learis wrote: ↑Sat Mar 23, 2024 4:06 pm
"This statement is not provable"
Definition of a Statement:
From Internet Encyclopedia of Philosophy:
A statement can be defined as a declarative sentence, or part of a sentence, that is capable of having a truth-value, such as being true or false.
A statement by definition must be capable of having a truth value. To state that this statement is not provable (aka not capable of having a truth value) is to go against the very definition of what a statement is.
So the original quote is absurd & malformed. There's nothing to debate.
"Provable" means that there exists a proof somewhere in the database of proofs for the statement at hand.
Imagine that you remove its proof from the database, does that mean that the statement suddenly becomes false? No, of course not! It remains true. It is just not provable anymore.
In 1931, Godel did of course not use the term "database", but he still went to great length to construct an abstract one, along with a lookup function. He used it to abstractly implement the predicate isProvable(s).
So, you try to look up the proof in the database. If it exists, then the statement is provable. Otherwise, it is not.
Hence, provability is clearly distinct from truth.
The confusion actually stems from Soundness theorem. We routinely assume Soundness theorem:
If a statement is provable from its theoretical context, then it is true in all its interpretations ("models").
For first-order logic, Soundness theorem is also confusingly known as
Godel's (semantic) completeness theorem.
The reverse is not necessarily the case.
It is not because a statement is true (in a particular interpretation) that it is provable from its theoretical context. This is bound to happen if the statement is true in one interpretation but not in another one. This is known as
incompleteness.
In first-order arithmetic, because of Lowenheim-Skolem theorem, we know that there are an infinite number of nonstandard models ("interpretations") besides the standard one, i.e. the natural numbers.
Therefore, the suspicion existed already that in first-order arithmetic there were statements true in one model but not in another (nonstandard) one. Godel finally proved this.
If you confuse truth with provability, you cannot possibly understand Godel's work.