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Simply defining Gödel Incompleteness away
Posted: Sat Jun 13, 2020 9:37 pm
by PeteOlcott
∀F ∈ Formal_Systems
∀X ∈ Language(F)
Gödel's entire proof depends on the definition that Undecidable(F,X)
means Incomplete(F).
When we define Undecidable(F,X) to mean Incorrect(X) his proof utterly
loses its entire "incompleteness" basis.
The key question is which of these two different definitions is the
correct foundational basis.
This question cannot be correctly answered by mathematics or logic
because both of these two fields never question the philosophical reasons
for their foundational basis.
Although it is nearly universally accepted that Undecidable(F, X) means
Incomplete(F) if instead it actually means Incorrect(X) then the
incompleteness aspect of the 1931 Incompleteness Theorem utterly loses
its entire basis causing the whole proof to totally fail.
Re: Simply defining Gödel Incompleteness away
Posted: Sun Jun 14, 2020 12:05 am
by Impenitent
even Ludwig finally decided, the thing in itself and its name are never the same...
-Imp
Re: Simply defining Gödel Incompleteness away
Posted: Mon Jun 15, 2020 7:25 pm
by Arising_uk
I thought he decided that we couldnt tell or say but there's always pointing.
Re: Simply defining Gödel Incompleteness away
Posted: Mon Jun 15, 2020 7:46 pm
by Skepdick
PeteOlcott wrote: ↑Sat Jun 13, 2020 9:37 pm
∀F ∈ Formal_Systems
∀X ∈ Language(F)
Gödel's entire proof depends on the definition that Undecidable(F,X)
means Incomplete(F).
When we define Undecidable(F,X) to mean Incorrect(X) his proof utterly
loses its entire "incompleteness" basis.
The key question is which of these two different definitions is the
correct foundational basis.
This question cannot be correctly answered by mathematics or logic
because both of these two fields never question the philosophical reasons
for their foundational basis.
Although it is nearly universally accepted that Undecidable(F, X) means
Incomplete(F) if instead it actually means Incorrect(X) then the
incompleteness aspect of the 1931 Incompleteness Theorem utterly loses
its entire basis causing the whole proof to totally fail.
And if you re-define Undecidable(F,X) to mean Unicorns(X) you've proven unicorns.
I keep explaining it to you. Maybe now you are ready to understand?
You need a
PURE type system. You need PURE lambda calculus. Like
System U
According to every Philosophical definition of "consistency" Python is inconsistent because it can express
Girard's paradox.
https://repl.it/repls/OutlandishSorrowfulCables
The foundational basis is ALWAYS self-reference which causes contradictions. The way to eradicate contradictions is to remove yourself from the equation.
Re: Simply defining Gödel Incompleteness away
Posted: Mon Jun 15, 2020 8:08 pm
by Skepdick
Arising_uk wrote: ↑Mon Jun 15, 2020 7:25 pm
I thought he decided that we couldnt tell or say but there's always pointing.
It's difficult (but not impossible) to point to abstract ideas in our heads.
Re: Simply defining Gödel Incompleteness away
Posted: Mon Jun 15, 2020 8:36 pm
by PeteOlcott
Skepdick wrote: ↑Mon Jun 15, 2020 7:46 pm
PeteOlcott wrote: ↑Sat Jun 13, 2020 9:37 pm
∀F ∈ Formal_Systems
∀X ∈ Language(F)
Gödel's entire proof depends on the definition that Undecidable(F,X)
means Incomplete(F).
When we define Undecidable(F,X) to mean Incorrect(X) his proof utterly
loses its entire "incompleteness" basis.
The key question is which of these two different definitions is the
correct foundational basis.
This question cannot be correctly answered by mathematics or logic
because both of these two fields never question the philosophical reasons
for their foundational basis.
Although it is nearly universally accepted that Undecidable(F, X) means
Incomplete(F) if instead it actually means Incorrect(X) then the
incompleteness aspect of the 1931 Incompleteness Theorem utterly loses
its entire basis causing the whole proof to totally fail.
You need a
PURE type system. You need PURE lambda calculus. Like
System U
I have already been saying that for a few years now.
That is why I invented Minimal Type Theory.
https://www.researchgate.net/publicatio ... y_YACC_BNF
Re: Simply defining Gödel Incompleteness away
Posted: Mon Jun 15, 2020 8:40 pm
by PeteOlcott
Skepdick wrote: ↑Mon Jun 15, 2020 7:46 pm
PeteOlcott wrote: ↑Sat Jun 13, 2020 9:37 pm
∀F ∈ Formal_Systems
∀X ∈ Language(F)
Gödel's entire proof depends on the definition that Undecidable(F,X)
means Incomplete(F).
When we define Undecidable(F,X) to mean Incorrect(X) his proof utterly
loses its entire "incompleteness" basis.
The key question is which of these two different definitions is the
correct foundational basis.
This question cannot be correctly answered by mathematics or logic
because both of these two fields never question the philosophical reasons
for their foundational basis.
Although it is nearly universally accepted that Undecidable(F, X) means
Incomplete(F) if instead it actually means Incorrect(X) then the
incompleteness aspect of the 1931 Incompleteness Theorem utterly loses
its entire basis causing the whole proof to totally fail.
And if you re-define Undecidable(F,X) to mean Unicorns(X) you've proven unicorns.
Yet the key difference is that:
Undecidable(F,X) means Incomplete(F) is epistemologically incorrect and
Undecidable(F,X) means Incorrect(X) is epistemologically correct.
