Discussion:
Refuting Incompleteness and Undefinability version(11)
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peteolcott
2018-11-03 20:27:48 UTC
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The first incompleteness theorem states that in any consistent formal system F within
which a certain amount of arithmetic can be carried out, there are statements of the
language of F which can neither be proved nor disproved in F. (Raatikainen Fall 2018)

Formalization of the above English:
∀F ∈ Formal_Systems
(
∃G ∈ F (G ↔ ∃Γ ⊆ F ~(Γ ⊢ G))
)

Proved(G) means the satisfaction of: (Γ ⊢ G)
thus the satisfaction of ∃Γ ⊆ F ~(Γ ⊢ G)
If G was Provable in F this contradicts its assertion: G is not Provable in F

Disproved(G) means the satisfaction of: (Γ ⊢ ~G)
thus the satisfaction of ∀Γ ⊆ F (Γ ⊢ G)
If ~G was Provable in F this contradicts its assertion: G is Provable in F

Taking as a hypothesis that the following correctly formalize the intuitive notions of True and False:
(1) ∀x (True(x) ↔ ⊢x)
(2) ∀x (False(x) ↔ ⊢~x)
Along with this common knowledge: ∀x (Logic_Sentence(x) ↔ (True(x) ∨ False(x))

We derive a formal expression of logic that can reject semantically ill-formed WFF:
∀x (Logic_Sentence(x) ↔ (⊢x ∨ ⊢~x))

Since neither G nor ~G is satisfiable therefore neither True(G) nor False(G)
∴ Logic_Sentence(G)

Copyright 2018 Pete Olcott

Raatikainen, Panu, "Gödel's Incompleteness Theorems",
The Stanford Encyclopedia of Philosophy (Fall 2018 Edition), Edward N. Zalta (ed.),
URL = <https://plato.stanford.edu/archives/fall2018/entries/goedel-incompleteness/>.
Peter Percival
2018-11-03 20:42:23 UTC
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    The first incompleteness theorem states that in any consistent
formal system F within
    which a certain amount of arithmetic can be carried out, there are
statements of the
    language of F which can neither be proved nor disproved in F.
(Raatikainen Fall 2018)
∀F ∈ Formal_Systems
(
  ∃G ∈ F (G ↔ ∃Γ ⊆ F ~(Γ ⊢ G))
)
Proved(G) means the satisfaction of: (Γ ⊢ G)
thus the satisfaction of ∃Γ ⊆ F ~(Γ ⊢ G)
If G was Provable in F this contradicts its assertion: G is not Provable in F
Disproved(G) means the satisfaction of: (Γ ⊢ ~G)
thus the satisfaction of ∀Γ ⊆ F (Γ ⊢ G)
If ~G was Provable in F this contradicts its assertion:  G is Provable in F
Taking as a hypothesis that the following correctly formalize the
(1) ∀x (True(x)    ↔ ⊢x)
(2) ∀x (False(x)   ↔ ⊢~x)
Along with this common knowledge: ∀x (Logic_Sentence(x) ↔ (True(x) ∨ False(x))
∀x (Logic_Sentence(x) ↔  (⊢x ∨ ⊢~x))
What is a logic sentence? My guess (why should I have to guess?) is
that it is a wff with no free variables. Look at the start of
Mendelson's Chapter 2. Is A_1^1(a_1) a logic sentence? Neither
|-A_1^1(a_1) nor |-~A_1^1(a_1). The easy answer to this problem is for
you to attend to what a formal systems is. If I understood a remark of
yours in another post ("The English does not say what a formal system
is, so I too did not.") you seem to think not knowing is a virtue.
Since neither G nor ~G is satisfiable therefore neither True(G) nor False(G)
∴  Logic_Sentence(G)
Copyright 2018 Pete Olcott
Raatikainen, Panu, "Gödel's Incompleteness Theorems",
The Stanford Encyclopedia of Philosophy (Fall 2018 Edition), Edward N. Zalta (ed.),
URL =
<https://plato.stanford.edu/archives/fall2018/entries/goedel-incompleteness/>.
peteolcott
2018-11-03 20:55:09 UTC
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     The first incompleteness theorem states that in any consistent formal system F within
     which a certain amount of arithmetic can be carried out, there are statements of the
     language of F which can neither be proved nor disproved in F. (Raatikainen Fall 2018)
∀F ∈ Formal_Systems
(
   ∃G ∈ F (G ↔ ∃Γ ⊆ F ~(Γ ⊢ G))
)
Proved(G) means the satisfaction of: (Γ ⊢ G)
thus the satisfaction of ∃Γ ⊆ F ~(Γ ⊢ G)
If G was Provable in F this contradicts its assertion: G is not Provable in F
Disproved(G) means the satisfaction of: (Γ ⊢ ~G)
thus the satisfaction of ∀Γ ⊆ F (Γ ⊢ G)
If ~G was Provable in F this contradicts its assertion:  G is Provable in F
(1) ∀x (True(x)    ↔ ⊢x)
(2) ∀x (False(x)   ↔ ⊢~x)
Along with this common knowledge: ∀x (Logic_Sentence(x) ↔ (True(x) ∨ False(x))
∀x (Logic_Sentence(x) ↔  (⊢x ∨ ⊢~x))
What is a logic sentence?  My guess (why should I have to guess?) is that it is a wff with no free variables.  Look at the start of Mendelson's Chapter 2.  Is A_1^1(a_1) a logic sentence?
https://en.wikipedia.org/wiki/Sentence_(mathematical_logic)
In mathematical logic, a sentence ... must be true or false.

