0

What is wrong with the following argument?

Let $F = T + ¬G_T$, where $T$ is an effectively generated formal system and $G_T$ is its Godel sentence.

Then, it is possible to prove within $F$ the First Incompleteness Theorem for $T$: $Con(T) \Rightarrow G_T$, just by utilising the axioms for $T$.

But from this the contrapositive follows: $¬G_T \Rightarrow ¬Con(T)$.

So in F it is possible to prove $¬Con(T)$, since $¬G_T$ is an axiom.

But it then follows that $¬Con(F)$, since the same contradiction that is derivable in $T$ is also derivable in $F$ by restricting ourselves to the axioms for $T$.

So any formal system that is of the form $F = T + ¬G_T$ is inconsistent.

J.-E. Pin
  • 40,163
Rojus Lukauskas
  • 11
  • it would be helpful if you explained what you mean by the = and + sign in your equation. you are mixing and matching operators and not forming well-formed formulas – RyRy the Fly Guy Nov 29 '23 at 17:28
  • @RyRytheFlyGuy F is the formal system that has the axioms of T plus the axiom of the negation of the Godel sentence of T. – Rojus Lukauskas Nov 29 '23 at 17:30
  • 1
    This is a good question (and absolutely should not have been downvoted, it's an important subtlety), but it's also a duplicate. See e.g. https://math.stackexchange.com/questions/2165704/puzzle-does-the-following-prove-that-pa-is-inconsistent, or https://math.stackexchange.com/questions/4179554/puzzle-does-the-following-prove-that-pa-is-inconsistent-part-2?noredirect=1&lq=1, or https://math.stackexchange.com/questions/2287867/how-to-think-about-theories-that-prove-their-own-inconsistency. – Noah Schweber Nov 29 '23 at 17:41
  • Actually that last one is not quite the same question, although it's closely related; my first link should have been my dupe target. – Noah Schweber Nov 29 '23 at 17:46

1 Answers1

2

Not exactly. You have shown that $F$ can formally prove $\neg \text{Con}(F)$ but you can't deduce from it that $F$ is inconsistent (which is a meta-statement). You seem to believe that the following meta-theorem is true :

"If $F \vdash \neg \text{Con}(F)$ then $F$ is inconsistent"

but how would you prove such a thing ?

C. Dubussy
  • 9,120