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Simply defining Gödel Incompleteness and Tarski Undefinability away V12
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olcott
Jun 26, 2020, 5:15:48 PM6/26/20
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<OVERVIEW-OVERVIEW>
If we simply assume this is the correct model of a universal truth
predicate because it implements the sound deductive inference model:
∀F ∈ Formal_Systems ∀𝒞 ∈ WFF(F) (((F⊢𝒞)) ↔ True(F, 𝒞))
Then:
(a) True(F, 𝒞) is always definable refuting Tarski Undefinability.
(b) True(F, 𝒞) can never diverge from Provable(F, 𝒞) refuting Gödel
Incompleteness.
As Wittgensteien and Curry both agree True(X) within a formal F system
can only be correctly construed as relative to the formal system thus on
this basis alone at least Tarski erred:
We shall show that the sentence x is actually undecidable
and at the same time true. ...
The formulas (8) and (9) together express the fact that x
is an undecidable sentence; moreover from (7) it follows
that x is a true sentence. ...
the proof of the sentence x given in the metatheory can
automatically be carried over into the theory itself: the
sentence x which is undecidable in the original theory
becomes a decidable sentence in the enriched theory. '
(Tarski, 1983 275-276)
http://www.liarparadox.org/Tarski_Proof_275_276.pdf
</OVERVIEW-OVERVIEW>
<OVERVIEW>
According to both Wittgenstein and Curry a WFF 𝒞 is only true in a
formal system F relative to that formal system F.
According to Wittgenstein:
'True in Russell's system' means, as was said: proved
in Russell's system; (Wittgenstein 1983,118-119)
We actually have to add that True(F,X) does not include every proof, it
only actually includes proofs having entirely true premises.
When we hypothesize axioms as a proxy for true premises then every proof
to theorem consequences forms an isomorphism to deduction from a sound
argument to a true conclusion:
∀F ∈ Formal_Systems ∀𝒞 ∈ WFF(F) (((F⊢𝒞)) ↔ True(F, 𝒞))
When we hypothesize that the sound deductive inference model provides
the architectural basis for a universal truth predicate then True(F, X)
is always definable refuting Tarski Undefinability and True(F, X) cannot
possibly diverge from Provable(F, X) refuting Gödel Incompleteness.
</OVERVIEW>
=====================================================================
(a) True premises of a sound deductive argument ≅ Axioms of a formal system.
(b) A valid argument of deduction ≅ formal proof of symbolic logic.
(c) The true conclusion of sound decduction ≅ theorem consequence of a
formal proof.
If we understand that true in a formal system is relative to that formal
system and we hypothesize that true in a formal system is always a
theorem of that formal system then true cannot possibly diverge from
provable within that same formal system and Gödel incompleteness does
not really exist.
Just like in sound deduction we cannot know that a conclusion is true
unless the argument is valid and all of the premises are true we cannot
know that a WFF C is a theorem of a formal system unless it is proven in
that same formal system: F ⊢ 𝒞.
According to Wittgenstein:
'True in Russell's system' means, as was said: proved
in Russell's system; and 'false in Russell's system'
means: the opposite has been proved in Russell's system.
(Wittgenstein 1983,118-119)
Formalized by Olcott as:
∀F ∈ Formal_Systems ∀𝒞 ∈ WFF(F) (((F⊢𝒞)) ↔ True(F, 𝒞))
∀F ∈ Formal_Systems ∀𝒞 ∈ WFF(F) (((F⊬𝒞)) ↔ ¬True(F, 𝒞))
∀F ∈ Formal_Systems ∀𝒞 ∈ WFF(F) (((F⊢¬𝒞)) ↔ False(F, 𝒞))
The terminology which has just been used implies that
the elementary statements are not such that their truth
and falsity are known to us without reference to {T}.
(Curry 1977:45)
Thus truth the Truth of a WFF of a formal system is relative to that
formal system. Although it may be true that "cats are animals" this is
not provable in first order logic, thus not true in first order logic.
(G) F ⊢ GF ↔ ¬ProvF(⌈GF⌉).
https://plato.stanford.edu/entries/goedel-incompleteness/#FirIncTheCom
(G) F ⊢ GF ↔ ¬ProvF(GF). // Simplified (remove arithmetization)
(G) F ⊢ GF ↔ (F ⊬ G). // Adapt syntax
// Adapt syntax & quantify
∃F ∈ Formal_Systems ∃G ∈ WFF(F) (G ↔ (F ⊬ G))
There exists a formal system F and a WFF G of formal system F that is
logically equivalent to a formal proof of its own unprovability.
Gödel proved that no such G exists and on this basis he concluded that
all sufficiently expressive formal systems are incomplete.