Closed WFF that are neither True nor False are semantically
ill-formed. Kurt and Alfred mistook semantically ill-formed WFF
as proof that logic is Incomplete and True() is Undefinable.

Copyright 2018 Pete Olcott
Peter Percival
2018-11-03 21:32:11 UTC
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Post by peteolcott
     The first incompleteness theorem states that in any consistent
formal system F within
     which a certain amount of arithmetic can be carried out, there
are statements of the
     language of F which can neither be proved nor disproved in F.
(Raatikainen Fall 2018)
∀F ∈ Formal_Systems
(
   ∃G ∈ F (G ↔ ∃Γ ⊆ F ~(Γ ⊢ G))
)
Proved(G) means the satisfaction of: (Γ ⊢ G)
thus the satisfaction of ∃Γ ⊆ F ~(Γ ⊢ G)
If G was Provable in F this contradicts its assertion: G is not Provable in F
Disproved(G) means the satisfaction of: (Γ ⊢ ~G)
thus the satisfaction of ∀Γ ⊆ F (Γ ⊢ G)
If ~G was Provable in F this contradicts its assertion:  G is Provable in F
Taking as a hypothesis that the following correctly formalize the
(1) ∀x (True(x)    ↔ ⊢x)
(2) ∀x (False(x)   ↔ ⊢~x)
Along with this common knowledge: ∀x (Logic_Sentence(x) ↔ (True(x) ∨ False(x))
∀x (Logic_Sentence(x) ↔  (⊢x ∨ ⊢~x))
What is a logic sentence?  My guess (why should I have to guess?) is
that it is a wff with no free variables.  Look at the start of
Mendelson's Chapter 2.  Is A_1^1(a_1) a logic sentence?
https://en.wikipedia.org/wiki/Sentence_(mathematical_logic)
In mathematical logic, a sentence ... must be true or false.
I'm not referring to true or false. I'm referring to _your_ ⊢x ∨ ⊢~x.
Is Mendelson's A_1^1(a_1) a logic sentence? Alternately, what do _you_
mean by a formal system?
Post by peteolcott
Closed WFF that are neither True nor False are semantically
ill-formed. Kurt and Alfred mistook semantically ill-formed WFF
as proof that logic is Incomplete and True() is Undefinable.
Copyright 2018 Pete Olcott
peteolcott
2018-11-03 21:56:26 UTC
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Post by peteolcott
(1) ∀x (True(x)    ↔ ⊢x)
(2) ∀x (False(x)   ↔ ⊢~x)
Along with this common knowledge: ∀x (Logic_Sentence(x) ↔ (True(x) ∨ False(x))
∀x (Logic_Sentence(x) ↔  (⊢x ∨ ⊢~x))
What is a logic sentence?  My guess (why should I have to guess?) is that it is a wff with no free variables.  Look at the start of Mendelson's Chapter 2.  Is A_1^1(a_1) a logic sentence?
https://en.wikipedia.org/wiki/Sentence_(mathematical_logic)
In mathematical logic, a sentence ... must be true or false.
I'm not referring to true or false.  I'm referring to _your_ ⊢x ∨ ⊢~x. Is Mendelson's A_1^1(a_1) a logic sentence?  Alternately, what do _you_ mean by a formal system?
I mean the exact same thing that Mendelson means in the first two pages of section 1.4.
I cannot find what you are referring to. I have both the 4th and 6th edition.
text line number on page number would narrow it down.
Peter Percival
2018-11-03 22:05:02 UTC
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Post by peteolcott
Post by peteolcott
Post by peteolcott
Taking as a hypothesis that the following correctly formalize the
(1) ∀x (True(x)    ↔ ⊢x)
(2) ∀x (False(x)   ↔ ⊢~x)
Along with this common knowledge: ∀x (Logic_Sentence(x) ↔ (True(x)
∨ False(x))
∀x (Logic_Sentence(x) ↔  (⊢x ∨ ⊢~x))
What is a logic sentence?  My guess (why should I have to guess?) is
that it is a wff with no free variables.  Look at the start of
Mendelson's Chapter 2.  Is A_1^1(a_1) a logic sentence?
https://en.wikipedia.org/wiki/Sentence_(mathematical_logic)
In mathematical logic, a sentence ... must be true or false.
I'm not referring to true or false.  I'm referring to _your_ ⊢x ∨ ⊢~x.
Is Mendelson's A_1^1(a_1) a logic sentence?  Alternately, what do
_you_ mean by a formal system?
I mean the exact same thing that Mendelson means in the first two pages of section 1.4.
I cannot find what you are referring to. I have both the 4th and 6th edition.
text line number on page number would narrow it down.
In the 5th edition section 1.4 is about propositional calculus. I see
no definition of logic sentence. I do see a definition of wf. A_1 is a
wf. So, according to you |-A_1 or |-~A_1. Please prove either A_1 or
~A_1. If you don't mean wf, what do you mean by 'logic sentence'?
Arnaud Fournet
2018-11-04 06:09:33 UTC
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Post by Peter Percival
Post by peteolcott
Post by peteolcott
Post by peteolcott
Taking as a hypothesis that the following correctly formalize the
(1) ∀x (True(x)    ↔ ⊢x)
(2) ∀x (False(x)   ↔ ⊢~x)
Along with this common knowledge: ∀x (Logic_Sentence(x) ↔ (True(x)
∨ False(x))
We derive a formal expression of logic that can reject semantically
∀x (Logic_Sentence(x) ↔  (⊢x ∨ ⊢~x))
What is a logic sentence?  My guess (why should I have to guess?) is
that it is a wff with no free variables.  Look at the start of
Mendelson's Chapter 2.  Is A_1^1(a_1) a logic sentence?
https://en.wikipedia.org/wiki/Sentence_(mathematical_logic)
In mathematical logic, a sentence ... must be true or false.
I'm not referring to true or false.  I'm referring to _your_ ⊢x ∨ ⊢~x.
Is Mendelson's A_1^1(a_1) a logic sentence?  Alternately, what do
_you_ mean by a formal system?
I mean the exact same thing that Mendelson means in the first two pages of section 1.4.
I cannot find what you are referring to. I have both the 4th and 6th edition.
text line number on page number would narrow it down.
In the 5th edition section 1.4 is about propositional calculus. I see
no definition of logic sentence. I do see a definition of wf. A_1 is a
wf. So, according to you |-A_1 or |-~A_1. Please prove either A_1 or
~A_1. If you don't mean wf, what do you mean by 'logic sentence'?
Please avoid propagating that garbage on sci.lang
Thanks.

