3
votes
1answer
113 views

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 ...
6
votes
2answers
294 views

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 ...
1
vote
1answer
51 views

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 ...
3
votes
1answer
113 views

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 ...
0
votes
1answer
141 views

Why isn't GL system of provability logic reflexive?

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 $p$), ...
0
votes
1answer
131 views

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 ...