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0 pointsjoe_the_user12y ago0 comments

Downvoted for your inaccessible link.

And while it is kinda nit-picky, the parent's statement is literally true (see my other post also).

Given any fixed axiom system, there will be true statements that aren't provable within the system (expand your axioms and you'll just have different true but provable statements in the expanded system). Now, Godel's completeness theorem shows that you construct complete mathematical system of true statements however such a system requires inserting an infinite number of arbitrarily choices among statements (and their negations) which aren't provable given the previous axioms. Since the framework of the article is finite, not infinite, I would claim the framework of the article, being finite, can't encompass all true statements of any given system, even if it an algorithm for producing axioms.

https://en.wikipedia.org/wiki/G%C3%B6del%27s_completeness_th...

Edit: I got through the pay-wall via Google but the discussion is somewhere between confused and confusing (the large part of post mostly meaningless speculation about the term "proved in an absolute sense", that he introduces without defining). The situation is really simple. All formal proof systems have hole (at least those of any reasonable "powerfulness"). Any formal proof system can be expanded indefinitely but at any point in that expansion will still have a hole.

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stiff12y ago

The parent comment seems to imply that there are some absolute mathematical truths and that there are some statements true in this absolute sense that can not be proved by mathematics. Goedels theorem shows something else: that starting from an axiom system there will be statements true in this axiom system that are not provable. I anyway doubt John Baez meant mapping all true sentences from all possible axiom systems in form of a graph...

teilo12y ago

Actually, my statement is limited to mathematical truths. I am not talking about truth in general. And yes, the nit-pick was out of place. And no, I am not an art major, and yes, I understand Goedel's theorem just fine.

I said, "Not all mathematical truths can be proven to be true." I don't know how you got from there to: "there are some statements true in the absolute sense that cannot be proved by mathematics".

My statement is equivalent to your statement: "Starting from an axiom system, there will be statements true in this axiom system that are not provable."

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