Philosophy Now Forum

Skepdick wrote: ↑Tue Apr 18, 2023 8:23 am

PeteOlcott wrote: ↑Mon Apr 17, 2023 4:47 am If we take the simplest possible essence of Gödel's logic sentence we have:
G asserts its own unprovability in F. This means that G is asserting
that there is no sequence of inference steps in F that derives G [something's missing here]

Yes, that's what it means. But you are missing somethign at the end.

Your statement (and understanding) is incomplete! And Gödel sure understood incompleteness.

The part that's mising is: "from other elements in F".

e.g G exists in F, but it is not connected to anything else in F.
G exists and G is not provable in F are both true.

PeteOlcott wrote: ↑Mon Apr 17, 2023 4:47 am For G to be proved in F requires a sequence of
inference steps in F that proves there is no such
sequence of inference steps in F.

This is like René Descartes saying: “I think therefore thoughts do not exist”

Why are you conflating Provable(G) with Exists(G) ?

Time to learn the difference between connected and disconnected spaces.

https://en.wikipedia.org/wiki/Totally_d ... cted_space

Skepdick wrote: ↑Tue Apr 18, 2023 4:11 pm

PeteOlcott wrote: ↑Tue Apr 18, 2023 4:09 pm Although this seems difficult
G = There is no sequence of inference steps in F that proves there is no such sequence of inference steps in F.

Compared to Gödel's arithmetization and diagonalization (that takes dozens of pages) the above expression is simple.

What sequence of steps do you expect to prove that no paths lead to G when it's true that no paths lead to G?

A->B->C->D G

PeteOlcott wrote: ↑Tue Apr 18, 2023 4:20 pm This means that it is asserting that there is no sequence of inference steps in F that derives G.

PeteOlcott wrote: ↑Tue Apr 18, 2023 4:20 pm The proof of G requires a sequence of inference steps in F that proves no such sequence of
inference steps exists in F.

PeteOlcott wrote: ↑Tue Apr 18, 2023 4:20 pm When we look at the barest essence of G behind the extraneous complexity of arithmetization
and diagonalization we see that G is merely self-contradictory and that is the reason that
G cannot be proved in F.

PeteOlcott wrote: ↑Tue Apr 18, 2023 4:24 pm The point is that G is self-contradictory and that is the only reason that G cannot be proved in F.

Gödel erroneously concludes that F is incomplete on the basis that F cannot prove a self-contradictory
expression.

Skepdick wrote: ↑Tue Apr 18, 2023 4:26 pm

PeteOlcott wrote: ↑Tue Apr 18, 2023 4:20 pm This means that it is asserting that there is no sequence of inference steps in F that derives G.

Sure. Another way to think about it is that in the graph - it's a node with no incoming arrows.

PeteOlcott wrote: ↑Tue Apr 18, 2023 4:20 pm The proof of G requires a sequence of inference steps in F that proves no such sequence of
inference steps exists in F.

Sure thing. Search the proof-space and convince yourself that no proofs lead to G.

PeteOlcott wrote: ↑Tue Apr 18, 2023 4:20 pm When we look at the barest essence of G behind the extraneous complexity of arithmetization
and diagonalization we see that G is merely self-contradictory and that is the reason that
G cannot be proved in F.

It's not self-contradictory. It's true.

No paths lead to G - it's a moat.

Skepdick wrote: ↑Tue Apr 18, 2023 4:28 pm

PeteOlcott wrote: ↑Tue Apr 18, 2023 4:24 pm The point is that G is self-contradictory and that is the only reason that G cannot be proved in F.

Gödel erroneously concludes that F is incomplete on the basis that F cannot prove a self-contradictory
expression.

You continue to confuse G's existence with G's provability.

Flannel Jesus wrote: ↑Tue Apr 18, 2023 4:54 pm "self contradictory" isn't the right word to describe G.

PeteOlcott wrote: ↑Tue Apr 18, 2023 4:51 pm No I don't.

G asserts its own unprovability in F

Skepdick wrote: ↑Tue Apr 18, 2023 5:19 pm

PeteOlcott wrote: ↑Tue Apr 18, 2023 4:51 pm No I don't.

G asserts its own unprovability in F

Yes, you do.

"G asserts its own unprovability" is not the same thing as "G is unprovable"
"0 asserts its own unprovability" vs "0 is unprovable"

You are anthropomorphising the elements.

PeteOlcott wrote: ↑Tue Apr 18, 2023 5:26 pm There are two distinct steps:
(1) G asserts its own unprovability in F
Which is the same thing as G asserts that there is no sequence of inference steps in F that derives G.

(2) The actual proof of G in F requires a sequence of inference steps in F
that proves that no such set of inference steps exists in F.

Skepdick wrote: ↑Tue Apr 18, 2023 5:30 pm

PeteOlcott wrote: ↑Tue Apr 18, 2023 5:26 pm There are two distinct steps:
(1) G asserts its own unprovability in F
Which is the same thing as G asserts that there is no sequence of inference steps in F that derives G.

(2) The actual proof of G in F requires a sequence of inference steps in F
that proves that no such set of inference steps exists in F.

It doesn't require any inference steps. The assertion is true but unprovable.

PeteOlcott wrote: ↑Tue Apr 18, 2023 5:35 pm

Skepdick wrote: ↑Tue Apr 18, 2023 5:30 pm

PeteOlcott wrote: ↑Tue Apr 18, 2023 5:26 pm There are two distinct steps:
(1) G asserts its own unprovability in F
Which is the same thing as G asserts that there is no sequence of inference steps in F that derives G.

(2) The actual proof of G in F requires a sequence of inference steps in F
that proves that no such set of inference steps exists in F.

It doesn't require any inference steps. The assertion is true but unprovable.

For G to assert that it is unprovable in F doesn't require any inference steps.

To prove in F that G is unprovable in F requires a sequence of inference steps
(all proofs always require a sequence of inference steps) in F that proves that
no such sequence of inference steps exists in F.

Skepdick wrote: ↑Tue Apr 18, 2023 5:49 pm

PeteOlcott wrote: ↑Tue Apr 18, 2023 5:35 pm

Skepdick wrote: ↑Tue Apr 18, 2023 5:30 pm
It doesn't require any inference steps. The assertion is true but unprovable.

For G to assert that it is unprovable in F doesn't require any inference steps.

To prove in F that G is unprovable in F requires a sequence of inference steps
(all proofs always require a sequence of inference steps) in F that proves that
no such sequence of inference steps exists in F.

If you prove in F that G is uprovable in F that's a contradiction.

PeteOlcott wrote: ↑Tue Apr 18, 2023 5:57 pm Exactly! So the reason that G cannot be proved in F is that G is contradictory
in F not that F is in any way incomplete.