Re: Simply defining Gödel Incompleteness away
Posted: Mon Jun 15, 2020 8:41 pm
by Skepdick
You aren't saying what you think you are saying.
The "Minimal" in your type theory is too much in the way of impurity. It's not minimal enough because it has too many rules.
Re: Simply defining Gödel Incompleteness away
Posted: Mon Jun 15, 2020 8:44 pm
by Skepdick
PeteOlcott wrote: ↑Mon Jun 15, 2020 8:40 pm
Undecidable(F,X) means Incomplete(F) is epistemologically incorrect and
Undecidable(F,X) means Incorrect(X) is epistemologically correct.
According to what standard for "correctness"?
The practical consequence of that which is said to be an "inconsistent system" is that it can prove ALL propositions.
Par for the course! I prefer a language that can prove too much over a language that can prove too little.
I want to be able to express ANYTHING and EVERYTHING. Freedom of speech and all that jazz...
Re: Simply defining Gödel Incompleteness away
Posted: Mon Jun 15, 2020 9:50 pm
by PeteOlcott
Skepdick wrote: ↑Mon Jun 15, 2020 8:41 pm
You aren't saying what you think you are saying.
The "Minimal" in your type theory is too much in the way of impurity. It's not minimal enough because it has too many rules.
Think of my system as a higher level language that translates into your system.
Writing in typed lambda calculus is like writing in assembly language.
Re: Simply defining Gödel Incompleteness away
Posted: Mon Jun 15, 2020 10:05 pm
by Skepdick
PeteOlcott wrote: ↑Mon Jun 15, 2020 9:50 pm
Think of my system as a higher level language that translates into your system.
Writing in typed lambda calculus is like writing in assembly language.
I understand your intention, but you are confused.
You have "high level" and "low level" languages mixed up.
Why must I write 20 lines of assembly to express what can be expressed in 1 line of Python?
Less is more.
Re: Simply defining Gödel Incompleteness away
Posted: Mon Jun 15, 2020 10:13 pm
by Impenitent
Skepdick wrote: ↑Mon Jun 15, 2020 10:05 pm
PeteOlcott wrote: ↑Mon Jun 15, 2020 9:50 pm
Think of my system as a higher level language that translates into your system.
Writing in typed lambda calculus is like writing in assembly language.
I understand your intention, but you are confused.
You have "high level" and "low level" languages mixed up.
Why must I write 20 lines of assembly to express what can be expressed in 1 line of Python?
Less is more.
writeln "This Parrot is Dead"
oh wait, that's Cobol... nevermind
-Imp
Re: Simply defining Gödel Incompleteness away
Posted: Mon Jun 15, 2020 10:23 pm
by PeteOlcott
Skepdick wrote: ↑Mon Jun 15, 2020 8:44 pm
PeteOlcott wrote: ↑Mon Jun 15, 2020 8:40 pm
Undecidable(F,X) means Incomplete(F) is epistemologically incorrect and
Undecidable(F,X) means Incorrect(X) is epistemologically correct.
According to what standard for "correctness"?
The practical consequence of that which is said to be an "inconsistent system" is that it can prove ALL propositions.
Par for the course! I prefer a language that can prove too much over a language that can prove too little.
I want to be able to express ANYTHING and EVERYTHING. Freedom of speech and all that jazz...
The reason why it is incorrect to define Unprovable(F, X) as Incomplete(F) instead of
defining it as Incorrect(X) is that Incorrect(X) detects that there the inference chain
from the premises to the conclusion is broken thus indicating that the argument is invalid.
The way that analytical truth really works is that truth preserving operations are
performed on expressions of language that are somehow known to be true** deriving
conclusions that are known to be true.
If there is any reason that the process does not proceed all the way from the
true premises to the conclusion then the conclusion is not derived and is
therefore untrue. There cannot possibly be any case of true and unprovable.
All that we are doing here is augmenting the notion of sound deduction:
A deductive argument is said to be valid if and only if it takes a form that makes
it impossible for the premises to be true and the conclusion nevertheless to be
false. Otherwise, a deductive argument is said to be invalid.
https://www.iep.utm.edu/val-snd/
The above definition allows the principle of explosion. This must be forbidden.
By requiring that truth preserving operations be performing on the premises to
derive the conclusion we implicitly require that the premises are semantically
relevant to the conclusion:
https://en.wikipedia.org/wiki/Relevance_logic
** An issue of foundationalism not relevant to the current point.
Copyright 2020 Pete Olcott
Re: Simply defining Gödel Incompleteness away
Posted: Mon Jun 15, 2020 10:26 pm
by PeteOlcott
Skepdick wrote: ↑Mon Jun 15, 2020 10:05 pm
PeteOlcott wrote: ↑Mon Jun 15, 2020 9:50 pm
Think of my system as a higher level language that translates into your system.
Writing in typed lambda calculus is like writing in assembly language.
I understand your intention, but you are confused.
You have "high level" and "low level" languages mixed up.
Why must I write 20 lines of assembly to express what can be expressed in 1 line of Python?
Less is more.
My end goal is to make a very easy way to formalize every subtle nuance of natural
language semantics. Certainly typed lambda Calculus is not this way. I know nothing
about Python.
Re: Simply defining Gödel Incompleteness away
Posted: Mon Jun 15, 2020 10:27 pm
by Skepdick
PeteOlcott wrote: ↑Mon Jun 15, 2020 10:23 pm
The above definition allows the principle of explosion. This must be forbidden.
You are really not hearing me!
The principle of explosion is what I WANT in my systems.
I prefer proving too much (everything even!) than proving too little.