He concluded that these formal systems are incomplete not just because
there are some WFF that they cannot prove, (first order logic cannot
proves that "cats are animals") but because these unprovable WFF of F
are true and unprovable.
It is true that G is not provable in F, yet because G is not provable in
F it is not true that G is not provable in F. As both Wittgenstein and
Curry pointed out the truth of an expression in a formal system is
relative to that formal system.
∀F ∈ Formal_Systems ∀𝒞 ∈ WFF(F) (((F⊢𝒞)) ↔ True(F, 𝒞))
Just like in sound deduction we cannot know that a conclusion is true
unless the argument is valid and all of the premises are true we cannot
know that a WFF C is a theorem of a formal system unless it is proven in
that same formal system: F ⊢ 𝒞.
As Tarski pointed out:
We shall show that the sentence x is actually undecidable
and at the same time true. ...
The formulas (8) and (9) together express the fact that x
is an undecidable sentence; moreover from (7) it follows
that x is a true sentence. ...
the proof of the sentence x given in the metatheory can
automatically be carried over into the theory itself: the
sentence x which is undecidable in the original theory
becomes a decidable sentence in the enriched theory. '
(Tarski, 1983 275-276)
He was assuming that
the proof of the sentence x given in the metatheory can
automatically be carried over into the theory itself.
It can't. The proof in the metatheory is a proof in different formal
system than a proof in the theory.
It sums up like this. Because there exists no formal proof in any formal
system that a WFF is unprovable in that formal system and
The terminology which has just been used implies that
the elementary statements are not such that their truth
and falsity are known to us without reference to {T}.
(Curry 1977:45)
The truth of a WFF is always relative to that formal system therefore
Gödel's famous true and unprovable does not really occur and thus
incompleteness is not proved.
Wittgenstein, Ludwig 1983. Remarks on the Foundations of Mathematics
(Appendix III), 118-119. Cambridge, Massachusetts and London, England:
The MIT Press (quoted in full below).
http://www.liarparadox.org/Wittgenstein.pdf
Curry, Haskell 1977. Foundations of Mathematical Logic. New York: Dover
Publications, 45
http://www.liarparadox.org/Haskell_Curry_45.pdf
Tarski, Alfred 1983. “The concept of truth in formalized languages” in
Logic Semantics, Metamathematics. Indianapolis: Hacket Publishing
Company, 275-276. http://www.liarparadox.org/Tarski_Proof_275_276.pdf
--
Copyright 2020 Pete Olcott
If we simply assume this is the correct model of a universal truth
predicate because it implements the sound deductive inference model:
∀F ∈ Formal_Systems ∀𝒞 ∈ WFF(F) (((F⊢𝒞)) ↔ True(F, 𝒞))
Then:
(a) True(F, 𝒞) is always definable refuting Tarski Undefinability.
(b) True(F, 𝒞) can never diverge from Provable(F, 𝒞) refuting Gödel
Incompleteness.
As Wittgensteien and Curry both agree True(X) within a formal F system
can only be correctly construed as relative to the formal system thus on
this basis alone at least Tarski erred:
We shall show that the sentence x is actually undecidable
and at the same time true. ...
The formulas (8) and (9) together express the fact that x
is an undecidable sentence; moreover from (7) it follows
that x is a true sentence. ...
the proof of the sentence x given in the metatheory can
automatically be carried over into the theory itself: the
sentence x which is undecidable in the original theory
becomes a decidable sentence in the enriched theory. '
(Tarski, 1983 275-276)
http://www.liarparadox.org/Tarski_Proof_275_276.pdf
</OVERVIEW-OVERVIEW>
<OVERVIEW>
According to both Wittgenstein and Curry a WFF 𝒞 is only true in a
formal system F relative to that formal system F.
According to Wittgenstein:
'True in Russell's system' means, as was said: proved
in Russell's system; (Wittgenstein 1983,118-119)
We actually have to add that True(F,X) does not include every proof, it
only actually includes proofs having entirely true premises.
When we hypothesize axioms as a proxy for true premises then every proof
to theorem consequences forms an isomorphism to deduction from a sound
argument to a true conclusion:
∀F ∈ Formal_Systems ∀𝒞 ∈ WFF(F) (((F⊢𝒞)) ↔ True(F, 𝒞))
When we hypothesize that the sound deductive inference model provides
the architectural basis for a universal truth predicate then True(F, X)
is always definable refuting Tarski Undefinability and True(F, X) cannot
possibly diverge from Provable(F, X) refuting Gödel Incompleteness.
</OVERVIEW>
=====================================================================
(a) True premises of a sound deductive argument ≅ Axioms of a formal system.
