Discussion:
Refuting Incompleteness and Undefinability Version(9) (relevant to linguistics)
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peteolcott
2018-11-02 15:06:02 UTC
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∀LP (LP ↔ ~⊢LP)
If LP was a theorem this contradicts its assertion: ~⊢LP
If ~LP was a theorem this contradicts its assertion: ⊢LP
∴ ~Logic_Sentence(LP)

Formalization followed by Simple English intuition
(1) ∀x ~True(x) ↔ ~Theorem(x) // x cannot be proven True from known facts.
(2) ∀x True(x) ↔ Theorem(x) // x can be proven True from known facts.
(3) ∀x False(x) ↔ Theorem(~x) // x can be proven False from known facts.

∀x (Logic_Sentence(x) ↔ (True(x) ∨ False(x)) ↔ (⊢x ∨ ⊢~x))

Axiom An axiom is a proposition regarded as self-evidently true without proof. (Weisstein 2018).
An Axiom from math is comparable to an established fact in English. To anchor the notion of Math(Axiom) and English(Fact) we can think of both of these as expressions of language that have been defined to have the semantic property of Boolean True.

Rules-of-Inference show how to correctly transform some expressions of language into other expressions of language.

Theorems are expressions of language proven entirely on the basis of Axioms, and Rules-of-Inference. (Mendelson 2015: 28). When the above definition of Axiom is used as the basis Theorems can be construed as the conclusions of sound deductive inference.

Copyright 2018 by Pete Olcott

Mendelson, Elliott 2015 Introduction to Mathematical logic Sixth edition
1.4 An Axiom System for the Propositional Calculus page 28

Weisstein, Eric W. 2018 “Axiom.” From MathWorld–A Wolfram Web Resource. http://mathworld.wolfram.com/Axiom.html

The Notion of Truth in Natural and Formal Languages
https://www.researchgate.net/publication/323866366_The_Notion_of_Truth_in_Natural_and_Formal_Languages

http://liarparadox.org/
Peter Percival
2018-11-02 17:26:07 UTC
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Post by peteolcott
∀LP (LP ↔ ~⊢LP)
If LP was a theorem this contradicts its assertion: ~⊢LP
If ~LP was a theorem this contradicts its assertion: ⊢LP
∴ ~Logic_Sentence(LP)
Formalization followed by Simple English intuition
(1) ∀x ~True(x)  ↔  ~Theorem(x)   // x cannot be proven True from known
facts.
(2) ∀x True(x)    ↔  Theorem(x)     // x can be proven True from known
facts.
Do you intend this bracketing: ∀x(~True(x) ↔ ~Theorem(x)) and
∀x(True(x) ↔ Theorem(x))? Are you aware that (~True(x) ↔
~Theorem(x)) and (True(x) ↔ Theorem(x)) are materially equivalent?
Post by peteolcott
(3) ∀x False(x)  ↔  Theorem(~x)   // x can be proven False from known
facts.
∀x (Logic_Sentence(x) ↔ (True(x) ∨ False(x)) ↔ (⊢x ∨ ⊢~x))
Axiom An axiom is a proposition regarded as self-evidently true without
proof. (Weisstein 2018).
An Axiom from math is comparable to an established fact in English. To
anchor the notion of Math(Axiom) and English(Fact) we can think of both
of these as expressions of language that have been defined to have the
semantic property of Boolean True.
Rules-of-Inference show how to correctly transform some expressions of
language into other expressions of language.
Theorems are expressions of language proven entirely on the basis of
Axioms, and Rules-of-Inference. (Mendelson 2015: 28). When the above
definition of Axiom is used as the basis Theorems can be construed as
the conclusions of  sound deductive inference.
Copyright 2018 by Pete Olcott
Mendelson, Elliott 2015 Introduction to Mathematical logic Sixth edition
1.4 An Axiom System for the Propositional Calculus page 28
Weisstein, Eric W. 2018 “Axiom.” From MathWorld–A Wolfram Web Resource.
http://mathworld.wolfram.com/Axiom.html
Have you considered the possibility that Mendelson and Weisstein don't
mean the same thing by 'axiom'?
Post by peteolcott
The Notion of Truth in Natural and Formal Languages
https://www.researchgate.net/publication/323866366_The_Notion_of_Truth_in_Natural_and_Formal_Languages
http://liarparadox.org/
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