[Paradox]How can Gödel prove that Gödel sentence is unprovable but true, if such proof itself proves that Gödel sentence is true?

Question

Isn't the proof that Gödel sentence is unprovable but true a proof itself that Gödel sentence is true?

Gödel in the preface of his proof remarked:

“From the remark that [the unprovable statement] asserts its own unprovability, it follows at once that [the unprovable statement] is correct, since [the unprovable statement] is certainly unprovable (because undecidable). So the proposition which is undecidable in the system PM yet turns out to be decided by meta-mathematical considerations.”

My question may be the example of what Gödel called "meta-mathematical considerations". It is hard to understand that the proposition which is undecidable in mathematics can be decided by meta-mathematics. What could be the explanation for this apparent paradox?

Answer 1

Roughly speaking, it does demonstrate it, but it does not prove it within a formal axiomatic system. It can only be concluded outside that system, so there is no paradox.

What Gödel's theorem actually asserts is that there is at least one proposition that is true but which lacks a proof within a particular axiomatic system. He was responding to Principia Mathematica, but the technique he employs can be applied "against" any sufficiently strong axiomatic system.

Therefore, the fact that one can conclude, outside that axiomatic system, that the proposition is true does not produce a paradox, does not disprove the proposition.

Answer 2

Godel produces a sentence $\varphi$. What Godel proves is that - assuming $PA$ is consistent - $\varphi$ is true but not provable in $PA$. (I'm assuming the theory we're looking at is "PA," here - but we can of course replace $PA$ with any sufficiently strong recursively axiomatized theory, such as PM, ZFC, NF, ...)

This proof goes through perfectly inside the theory $PA$. There's no contradiction, though, because - in order to conclude that $\varphi$ is true - $PA$ would have to know that $PA$ is consistent. So, instead of a paradox, we get Godel's second incompleteness theorem: that, if $PA$ is consistent, $PA$ doesn't prove "$PA$ is consistent."

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I'm being ahistorical here - in fact, what Godel proved was slightly weaker, and Rosser was the one who brought the hypothesis down to "$PA$ is consistent" - but this is the meat of the situation.

Answer 3

You changed "undecidable in the system PM" into "undecidable in mathematics" in your paraphrase and so the explanation is that you introduced the apparent paradox.

[Aside: the quotation is correct in saying "PM" rather than "PA": Gödel’s original paper on the incompleteness theorem deals with the system of Russell and Whitehead's Principia Mathematica and not Peano Arithmetic.]