(b) A valid argument of deduction ≅ formal proof of symbolic logic.
(c) The true conclusion of sound decduction ≅ theorem consequence of a
formal proof.
If we understand that true in a formal system is relative to that formal
system and we hypothesize that true in a formal system is always a
theorem of that formal system then true cannot possibly diverge from
provable within that same formal system and Gödel incompleteness does
not really exist.
Just like in sound deduction we cannot know that a conclusion is true
unless the argument is valid and all of the premises are true we cannot
know that a WFF C is a theorem of a formal system unless it is proven in
that same formal system: F ⊢ 𝒞.
According to Wittgenstein:
'True in Russell's system' means, as was said: proved
in Russell's system; and 'false in Russell's system'
means: the opposite has been proved in Russell's system.
(Wittgenstein 1983,118-119)
Formalized by Olcott as:
∀F ∈ Formal_Systems ∀𝒞 ∈ WFF(F) (((F⊢𝒞)) ↔ True(F, 𝒞))
∀F ∈ Formal_Systems ∀𝒞 ∈ WFF(F) (((F⊬𝒞)) ↔ ¬True(F, 𝒞))
∀F ∈ Formal_Systems ∀𝒞 ∈ WFF(F) (((F⊢¬𝒞)) ↔ False(F, 𝒞))
The terminology which has just been used implies that
the elementary statements are not such that their truth
and falsity are known to us without reference to {T}.
(Curry 1977:45)
Thus truth the Truth of a WFF of a formal system is relative to that
formal system. Although it may be true that "cats are animals" this is
not provable in first order logic, thus not true in first order logic.
(G) F ⊢ GF ↔ ¬ProvF(⌈GF⌉).
https://plato.stanford.edu/entries/goedel-incompleteness/#FirIncTheCom
(G) F ⊢ GF ↔ ¬ProvF(GF). // Simplified (remove arithmetization)
(G) F ⊢ GF ↔ (F ⊬ G). // Adapt syntax
// Adapt syntax & quantify
∃F ∈ Formal_Systems ∃G ∈ WFF(F) (G ↔ (F ⊬ G))
There exists a formal system F and a WFF G of formal system F that is
logically equivalent to a formal proof of its own unprovability.
Gödel proved that no such G exists and on this basis he concluded that
all sufficiently expressive formal systems are incomplete.
He concluded that these formal systems are incomplete not just because
there are some WFF that they cannot prove, (first order logic cannot
proves that "cats are animals") but because these unprovable WFF of F
are true and unprovable.
It is true that G is not provable in F, yet because G is not provable in
F it is not true that G is not provable in F. As both Wittgenstein and
Curry pointed out the truth of an expression in a formal system is
relative to that formal system.
∀F ∈ Formal_Systems ∀𝒞 ∈ WFF(F) (((F⊢𝒞)) ↔ True(F, 𝒞))
Just like in sound deduction we cannot know that a conclusion is true
unless the argument is valid and all of the premises are true we cannot
know that a WFF C is a theorem of a formal system unless it is proven in
that same formal system: F ⊢ 𝒞.
As Tarski pointed out:
We shall show that the sentence x is actually undecidable
and at the same time true. ...
The formulas (8) and (9) together express the fact that x
is an undecidable sentence; moreover from (7) it follows
that x is a true sentence. ...
the proof of the sentence x given in the metatheory can
automatically be carried over into the theory itself: the
sentence x which is undecidable in the original theory
becomes a decidable sentence in the enriched theory. '
(Tarski, 1983 275-276)
He was assuming that
the proof of the sentence x given in the metatheory can
automatically be carried over into the theory itself.
It can't. The proof in the metatheory is a proof in different formal
system than a proof in the theory.
It sums up like this. Because there exists no formal proof in any formal
system that a WFF is unprovable in that formal system and
The terminology which has just been used implies that
the elementary statements are not such that their truth
and falsity are known to us without reference to {T}.
(Curry 1977:45)
The truth of a WFF is always relative to that formal system therefore
Gödel's famous true and unprovable does not really occur and thus
incompleteness is not proved.
