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> The consequences of the Incompleteness theorems are often overstated, but they do falsify the idea that mathematics is logically consistent, or that a set of

> The consequences of the Incompleteness theorems are often overstated, but they do falsify the idea that mathematics is logically consistent, or that a set of symbols, however clear and unambiguous, will prevent basic misunderstandings in even the most fertile minds. The incompleteness theorems don't falsify the idea that mathematics is logically consistent; they do falsify the idea that any reasonable mathematical theory could prove its own logical consistency. They also don't remove the possibility that there is some metamathematical justification for an unambiguous interpretation of the concept of the natural numbers or even of a set (see all of the work in large cardinals, culminating in Hugh Woodin's recent work on extender models for supercompact cardinals).

lutusp 14y ago
> The incompleteness theorems don't falsify the idea that mathematics is logically consistent; they do falsify the idea that any reasonable mathematical theory could prove its own logical consistency. Yes, true, but I think that amounts to the same thing. Our inability to prove the thesis casts in doubt our right to assert it at all. That's certainly true for any other mathematical idea. No one was willing to say that Fermat's Last Theorem, or the Four-Color Map Theorem, were proven, until they were. Nevertheless, I shouldn't have said that the Theorems "falsify" the logical consistency of mathematics. They prevent the notion from being demonstrated, but doesn't invalidate its existence as a hypothesis. > They also don't remove the possibility that there is some metamathematical justification for ... Yes, but that's not a positive claim, it's the assertion that it can't be ruled out. And such an effort might fall afoul of the "sufficiently complex" criterion of Godel's Theorems, which brings us full circle.

lutusp 14y ago

> The incompleteness theorems don't falsify the idea that mathematics is logically consistent; they do falsify the idea that any reasonable mathematical theory could prove its own logical consistency. Yes, true, but I think that amounts to the same thing. Our inability to prove the thesis casts in doubt our right to assert it at all. That's certainly true for any other mathematical idea. No one was willing to say that Fermat's Last Theorem, or the Four-Color Map Theorem, were proven, until they were. Nevertheless, I shouldn't have said that the Theorems "falsify" the logical consistency of mathematics. They prevent the notion from being demonstrated, but doesn't invalidate its existence as a hypothesis. > They also don't remove the possibility that there is some metamathematical justification for ... Yes, but that's not a positive claim, it's the assertion that it can't be ruled out. And such an effort might fall afoul of the "sufficiently complex" criterion of Godel's Theorems, which brings us full circle.