Re: ZFC solution to incorrect questions: reject them --Gödel--

On 3/12/2024 11:36 PM, Richard Damon wrote:

On 3/12/24 8:14 PM, olcott wrote:

On 3/12/2024 9:51 PM, Richard Damon wrote:

On 3/12/24 4:14 PM, olcott wrote:

On 3/12/2024 6:00 PM, Richard Damon wrote:

On 3/12/24 2:44 PM, olcott wrote:

On 3/12/2024 4:31 PM, Richard Damon wrote:

On 3/12/24 1:38 PM, olcott wrote:

On 3/12/2024 3:31 PM, immibis wrote:

On 12/03/24 20:02, olcott wrote:

On 3/12/2024 1:31 PM, immibis wrote:

On 12/03/24 19:12, olcott wrote:

∀ H ∈ Turing_Machine_Deciders
∃ TMD ∈ Turing_Machine_Descriptions  |
Predicted_Behavior(H, TMD) != Actual_Behavior(TMD)
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There is some input TMD to every H such that
Predicted_Behavior(H, TMD) != Actual_Behavior(TMD)

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And it can be a different TMD to each H.
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When we disallow decider/input pairs that are incorrect
questions where both YES and NO are the wrong answer

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Once we understand that either YES or NO is the right answer, the whole rebuttal is tossed out as invalid and incorrect.
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Ĥ.q0 ⟨Ĥ⟩ ⊢* Ĥ.Hq0 ⟨Ĥ⟩ ⟨Ĥ⟩ ⊢* Ĥ.Hqy ∞ // Ĥ applied to ⟨Ĥ⟩ halts
Ĥ.q0 ⟨Ĥ⟩ ⊢* Ĥ.Hq0 ⟨Ĥ⟩ ⟨Ĥ⟩ ⊢* Ĥ.Hqn   // Ĥ applied to ⟨Ĥ⟩ does not halt
BOTH YES AND NO ARE THE WRONG ANSWER FOR EVERY Ĥ.H ⟨Ĥ⟩ ⟨Ĥ⟩
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Once we understand that either YES or NO is the right answer, the whole rebuttal is tossed out as invalid and incorrect.
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Does the barber that shaves everyone that does not shave
themselves shave himself? is rejected as an incorrect question.

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The barber does not exist.

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Russell's paradox did not allow this answer within Naive set theory.

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Naive set theory says that for every predicate P, the set {x | P(x)} exists. This axiom was a mistake. This axiom is not in ZFC.
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In Turing machines, for every non-empty finite set of alphabet symbols Γ, every b∈Γ, every Σ⊆Γ, every non-empty finite set of states Q, every q0∈Q, every F⊆Q, and every δ:(Q∖F)×Γ↛Q×Γ×{L,R}, ⟨Q,Γ,b,Σ,δ,q0,F⟩ is a Turing machine. Do you think this is a mistake? Would you remove this axiom from your version of Turing machines?
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(Following the definition used on Wikipedia: https://en.wikipedia.org/wiki/Turing_machine#Formal_definition)
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The following is true statement:
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∀ Barber ∈ People. ¬(∀ Person ∈ People. Shaves(Barber, Person) ⇔ ¬Shaves(Person, Person))
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The following is a true statement:
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¬∃ Barber ∈ People. (∀ Person ∈ People. Shaves(Barber, Person) ⇔ ¬Shaves(Person, Person))
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That might be correct I did not check it over and over
again and again to make sure.
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The same reasoning seems to rebut Gödel Incompleteness:
...We are therefore confronted with a proposition which
asserts its own unprovability. 15 ... (Gödel 1931:43-44)
¬∃G ∈ F | G := ~(F ⊢ G)
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Any G in F that asserts its own unprovability in F is
asserting that there is no sequence of inference steps
in F that prove that they themselves do not exist in F.

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The barber does not exist and the proposition does not exist.
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When we do this exact same thing that ZFC did for self-referential
sets then Gödel's self-referential expressions that assert their
own unprovability in F also cease to exist.
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And you end up with a very weak logic system that can't even have the full properties of the Natuarl Numbers.

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Natural numbers never really did have the property of provability.
This was something artificially contrived that never really belonged
to them.
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No, Godel showed (or maybe used a previous proof) that you can use the Mathematics of Natural Numbers to test if a proof is valid.
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You just don't understand it. It really is very related to how Turing Machines work, which can be converted to a mathematical model.
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There is a field that looks at the comparability of Computations to Logic, so they are all really quite related.

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This is refuted.
...We are therefore confronted with a proposition which
asserts its own unprovability. 15 ...(Gödel 1931:43-44)

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Right, seen in the meta-Theory from F.
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based on immblis Russell's Paradox reply
¬∃G ∈ F | G ↔ ~(F ⊢ G)  // is simply false

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Then a proof must exist in F that G is True, So G can't be false.
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Note, The statement G is NOT a statement about itself being provable, that is only a semantic revealed in the RIGHT meta-theory.
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*Not in my actual example where F has its own provability operator*

 CAN'T EXIST.
 Provability in uncomputable.

