bad examples) to suit the limited time given in school. He then asks if/where we really need them at all, and has yet to be answered convincingly enough (probably only because nobody cares).

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