Discussion:
Refuting Incompleteness and Undefinability --- Version (5)
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peteolcott
2018-10-26 17:03:12 UTC
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A Theorem is essentially nothing more than an expression of language that can be proven True entirely on the basis on known facts, thus the conclusion of sound deductive inference.

More formally Theorems are mathematical proofs based on axioms and rules-of-inference only without specifying any premises, thus forming sound deductive inference.

Axioms can be thought of as expressions of language that have been expressly 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.

Any expression of language not having a Boolean Property with a value of True or False is not a correct Declarative sentence / sentence of logic.

This universal Truth predicate rejects as malformed any expressions of language that would otherwise show Incompleteness or Undefinability: ∀x ∈ F True(F, x) ↔ Theorem(F, x)

For all expressions x in language L, x is True in L if and only if x is a Theorem of L

https://plato.stanford.edu/entries/goedel-incompleteness/#FirIncTheCom
(G) F ⊢ GF ↔ ¬ProvF(⌈GF⌉).

Simplified to this:
G ↔ G ∈ F ~Provable(F, G)

“This sentence is not provable”
G ≡ ~Provable(G) // To avoid clutter F is implied rather than stated

G asserts that G is not Provable. If G was a theorem (thus true) then it would contradict its assertion that G is not Provable.

~G asserts that G is Provable. If ~G was a theorem (thus true) then it would contradict its assertion that G is Provable.

∴ Since Neither G nor ~G is a theorem (true) G is rejected as an incorrect Boolean Proposition because it is neither True nor False.

Copyright 2018 by Pete Olcott
Arnaud Fournet
2018-10-26 19:51:22 UTC
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Post by peteolcott
A Theorem is essentially nothing more than an expression of language that can be proven True entirely on the basis on known facts, thus the conclusion of sound deductive inference.
More formally Theorems are mathematical proofs based on axioms and rules-of-inference only without specifying any premises, thus forming sound deductive inference.
Axioms can be thought of as expressions of language that have been expressly 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.
Any expression of language not having a Boolean Property with a value of True or False is not a correct Declarative sentence / sentence of logic.
This universal Truth predicate rejects as malformed any expressions of language that would otherwise show Incompleteness or Undefinability: ∀x ∈ F True(F, x) ↔ Theorem(F, x)
For all expressions x in language L, x is True in L if and only if x is a Theorem of L
https://plato.stanford.edu/entries/goedel-incompleteness/#FirIncTheCom
(G) F ⊢ GF ↔ ¬ProvF(⌈GF⌉).
G ↔ G ∈ F ~Provable(F, G)
“This sentence is not provable”
G ≡ ~Provable(G) // To avoid clutter F is implied rather than stated
G asserts that G is not Provable. If G was a theorem (thus true) then it would contradict its assertion that G is not Provable.
~G asserts that G is Provable. If ~G was a theorem (thus true) then it would contradict its assertion that G is Provable.
∴ Since Neither G nor ~G is a theorem (true) G is rejected as an incorrect Boolean Proposition because it is neither True nor False.
Copyright 2018 by Pete Olcott
Is this garbage a new thread on sci.lang ??
peteolcott
2018-10-27 05:00:53 UTC
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Post by peteolcott
A Theorem is essentially nothing more than
FUCK you.
The word HAS *A*DEFINITION*.
It's ALREADY IN the dictionary.
You DON'T get an OPINION.
Now that I double checked my most recent revisions, my
English definitions of these terms seem more succinct
and accurate than any others that I have ever seen.

I challenge anyone to show that any other English definitions
of these terms are in any way better than mine.

Mendelson had a great mathematical definition of Theorem.
It is this definition that I refer to in my universal Truth
predicate. It is his text that allowed me to finally translate
my thirty year long intuitions into conventional terms of the art.

My English definition of Theorem equally apply to both formal
and natural languages (Mendelson's does not):

A Theorem is essentially nothing more than an expression
of language that can be proven True on the basis on known
facts (axioms), thus the conclusion of sound deductive inference.

Since it is unconventional (although more accurate) to think
of an Axiom as a known truth, my English definition of Theorem
requires my definition of Axiom (see below) as its basis.

No one seemed to have any decent definition of Axiom,
everywhere I looked it was always some sort of vague notion.

If we understand an Axiom to be simply be an expression of
language that has been expressly defined to have the semantic
property of Boolean True, then this otherwise vague notion
is finally totally anchored coherently.

Rules-of-inference show how to correctly transform some
expressions of language into other expressions of language.
Again this is generic to both formal and natural language.

