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I just started to study Gödel's Incompleteness Theorems and have a doubt.

I know that, if Math is inconsistent, than every statement can be proven to be True. This happens because if we can prove that (a) $X$ is True and that (b) $X$ is False, than "$X$ or $Y$" is True by (a) and hence, $Y$ is True by (b), and this for all statements $Y$.

My doubt is that, by all the defined axioms, I think we can certainly prove that some statements are False and only False, like "$5$ is even". This way, for me, not every statement can be proven to be True, which would mean that Math is not inconsistent. Therefore, Math would be consistent.

Obviously Gödel Second Theorem stops me to arrive at this conclusion, but I still can't get what exactly is the confusion in my thinking.

Alexandre Tourinho
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    You're saying that it's possible to prove that "5 is even" is not provable. How exactly would you do that? – Chris Culter Jun 12 '20 at 21:34
  • Assume we work in an inconsistent axiom system, then there is a statement $X$ which is true and false. Now assume "5 is not even", then $X$ is true. But this is a contradiction with the fact that $X$ is false. Conclusion: "5 is even", proof by contradiction. – Jens Renders Jun 12 '20 at 21:37
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    You can certainly prove that $5$ is not even, but how do you know that you can't prove that $5$ is even? You only know that if you know that your axiom system is consistent. – Robert Israel Jun 12 '20 at 22:03
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    Very important to understand Goedel's results is the difference between "true" and "provable". The true statements are NOT equivalent to the provable statements. Assuming consistency , a statement is provable if and only if it is true in every interpretation. "True" and "False" only make sense in an interpretation. If the theory is inconsistent, it is utterly useless because it can prove (and also disprove!) every statement. – Peter Jun 14 '20 at 18:01
  • If we assume consistency, and the theory is strong enough for the representation theorem, some statements will be neither provable nor disprovable. But this does not mean that we can prove/disprove nothing. Universally true statements (true in every interpretation) are as said provable. – Peter Jun 14 '20 at 18:05
  • "if Math is inconsistent, than every statement can be proven... fullstop". This does not mean that what we prove is True. If we assume that True/False refer to some mathematical reality "out there", what is inconsistent is not the world, but our description of it (the theory). – Mauro ALLEGRANZA Jun 16 '20 at 08:43
  • We can prove, from usual arithmetical axioms, that "$5$ is not Even", assuming that $\text {Even}(n) \leftrightarrow \exists k (n=2 \times k)$, because $5=2 \times 2 +1$. The issue is that, if our "usual arithmetical axioms" are inconsistent, then we can prove also that "$5$" is Even". – Mauro ALLEGRANZA Jun 16 '20 at 08:46

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