Philosophy Now Forum

∃G ∈ F (G ↔ (F ⊬ G))
There exists a G in F such that G is logically equivalent to its own unprovability in F

Logical equality
p---q---p ↔ q
T---T------T // G is true if and only if G is Unprovable.
T---F------F //
F---T------F //
F---F------T // G is false if and only if G is Provable.

https://en.wikipedia.org/wiki/Truth_tab ... l_equality

Row(1) There exists a G in F such that G is true if and only if G is
unprovable in F making G unsatisfied thus untrue in F.

Row(4) There exists a G in F such that G is false if and only if G is
provable in F making G satisfied thus true in F.

If either Row(1) or Row(4) are unsatisfied then ↔ is false and there
is no such G in F.

Because the RHS of ↔ contradicts the LHS there is no such G in F.
Thus the above G simply does not exist in F.

When we take the informal version of this where
G asserts its own unprovability in F

The proof of G in F requires a sequence of inference
steps in F that prove that they themselves do not exist.

PeteOlcott wrote: ↑Sat Apr 22, 2023 2:59 pm ∃G ∈ F (G ↔ (F ⊬ G))
There exists a G in F such that G is logically equivalent to its own unprovability in F

Logical equality
p---q---p ↔ q
T---T------T // G is true if and only if G is Unprovable.
T---F------F //
F---T------F //
F---F------T // G is false if and only if G is Provable.

https://en.wikipedia.org/wiki/Truth_tab ... l_equality

Row(1) There exists a G in F such that G is true if and only if G is
unprovable in F making G unsatisfied thus untrue in F.

Row(4) There exists a G in F such that G is false if and only if G is
provable in F making G satisfied thus true in F.

If either Row(1) or Row(4) are unsatisfied then ↔ is false and there
is no such G in F.

Because the RHS of ↔ contradicts the LHS there is no such G in F.
Thus the above G simply does not exist in F.

Olcott, you are a fucking idiot!

You still don't understand the difference between existence and provability.

ALL elements of F exist.

B,C,D and E are provable.
G is not provable.

Thank God for PeteOlcott!

Keeping the flame of mathematical philosophy going is not exactly a walk in the park. Not after Kurt Gödel broke up the party and sent everybody packing.

My doubt is very simple: Since when did we start entertaining liars?