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)