G asserts its own unprovability in F

[PeteOlcott]

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.

It's the same thing. The completion of F will result in a contradiction in F

We could say that a baker lacks sufficient baking skill if they cannot
bake an angel food cake using only house bricks for ingredients.

Saying that a formal system is "incomplete" on the basis that it cannot
prove a contradictory expression is the same sort of thing. Formal
systems are not supposed to prove contradictory expressions.

[Skepdick]

We could say that a baker lacks sufficient baking skill if they cannot
bake an angel food cake using only house bricks for ingredients.

Saying that a formal system is "incomplete" on the basis that it cannot
prove a contradictory expression is the same sort of thing. Formal
systems are not supposed to prove contradictory expressions.

[redacted} That's literally the implication of Godel's theorems.

A system can be consistent AND incomplete; or complete AND inconsistent.


[Edited by imod}

[PeteOlcott]

We could say that a baker lacks sufficient baking skill if they cannot
bake an angel food cake using only house bricks for ingredients.

Saying that a formal system is "incomplete" on the basis that it cannot
prove a contradictory expression is the same sort of thing. Formal
systems are not supposed to prove contradictory expressions.

[redacted}. That's literally the implication of Godel's theorems.

A system can be consistent AND incomplete; or complete AND inconsistent.

The site crashed.
It seems that you are failing to comprehend that a contradictory expression is
an erroneous expression. It is like you believe that the liar paradox is true.

[Skepdick]

The site crashed.
It seems that you are failing to comprehend that a contradictory expression is
an erroneous expression. It is like you believe that the liar paradox is true.

You are faiing to comprehend that the completeness property doesn't care which property causes contradiction.

A system is complete if every formula posessing that property is derrivable within the system.

The property of interest is the non-provability of one or more formulas in F. Can you derrive all non-provable formulas within the system? No, you can't. So the system is not complete.

[PeteOlcott]

The site crashed.
It seems that you are failing to comprehend that a contradictory expression is
an erroneous expression. It is like you believe that the liar paradox is true.

You are faiing to comprehend that the completeness property doesn't care which property causes contradiction.

That is its mistake. It is the same thing as saying that a baker
is not a proper baker unless they can bake an angel food cake
using only house bricks for ingredients.

If they cannot do this then their baking skills are deemed "incomplete".

[Skepdick]

That is its mistake. It is the same thing as saying that a baker
is not a proper baker unless they can bake an angel food cake
using only house bricks for ingredients.

If they cannot do this then their baking skills are deemed "incomplete".

It's not a mistake.

Can you derrive all forlmulas posessing the "primeness" property?
Can you derrive all forlmulas posessing the "oddness" property?
Can you derrive all forlmulas posessing the "less than 75 characters long" property?

There are infinite number of properties of interest that lead to non-contradictory results.

The system may or may not be complete with respect to those properties. The contradiction is a corner case of the "unprovability" property.

Every system has a corner case. So what?

[Flannel Jesus]

"self contradictory" isn't the right word to describe G.

G asserts its own unprovability in F

The reason why G cannot be proved in F is that this proof requires a
sequence of inference steps in F that proves there is no such set of
inference steps in F.

Yes, I understand. I don't call that "self contradictory".

It's a statement about the relationship between g and f. It's not saying g is false - THAT would be self contradictory.

If I said "iambiguous doesn't like me", would that statement be me hating myself?

[PeteOlcott]

"self contradictory" isn't the right word to describe G.

G asserts its own unprovability in F

The reason why G cannot be proved in F is that this proof requires a
sequence of inference steps in F that proves there is no such set of
inference steps in F.

Yes, I understand. I don't call that "self contradictory".

It's a statement about the relationship between g and f. It's not saying g is false - THAT would be self contradictory.

If I said "iambiguous doesn't like me", would that statement be me hating myself?

When G asserts its own unprovability in F the 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.

This means that G is not provable in F because there is something wrong with G not
that there is something wrong with F.

[Skepdick]

When G asserts its own unprovability in F the 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.

You are working over-time to self-sabotage yourself.

You are the one asserting that G is asserting its unprovability.
You are the one laying down the requirements and requiring a proof that G is unprovable.

And finally - you are the one who throws his toys out the cot because the proof you demand (a proof of unprovability) would result in a contradiction.

It's like you don't know what [redacted} you want.

Consistency OR completeness. Choose one and accept the consequences!


[Edited by iMod}

[PeteOlcott]

When G asserts its own unprovability in F the 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.

You are the one asserting that G is asserting its unprovability.

Gödel sums up his own G as:
"...a proposition which asserts its own unprovability. 15" (Gödel 1931:39-41)

Gödel, Kurt 1931.
On Formally Undecidable Propositions of Principia Mathematica And Related Systems

[Skepdick]

When G asserts its own unprovability in F the 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.

You are the one asserting that G is asserting its unprovability.

Gödel sums up his own G as:
"...a proposition which asserts its own unprovability. 15" (Gödel 1931:39-41)

Gödel, Kurt 1931.
On Formally Undecidable Propositions of Principia Mathematica And Related Systems

Sure. And you are the one who demands that a proposition which asserts its own unprovability be proven.

I am unprovable. Prove me!

What is it that you expect to happen?

[PeteOlcott]

You are the one asserting that G is asserting its unprovability.

Gödel sums up his own G as:
"...a proposition which asserts its own unprovability. 15" (Gödel 1931:39-41)

Gödel, Kurt 1931.
On Formally Undecidable Propositions of Principia Mathematica And Related Systems

Sure. And you are the one who demands that a proposition which asserts its own unprovability be proven.

I am the one that demands that the reason why G is unprovable in F be correctly
acknowledged and not hidden behind arithmetization and diagonalization.

When the reason why G cannot be proved in F is that G is erroneous in F then
the reason is not that F is incomplete.

[Skepdick]

Gödel sums up his own G as:
"...a proposition which asserts its own unprovability. 15" (Gödel 1931:39-41)

Gödel, Kurt 1931.
On Formally Undecidable Propositions of Principia Mathematica And Related Systems

Sure. And you are the one who demands that a proposition which asserts its own unprovability be proven.

I am the one that demands that the reason why G is unprovable in F be correctly
acknowledged and not hidden behind arithmetization and diagonalization.

When the reason why G cannot be proved in F is that G is erroneous in F then
the reason is not that F is incomplete.

The reason that G is unprovable is because G is unprovable.
That's why we say that F is incomplete.

And if you were to provide a completion of F (such that G becomes provable) you would contradict the unprovability of G.

What's confusing you about this?

[PeteOlcott]

Sure. And you are the one who demands that a proposition which asserts its own unprovability be proven.

I am the one that demands that the reason why G is unprovable in F be correctly
acknowledged and not hidden behind arithmetization and diagonalization.

When the reason why G cannot be proved in F is that G is erroneous in F then
the reason is not that F is incomplete.

The reason that G is unprovable is because F is incomplete.

And if you were to provide a completion of F (such that G becomes provable) you would contradict the system.

A formal system is not supposed to prove contradictory expressions.
The fact that the proof of G is contradictory in F excludes G from membership in F.

[Skepdick]

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?