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Problem is
Authored by: PolR on Monday, May 28 2012 @ 10:59 PM EDT
You forget the hypotheses of the theorems.

First these are not a theorems about math. These are theorems about formal systems with the capability of reproducing Peano arithmetic. This is a large portion of math but this is not all of math. These theorems don't apply to Boolean algebra for example.

Second, if you don't have consistency, you don't have math. The incompleteness theorems work because consistency is assumed in the hypotheses. Without an assumption of consistency the proofs mean nothing.

You wrote literally:

maths in its very foundations is built on incompleteness and inconsistency.
Maybe you wrote something that means more than you intended, but it is not possible to build math on inconsistency. You can't be confident your proofs mean anything when the conclusions are inconsistent. What Gödel's second incompleteness theorem said is you can't prove the consistency of Peano arithmetic from within the same system. This doesn't mean it isn't consistent, just that the proof is unavailable from within the same system. This is an incompleteness theorem because it is about a proof that can't be done. This is not about (in)consistency itself.

FWIW Gentzen has managed to prove the consistency of first order arithmetic using transfinite induction.

What Gödel proved is not that you cannot have a solid foundation underpinning mathematics. He proved that you cannot build this foundation and prove this it is solid (ie consistent) using the methods envisioned by David Hilbert. The possibility of having a solid foundation you can't prove to be solid is left open. Then empirical evidence of solidity may be available. If after a few decades of searching for the inconsistencies none are being found, mathematicians may have grounds to believe their foundations are consistent.

Mathematicians can't imagine a system of logic for writing some foundations of mathematics which won't be subject to the methods of Gödel. But how can we know whether this is a failure of human imagination or a fundamental limitation of mathematics? Most mathematicians think it is a fundamental limitation on the grounds that all attempts to find a work around have failed. This is drawing conclusions from strong empirical evidence. This is different from proven mathematical knowledge.

As Gentzen has illustrated, it may still be possible to discover some principles of logic or mathematics which will surprise us. Gentzen proof has its own assumptions and I am not familiar with it. I don't know how big of a work around Gödel it offers. But I think it hints that we should be careful not to overstate what Gödel theorems actually mean. They are very strong theorems but they are not without limits. And I also think you tend to overstate them because you are not familiar with their limits.

[ Reply to This | Parent | # ]

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