Wittgenstein, Ludwig 1983. Remarks on the Foundations of Mathematics
(Appendix III), 118-119. Cambridge, Massachusetts and London, England:
The MIT Press (quoted in full below).
http://www.liarparadox.org/Wittgenstein.pdf
Curry, Haskell 1977. Foundations of Mathematical Logic. New York: Dover
Publications, 45
http://www.liarparadox.org/Haskell_Curry_45.pdf
Tarski, Alfred 1983. “The concept of truth in formalized languages” in
Logic Semantics, Metamathematics. Indianapolis: Hacket Publishing
Company, 275-276. http://www.liarparadox.org/Tarski_Proof_275_276.pdf
--
Copyright 2020 Pete Olcott
olcott
Jun 26, 2020, 7:29:21 PM6/26/20
to
On 6/26/2020 4:15 PM, olcott wrote:
> <OVERVIEW-OVERVIEW>
> If we simply assume this is the correct model of a universal truth
> predicate because it implements the sound deductive inference model:
>
> ∀F ∈ Formal_Systems ∀𝒞 ∈ WFF(F) (((F⊢𝒞)) ↔ True(F, 𝒞))
𝒞 := (G ↔ (F ⊬ G))
> <OVERVIEW-OVERVIEW>
> If we simply assume this is the correct model of a universal truth
> predicate because it implements the sound deductive inference model:
>
> ∀F ∈ Formal_Systems ∀𝒞 ∈ WFF(F) (((F⊢𝒞)) ↔ True(F, 𝒞))
∃F ∈ Formal_Systems ∃𝒞 ∈ WFF(F) (True(F, 𝒞)) becomes
∃F ∈ Formal_Systems ∃G ∈ WFF(F) (True(F, (G ↔ (F ⊬ G))))
It is true in F that there exists a WFF G of formal system F that is
unprovable in F.
All of the proofs of incompleteness utterly ignored the key subtle
nuance that a WFF must be both true and unprovable in the same formal
system for the formal system to actually be incomplete.
Within the sound deductive inference model this is impossible because
True(F, X) is defined as Theorem(F, X).
So I went around and around and came back to almost exactly where I have
been for a few years.
The very slight distinction might be that I did not quite so clearly
state that within the hypothesis that True(F, X) is exactly Theorem(F,X)
both Gödel Incompleteness and Tarski Undefinability are defined away.
olcott
Jun 26, 2020, 8:58:07 PM6/26/20
to
According to Wittgenstein:
'True in Russell's system' means, as was said: proved in Russell's
system; and 'false in Russell's system' means: the opposite has been
proved in Russell's system. (Wittgenstein 1983,118-119)
Formalized by Olcott as:
'True in Russell's system' means, as was said: proved in Russell's
system; and 'false in Russell's system' means: the opposite has been
proved in Russell's system. (Wittgenstein 1983,118-119)
Formalized by Olcott as:
∀F ∈ Formal_Systems ∀𝒞 ∈ WFF(F) (((F⊢𝒞)) ↔ True(F, 𝒞))
∀F ∈ Formal_Systems ∀𝒞 ∈ WFF(F) (((F⊬𝒞)) ↔ ¬True(F, 𝒞))
∀F ∈ Formal_Systems ∀𝒞 ∈ WFF(F) (((F⊢¬𝒞)) ↔ False(F, 𝒞))
∀F ∈ Formal_Systems ∀𝒞 ∈ WFF(F) (((F⊬𝒞)) ↔ ¬True(F, 𝒞))
We had to add that the proofs referred to by Wittgenstein must be to
theorem consequences thus requiring the axioms of formal proofs to act
as a proxy for the true premises of sound deduction.
If we simply construe a formal proof to theorem consequences as
isomorphic to deduction from a sound argument to a true conclusion. This
requires the axioms of formal systems to be construed as the true
premises of sound deduction.
[Simplified Gödel
Sentence](https://plato.stanford.edu/entries/goedel-incompleteness/#FirIncTheCom)
<br>
$(G) \ F ⊢ G{_F} ↔ ¬Prov_F(⌈G_F⌉).$ **// Original <br>**
$(G) \ F ⊢ G{_F} ↔ ¬Prov_F(G_F).\ \ \ $ **// Remove arithmetization<br>**
// Adapt syntax and quantify:
∃F ∈ Formal_Systems ∃G ∈ WFF(F) (G ↔ (F ⊬ G))
**The definition of incompleteness:**<br> A theory T is incomplete if
and only if there is some sentence φ such that (T ⊬ φ) and (T ⊬ ¬φ).
When we adopt sound deductive inference as our architectural basis of
the formalized notion of truth then:
(T ⊬ φ) means ¬True(T, φ) and (T ⊬ ¬φ) means ¬False(T, φ) and (¬True(T,
φ) and ¬False(T, φ)) means ¬Boolean(T, φ)).
Within the architectural assumption ¬Boolean(T, φ) would not be
construed as Incomplete(T). Instead it would be construed as
Syntactically_WFF(T, φ) is ¬Semantically_WFF(T, φ).
Copyright 2020 PL Olcott
Wittgenstein, Ludwig 1983. Remarks on the Foundations of Mathematics
(Appendix III), 118-119. Cambridge, Massachusetts and London, England:
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