It this was true then it would be a little nuts for anyone
to expect that G can be proved in F or that F is incomplete
if it can't do this. It is like expecting your washing machine
to drive you to the store and then blaming it for not doing this.

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Any G in F that asserts its own unprovability in F is
asserting that there is no sequence of inference steps
in F that prove that they themselves do not exist in F.
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No FINITE sequence of inference steps.
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No one can prove that they themselves do not exist.
Thus G cannot possibly derive a sequence of inference
steps that prove that they themselves do not exist.

 It doesn't.
 

This requires that:
...We are therefore confronted with a proposition which
asserts its own unprovability. 15 ... (Gödel 1931:43-44)

It shows that no proof exists of its truth.
It also shows that it can't be false, as if it were, then it would be proven to be true, and you can't prove a false statement true in a consistant system.
 

There cannot possibly be any set of inference steps that prove
that they themselves do not exist, thus under existential
quantification we get false.

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Note, the key is that G doesn't assert that, G is a statement about math, that only when interpreted in a meta-theory sees that.
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I am not referring to that one.

 Then you aren't talking about Godel, who you mentioned just before.
 

I am talking about this.
...We are therefore confronted with a proposition which
asserts its own unprovability. 15 ... (Gödel 1931:43-44)

So, you LIE.
 

*That lie might possibly cost you your soul*

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G is actually a statment about the existance of a number that matches a complected and carefully constructed relationship, that is fully computable. The Existance or non-existance of such a number is a pure binary thing, either it WILL or it WON'T.
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The complicated relationship deals with a way to encode as a number ANY analyitic statement in F (since all statements are just strings, and strings can be encoded into a number), and a calculation to see if that statement actually is a proof starting with the enumerated truth makers of F, through the valid and allowed logical operatons to the statement of G. A number that satisfies this relation, encodes a proof of G, and any such proof, can always be encoded in to a number.
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No formal system can possibly have any sequence of inference
step that prove that they themselves do not exist because
this is self-contradictory, not because the system is incomplete.
In my G this is obvious.

 Right, You can't have, IN THE SYSTEM, a statement that something must be true and unprovable, as that is, itself, a contradiction.
 

Great finally some more mutual agreement.

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Whether or not Gödel's G is isomorphic to mine I have proved
this this is false by providing the counter-example of my G.

 No, you have proved that a direct statement in F, can't assert that a statement is True but Unprovable.
 

When true means provable from axioms then unprovable means
not true in the system. Both Wittgenstein and Curry agree on this.

But Godel G, doesn't directly assert it, but creates mathematics based on a meta-theory that establish it.
 

*This directly asserts it*
...We are therefore confronted with a proposition which
asserts its own unprovability. 15 ... (Gödel 1931:43-44)

That is the power of Mathematics.
 

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...14 Every epistemological antinomy can likewise be used for a similar undecidability proof...(Gödel 1931:43-44)

 Which you don't understand, and is refering to logic in Meta-F, which you don't handle at all.
 

That hardly anyone knows that epistemological antinomies
are not truth bearers does not make me wrong. The leading
author on truthmaker maximalism agrees that the Liar Paradox
does not express a proposition.

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G is the statement that no such number exist. So, if G is false, then such a number exists, and then in the Meta-Theory, we can decode that number into a proof in F that G must be true.
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If this is the case, then F must be inconsistant, and the proof starts with the presumption that we are dealing with a consistant logic system.
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Or it could be the epistemological antinomy of trying to find a
sequence of inference steps in F that prove that they themselves
do not exist.

 It shows that F defines an epistemological statement to be true, which is equivent to F being inconsistant.
 

...We are therefore confronted with a proposition which
asserts its own unprovability. 15 ... (Gödel 1931:43-44)
Any G in F that asserts its own unprovability in F is
asserting that there is no sequence of inference steps
in F that prove that they themselves do not exist in F.

As you point out, you can't make the statement in F, you need a level of indirection with a meta-system to make it work.
 

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But, if G is true, then there can be no such number, and thus from our knowledge in the mete-theory, we know that no proof CAN exist for that fact.
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Thus, the only possiblity for this statement, is to be True, but unprovable.
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Or it could be the epistemological antinomy of trying to find a
sequence of inference steps in F that prove that they themselves
do not exist.

 You are repeating yourself, showing you self to be just the CHILDISH pathetic ignorant hypocritical pathological lying idiot.
 

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Meta-F can see this and F is not incomplete if it cannot accomplish
an epistemological antinomy.
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The big part of the theory is showing that such a relationship can be made, and that shows that proof checkers are a computable function.
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Given a system, and an enumeration of its basic truths, we can check if a proof present actually proves the result it claims to.

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--
Copyright 2024 Olcott "Talent hits a target no one else can hit; Genius
hits a target no one else can see." Arthur Schopenhauer