Copyright 2018 Pete Olcott
Peter Percival
2018-11-01 16:24:25 UTC
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Post by peteolcott
Post by peteolcott
A Theorem is essentially nothing more than
FUCK you.
The word HAS *A*DEFINITION*.
It's ALREADY IN the dictionary.
You DON'T get an OPINION.
Now that I double checked my most recent revisions, my
English definitions of these terms seem more succinct
and accurate than any others that I have ever seen.
I challenge anyone to show that any other English definitions
of these terms are in any way better than mine.
Mendelson had a great mathematical definition of Theorem.
It is this definition that I refer to in my universal Truth
predicate. It is his text that allowed me to finally translate
my thirty year long intuitions into conventional terms of the art.
Thirty years! And to think, you could have been doing something useful,
like sweeping the streets or cleaning pubic toilets, to earn a few dollars.
Post by peteolcott
My English definition of Theorem equally apply to both formal
A Theorem is essentially nothing more than an expression
of language that can be proven True on the basis on known
facts (axioms), thus the conclusion of sound deductive inference.
Since it is unconventional (although more accurate) to think
of an Axiom as a known truth
Known to whom, and how?
Post by peteolcott
, my English definition of Theorem
requires my definition of Axiom (see below) as its basis.
No one seemed to have any decent definition of Axiom,
everywhere I looked it was always some sort of vague notion.
If you ever read a maths or logic book, you will find that the axioms
are spelt out with admirable clarity. I do not claim that 'with
admirable clarity' means anything like 'of interest to Pete Olcott' or
'understandable by Pete Olcott'.
Post by peteolcott
If we understand an Axiom to be simply be an expression of
language that has been expressly defined to have the semantic
property of Boolean True, then this otherwise vague notion
is finally totally anchored coherently.
No it isn't. I define X to be an axiom, you define not-X to be an
axiom. Does it matter? Sometimes yes, sometimes no. You cannot even
exemplify the cases.
Post by peteolcott
Rules-of-inference show how to correctly transform some
expressions of language into other expressions of language.
Waffle. You don't know what 'correct' and 'expression' mean in this
context.
Post by peteolcott
Again this is generic to both formal and natural language.
Copyright 2018 Pete Olcott
Those few dollars that I referred to above - you could use them to buy a
few cans of beer or cheap wine and you could sit in the park sharing it
with your buddies and have a great time. But you choose to waste your
time posting to sci.logic... I don't know...

peteolcott
2018-10-27 07:01:49 UTC
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Post by peteolcott
A Theorem is essentially nothing more than
FUCK you.
The word HAS *A*DEFINITION*.
It's ALREADY IN the dictionary.
You DON'T get an OPINION.
The greatest thing about my current paper is that I am using all of the
technical terms intending their precise conventional meaning.
This is not actually possible without first understanding those precise
conventional meanings.
The glossary of terms is only for the purpose of explaining these technical
terms in plain English to people unfamiliar with these terms.
People unfamiliar with these terms would be much better off turning to a
dictionary than to your 'definition', which is entirely incorrect.
Andre
Apparently you are wrong about that and especially wrong
when it comes to the definition the term Axiom.

No one seemed to have any decent definition of Axiom,
everywhere I looked it was always some sort of vague notion.

If we understand an Axiom to be simply be an expression of
language that has been expressly defined to have the semantic
property of Boolean True, then this otherwise vague notion
is finally totally anchored coherently.

Copyright 2018 Pete Olcott
Franz Gnaedinger
2018-10-27 07:58:43 UTC
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Post by peteolcott
Copyright 2018 by Pete Olcott
Peter Olcott knows the absolute and complete and total truth, he is the author
of life and creator of life, he has hundred reasons to assume that he is God,
he creates our future minds in order that we can go on existing (thank you very
much, so I can write this reply), he is a human being and God in personal union,
he is the one Creator of the Universe (claims he made in sci.lang). However,
Allgod failed as creator of the universe and life, since the universe and life
produced Goedel who evaporated what Allgod writes already before Allgod
descended on Earth.
Peter Percival
2018-11-01 16:09:59 UTC
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Post by peteolcott
A Theorem is essentially nothing more than an expression of language
that can be proven True entirely on the basis on known facts, thus the
It would be better to say just "proven" rather than "proven True" (why
the capital T? I don't know) because there is also a definition of true
and it might be that what is proven and what is true do not coincide.
It turns out that there are theories (the word has a precise definition)
for which they do coincide and theories for which they do not. Such a
distinction is invisible to you, but that doesn't mean that it doesn't
exist.
Post by peteolcott
conclusion of sound deductive inference.
More formally Theorems are mathematical proofs based on axioms and
rules-of-inference only without specifying any premises, thus forming
sound deductive inference.
What follows is rot...
Post by peteolcott
Axioms can be thought of as expressions of language that have been
expressly defined to have the semantic property of Boolean True.
... what's to stop someone from saying that X is an axiom and someone
else saying that not-X is an axiom?
Post by peteolcott
Rules-of-inference show how to correctly transform some expressions of
language into other expressions of language.
... waffle. Read a logic text if you want to know what rules of
inference are.
Post by peteolcott
Any expression of language not having a Boolean Property with a value of
True or False is not a correct Declarative sentence / sentence of logic.
... grammar texts and logic texts will tell you what correct sentences
(in the everyday sense and in the logical sense respectively) are.
Post by peteolcott
This universal Truth predicate rejects as malformed any expressions of
language that would otherwise show Incompleteness or Undefinability: ∀x
∈ F True(F, x) ↔ Theorem(F, x)
For all expressions x in language L, x is True in L if and only if x is a Theorem of L
https://plato.stanford.edu/entries/goedel-incompleteness/#FirIncTheCom
(G) F ⊢ GF ↔ ¬ProvF(⌈GF⌉).
G ↔ G ∈ F ~Provable(F, G)
“This sentence is not provable”
G ≡ ~Provable(G) // To avoid clutter F is implied rather than stated
G asserts that G is not Provable. If G was a theorem (thus true) then it
would contradict its assertion that G is not Provable.
~G asserts that G is Provable. If ~G was a theorem (thus true) then it
would contradict its assertion that G is Provable.
∴ Since Neither G nor ~G is a theorem (true) G is rejected as an
incorrect Boolean Proposition because it is neither True nor False.
Copyright 2018 by Pete Olcott
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