Talk:Gödel's Incompleteness Theorems/First

This Proof is Circular

The proof depends on the contrapositive of the proof. That is not a rigourous proof, if you ask me.

Not sure I follow. Are you a follower of the intuitionist school? Works as:

Suppose we have an object $x$ which possesses property $A$. In this context, $x$ is a set of theorems. Property $A$ is that of being both consistent and complete.

From subtheorems that we have deduced, we demonstrate that if $x$ has property $A$, a false statement can be deduced. (Strictly speaking, a contradiction can be deduced.)

Because a falsehood, or a contradiction, results in mathematics itself being inconsistent, we cannot allow within the rules of mathematics within which we work this statement that "$x$ has property $A$" which, in this case, is "A set of theorems can be both complete and consistent."

That is, "It is not the case that a set of theorems can be both complete and consistent." Because if it were, it would be outside the rules of mathematics.

So what are these "rules of mathematics"? Can we expand them to take the contradiction on board? Oh, we can? If you can't, mathematics is inconsistent. If you can, mathematics is incomplete.

Pick any one -- you can't have both. --prime mover (talk) 23:32, 25 March 2023 (UTC)

One more thing: Please sign your posts. --prime mover (talk) 23:35, 25 March 2023 (UTC)

Mathematical logic can be very unintuitive, since there are so many moving parts. There are three separate versions of the "rules of mathematics" at play in a theorem like this.

1. The logical system in which we, the mathematicians, work. In this world, we apply reasoning in order to argue that certain things are true. We have to assume that, if our reasoning is sound, everything we prove in this system is correct.
2. The formal system we are analyzing. This is $T$ in the theorem. It is a set of precise axioms and deduction rules in a formal language, which we can study as a mathematical object.
3. The interpretation we give to the formal system. We want the formal system to actually represent the fact that some statements are "true" and others "false," and we want this to match our own logical system.

What the Incompleteness Theorem says is, in essence, is that such a formal system can never fully capture the concept of truthhood. No matter what we do, there will always be pieces missing. This has an unfortunate implication for the logical system we build mathematics in, since the Formalists believe it can be captured with such an object. The conclusion is that our own logic isn't enough to define what is true and false; at least, not without adding on more and more assumptions as we run into problems, putting us at risk of causing a contradiction.

My terminology here is probably all wrong, and I'm certainly not an expert, but that's my interpretation of the subject. I hope this helps. --CircuitCraft (talk) 04:03, 26 March 2023 (UTC)

What he said, possibly apart from what the Formalists believe. --prime mover (talk) 07:44, 26 March 2023 (UTC)

Forgive me, I was being stupid. When I reviewed this theorem, for some odd reason, I read "Corollary" as "Lemma." --PeterJohnson (talk) 16:18, 26 March 2023 (UTC)