# Tagged Questions

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### What other unprovable theorems are there? [duplicate]

Gödel's incompleteness theorem presents us with the possibility of having theorems that are neither provable nor disprovable in a given axiomatic set. Already we have the continuum hypothesis which ...
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### Can an unprovable statement be interpreted as being generally true in some cases?

For example, let's say that Goldbach's conjecture turns out to be unprovable. This would mean that a program cannot devise a way to check whether any counterexample exists. This seems to mean that ...
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### Can ZFC+A and ZFC+negation of A be both inconsistent where A is some conjecture?

So I know that a conjecture/statement or negation of it plus ZFC can both turn out to be consistent, which means that a statement is not provable. But I would like to go opposite way - and let's say ...
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### properties of the provability predicate applied to open formulas

Good day! Let $\mathrm{T}$ be a first-order theory which contains the Peano arithmetic and has a recursively enumerable set of axioms. It is well known that one can construct a predicate ...
Formula $\square p \rightarrow p$ (axiom T; corresponding to reflexive modal frames) is interpreted as "if p is provable, then p", or more precisely: for all realizations (all substitutions for $p$), ...
### Uniqueness of super godel numbers of $\varphi$ and $\neg \varphi$
Let $e_{0},e_{1},...,e_{n}$ be a sequence of wffs or other expressions. Code each $e_{i}$ by a regular godel number $g_{i}$, to yield a sequence of numbers $g_{0},g_{1},...,g_{n}$. Then encode this ...