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[bloop_bleep]

(confound it, sorry for the double-post)

Not true, actually. See Gödel's incompleteness theorem. You can have statements that are true that cannot be proven to be true. Toss in the negative proof of the Entscheidungsproblem and you have that there is no way to actually prove that a statement cannot be proven given a particular set of axioms.

So, it is true that any proposition in mathematical logic can be true or false. It is not true that there necessarily exists a proof of the truth or falsehood of a statement that is true or false, or whether there's a way to tell whether such a proof exists in the first place. Proof implies truth. Truth does not imply the existence of a proof.

(Ispil, does your signature imply that philosophy can turn you to evil? )

Wait, any proposition? Even those that have been proven?

I mean the definition of a proposition- a statement which can be either true or false. They are not to be confused with facts, which cannot be false. They are truth-bearers.

Ah, I was going by the layperson's definition ("a statement or assertion that expresses a judgment or opinion"), but you meant the more specific philosophical term, of which I was not aware. Okay.

So if I say something mathematical and it has been proved, it is a fact. Otherwise, it is a proposition. Is that the distinction? And does that mean that propositions become facts after being proven?

Wait, but... You say that "there is no way to actually prove that a statement cannot be proven given a particular set of axioms", but what if I have proven that "'1+1=2' is true" is true? Doesn't that mean that "'1+1=2' is false" is false and thus disproven, then?

Given a proposition, there is no way to prove that a proof or disproof exists without actually finding said proof or disproof. Think about it like the halting problem (since it's just a variation on it). If a proof does exist for a proposition, that makes the proposition true for that given set of axioms. It does not necessarily make it fact. If you want, you can take a look into proof theory to get an idea of what a proof entails in a math-theoretic sense.

There's no distinction between a "mathematical proposition" and a proposition. All that's required is that a proposition is a closed sentence. Closed sentence being the mathematical logic definition of a sentence; that is, a boolean-expressible formula with no free variables. As contrasted with an open sentence, which contains free variables.

That just seems a bit tautological. Isn't "true under these axioms" the same as "proved within these axioms", by definition?

Godel is some weird shit, man. Like much of the rest of mathematics.