∃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.
∃G ∈ F (G ↔ (F ⊬ G))
-
PeteOlcott
- Posts: 1514
- Joined: Mon Jul 25, 2016 6:55 pm
∃G ∈ F (G ↔ (F ⊬ G))
Last edited by PeteOlcott on Sun Apr 23, 2023 1:07 am, edited 2 times in total.
Re: ∃G ∈ F (G ↔ (F ⊬ G))
Olcott, you are a fucking idiot!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.
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.
- IMG_5510.png (93.19 KiB) Viewed 2130 times
-
Agent Smith
- Posts: 1442
- Joined: Fri Aug 12, 2022 12:23 pm
Re: ∃G ∈ F (G ↔ (F ⊬ G))
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?
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?
