G asserts its own unprovability in F

[PeteOlcott]

A formal system is not supposed to prove contradictory expressions.

It can.

If you want completeness you'll have to sacrifice consistency.
If you want consistency you'll have to sacrifice completeness.

What's confusing you about how choice works?

That is not how it actually works. We simply toss G out on its ass
(as erroneous) then F remains complete and consistent.

[Skepdick]

That is not how it actually works. We simply toss G out on its ass
(as erroneous) then F remains complete and consistent.

Then that's not F anymore. It's F without G.

It's a bit like tossing 0 on its ass as erroneous.

So you have 1,2,3,... left

[PeteOlcott]

That is not how it actually works. We simply toss G out on its ass
(as erroneous) then F remains complete and consistent.

Then that's not F anymore.

It's F without G.

G never was a member of F.
A formal system of sufficient expressive power can detect this and reject G.

[Skepdick]

G never was a member of F.
A formal system of sufficient expressive power can detect this and reject G.

So you were allways confused about G's existence in F?

[PeteOlcott]

G never was a member of F.
A formal system of sufficient expressive power can detect this and reject G.

So you were allways confused about G's existence in F?

I don't see how you can't understand that any proof of G in F requires a sequence
of inference steps in F that proves there is no such sequence of inference steps in F.

Do you understand that the Liar Paradox is not true?

[Skepdick]

I don't see how you can't understand that any proof of G in F requires a sequence
of inference steps in F that proves there is no such sequence of inference steps in F.

Do you understand that the Liar Paradox is not true?

Why are you conflating poofs of G in F with the existence of G in F?

G can exist in F while being unprovable.

[PeteOlcott]

I don't see how you can't understand that any proof of G in F requires a sequence
of inference steps in F that proves there is no such sequence of inference steps in F.

Do you understand that the Liar Paradox is not true?

Why are you conflating poofs of G in F with the existence of G in F?

G can exist in F while being unprovable.

The Liar Paradox is not a member of any formal system because
formal systems only include truth bearers and the Liar Paradox
is not a truth bearer. The proof of LP and the proof of ~LP are
both unsatisfiable.

According to math this would make the set of all truth incomplete yet
this is impossible because the set of all truth is stipulated to be all truth.
This proves that the math notion of incomplete is incoherent.

[Skepdick]

The Liar Paradox is not a member of any formal system because
formal systems only include truth bearers and the Liar Paradox
is not a truth bearer. The proof of LP and the proof of ~LP are
both unsatisfiable.

According to math this would make the set of all truth incomplete yet
this is impossible because the set of all truth is stipulated to be all truth.
This proves that the math notion of incomplete is incoherent.

OK, but there are other situations in which elements of a system can exist without being provable.

0 exists in ℕ.
0 is not provable in ℕ.

If you prove that 0 is not provable you also get a contradiction.

Do you want to toss 0 out also?

[PeteOlcott]

The Liar Paradox is not a member of any formal system because
formal systems only include truth bearers and the Liar Paradox
is not a truth bearer. The proof of LP and the proof of ~LP are
both unsatisfiable.

According to math this would make the set of all truth incomplete yet
this is impossible because the set of all truth is stipulated to be all truth.
This proves that the math notion of incomplete is incoherent.

OK, but there are other situations in which elements of a system can exist without being provable.

0 exists in ℕ.
0 is not provable in ℕ.

If you prove that 0 is not provable you also get a contradiction.

Do you want to toss 0 out also?

Some definitions of ℕ include 0 and others exclude it.
https://en.wikipedia.org/wiki/Natural_number

Those definitions of ℕ that include 0 prove that 0 is an element by definition.
I remember high school geometry included citing the definition as a proof step.

Much more recently it occurred to me that every element of the set of analytic
truth is semantically derived from expressions of language that have been
stipulated to be true. Such a system eliminates undecidabilty and incompleteness.

[Skepdick]

Those definitions of ℕ that include 0 prove that 0 is an element by definition.

No. A definition is not a proof. If that were the case then everything which exists proves itself because X -> X for all X.

You've wasted my time explaining this already.

[PeteOlcott]

Those definitions of ℕ that include 0 prove that 0 is an element by definition.

No. A definition is not a proof.

The entire body of analytic truth is entirely comprised of expressions of language
that are stipulated to be true and expressions of language that are semantically
derived from these expressions.

If a definition is not a proof then we have no idea how to show that cats are animals
because this is only true by definition (AKA stipulated truth)

[Skepdick]

The entire body of analytic truth is entirely comprised of expressions of language
that are stipulated to be true and expressions of language that are semantically
derived from these expressions.

If a definition is not a proof then we have no idea how to show that cats are animals
because this is only true by definition.

There you are conflating true and provable again.

There is no way to prove that 0 is a number!

It's true by definition. It requires no proof. You accept it on faith.

[PeteOlcott]

The entire body of analytic truth is entirely comprised of expressions of language
that are stipulated to be true and expressions of language that are semantically
derived from these expressions.

If a definition is not a proof then we have no idea how to show that cats are animals
because this is only true by definition.

There you are conflating true and provable again.

There is no way to prove that 0 is a number!

It's true by definition. It requires no proof. You accept it on faith.

It is stipulated to be true.
Haskell Curry understands that the elementary theorems (AKA axioms) of
a formal system are true within this formal system.
https://www.liarparadox.org/Haskell_Curry_45.pdf

[Skepdick]

It is stipulated to be true.
Haskell Curry understands that the elementary theorems (AKA axioms) of
a formal system are true within this formal system.
https://www.liarparadox.org/Haskell_Curry_45.pdf

[redacted}. Truth and provability are different properties!

An axiom has the property of being true.
An axiom doesn't have the property of being provable.


[Edited by iMod}

[PeteOlcott]

It is stipulated to be true.
Haskell Curry understands that the elementary theorems (AKA axioms) of
a formal system are true within this formal system.
https://www.liarparadox.org/Haskell_Curry_45.pdf

Idiot. Truth and provability are different properties!

An axiom has the property of being true.
An axiom doesn't have the property of being provable.

An axiom is proven true on the basis of it being an axiom.
Mendelson represents this as ⊢𝒞 provable on the basis of an empty set of premises.
https://books.google.com/books?id=ZO1p4 ... &q&f=false