Re: what makes it true?

mensanator@xxxxxxxxxxx wrote:
> I said if it were false, a counterexample would exist.

That's not necessarily true. For example, if the false statement is
of the form "there exists y such that P(y)", what counterexample can
you find?


> Complete means everything that's true is provable. Consistent means
> everything that's provable is true. You cannot have both. It's one
> or the other.

You can have both, if your system does not include sufficiently
expressive arithmetic. For example, certain axiomatizations of
Euclidean geometry.


>> but not the other way around. One should mention here Goedel's completeness
>> theorem which states: if a statement is true in any model then it has a proof.
>
> It doesn't say that. It states that there are undecidable
> statements. Or that only an inconsistent system can prove its own
> consistency.

That's Goedel's INcompleteness theorem, which is better-known but
certainly not the only theorem he ever published.


- Tim
.

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