Peter Percival
2018-11-03 20:52:40 UTC
Reply
Permalink
    The first incompleteness theorem states that in any consistent
formal system F within
    which a certain amount of arithmetic can be carried out, there are
statements of the
    language of F which can neither be proved nor disproved in F.
(Raatikainen Fall 2018)
∀F ∈ Formal_Systems
(
  ∃G ∈ F (G ↔ ∃Γ ⊆ F ~(Γ ⊢ G))
)
Proved(G) means the satisfaction of: (Γ ⊢ G)
thus the satisfaction of ∃Γ ⊆ F ~(Γ ⊢ G)
If G was Provable in F this contradicts its assertion: G is not Provable in F
Disproved(G) means the satisfaction of: (Γ ⊢ ~G)
thus the satisfaction of ∀Γ ⊆ F (Γ ⊢ G)
The only formulae that follow from all sets of premisses are the
validities. The Gödel sentence is not a validity. So your formula has
got nothing to with Gödel's work.
If ~G was Provable in F this contradicts its assertion:  G is Provable in F
Taking as a hypothesis that the following correctly formalize the
(1) ∀x (True(x)    ↔ ⊢x)
(2) ∀x (False(x)   ↔ ⊢~x)
And what if one doesn't take it as an hypothesis? For some calculi it
it the case that being true and being a theorem are co-extensive. But
that's something to be proved, it is not an hypothesis. For other
calculi, they are not co-extensive.

[An hypothesis, a hypothesis? An hotel, a hotel?]
Along with this common knowledge: ∀x (Logic_Sentence(x) ↔ (True(x) ∨ False(x))
∀x (Logic_Sentence(x) ↔  (⊢x ∨ ⊢~x))
Since neither G nor ~G is satisfiable therefore neither True(G) nor False(G)
∴  Logic_Sentence(G)
Copyright 2018 Pete Olcott
Raatikainen, Panu, "Gödel's Incompleteness Theorems",
The Stanford Encyclopedia of Philosophy (Fall 2018 Edition), Edward N. Zalta (ed.),
URL =
<https://plato.stanford.edu/archives/fall2018/entries/goedel-incompleteness/>.
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