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Refuting Incompleteness and Undefinability
Posted: Thu Mar 21, 2019 10:43 pm
by PeteOlcott
https://www.researchgate.net/publicatio ... finability
Philosophy of Logic – Reexamining the Formalized Notion of Truth
I update this linked paper all the time: I just added an explanation for laymen:
https://philpapers.org/archive/OLCPOL.pdf
Re: Refuting Incompleteness and Undefinability
Posted: Fri Mar 22, 2019 6:21 am
by Logik
Ohhhh, what a lovely, elegant and brief paper!!! Well done Pete!
Turing-completeness/Curry-Howard/Chomsky Hierarchy wins the day again. Proofs compute
I see that you have published other papers on MTT before. Is there an actual compiler for the grammar you have invented? if not - I trust you will make a lot of waves in the Mathematics universe if you re-write your proof in a language that is seen as an "authority" in that community. Coq/Agda etc.
Re: Refuting Incompleteness and Undefinability
Posted: Fri Mar 22, 2019 4:19 pm
by PeteOlcott
Yes there is a compiler. The earlier version translated into an acyclic directed graph.
Since I redesigned the syntax the latest compiler merely creates an XML abstract syntax tree.
Its final form is intended to be a directed acyclic graph, knowledge ontology.
Re: Refuting Incompleteness and Undefinability
Posted: Fri Mar 22, 2019 4:56 pm
by Logik
PeteOlcott wrote: ↑Fri Mar 22, 2019 4:19 pm
Yes there is a compiler. The earlier version translated into an acyclic directed graph.
Since I redesigned the syntax the latest compiler merely creates an XML abstract syntax tree.
Its final form is intended to be a directed acyclic graph, knowledge ontology.
So you are merely handling the lexical/semantic parsing so far?
Any reason you chose to go this way rather than use one of the existing tools/languages?
Re: Refuting Incompleteness and Undefinability
Posted: Fri Mar 22, 2019 9:08 pm
by PeteOlcott
I use the approach pioneered by Rudolf Carnap in his 1952 Meaning Postulates
to specify semantics directly in the syntax of the formal system
I elaborate the foundational link between the syntax of symbolic logic
and the semantics of sound deductive inference in this paper:
https://philpapers.org/archive/OLCETD.pdf
Semantic meaning of English sentences specified in First Order Logic
(from Meaning Postulates Rudolf_Carnap, 1952)
(1) Bachelor(Jack) → ~Married(Jack)
(2) Black(Fido) ∨ ~Black(Fido)
Expressing the (a) transitivity (b) irreflexivity and (c) asymmetry relations of the English word Warmer:
(a) (x)(y)(z) (Warmer(x,y) ∧ Warmer(y,z) → Warmer(x,z))
(b) (x) ~Warmer(x,x)
(c) (x)(y) (Warmer(x,y) → ~Warmer(y,x)
The reason that I created MTT was to provide a simple universal Tarski Meta-language so that
these things could be expressed and directly evaluated within a single fully elaborated formal language.
Tarski keeps going back and forth, mixing and matching between his unspecified metalanguage
and object language. He force fits metalanguage variables directly into his object language.
A single universal Tarski metalanguage totally eliminates all of the extraneous complexity.
Re: Refuting Incompleteness and Undefinability
Posted: Sat Mar 23, 2019 1:12 am
by Logik
PeteOlcott wrote: ↑Fri Mar 22, 2019 9:08 pm
The reason that I created MTT was to provide a simple universal Tarski Meta-language so that
these things could be expressed and directly evaluated within a single fully elaborated formal language.
Tarski keeps going back and forth, mixing and matching between his unspecified metalanguage
and object language. He force fits metalanguage variables directly into his object language.
A single universal Tarski metalanguage totally eliminates all of the extraneous complexity.
Have you yet connected the dots between Tarski's concept of metalanguages and recursively-enumerable languages like those addressed in the Chomsky's hierarchy? They are functionally equivalent.
https://en.wikipedia.org/wiki/Recursive ... e_language
The significance of Chomsky's work puts us in a universe where all recursively-enumerable languages are Turing-complete and thus can be formalized in Lambda calculus. If MTT is Turing-complete then it's no different to any other modern Turing-complete programming language.
Sadly - you don't end up with a "single" metalanguage. You end up with ALL the metalanguages
Re: Refuting Incompleteness and Undefinability
Posted: Sat Mar 23, 2019 5:39 am
by PeteOlcott
My fundamental design criteria for MTT was to make sure that its parsed expressions
decomposed into a directed acyclic graph. This would seem to be expressible in
lambda calculus.
Because MTT is an extension of First Order Logic it can express complex relationships
in a way that is much easier for the limitations of the human mind. It uses conventional
logic symbols with named predicates and functions.
In terms of human cognition MTT is the simplest possible language that can completely
express any (one or more) elements of the currently existing finite set of human
concepts. In this regard it is analogous to the slightly more complex CycL of Cycorp.
So although it is possible to magnify extraneous complexity of languages without limit
it may be impossible to derive any language with the expressive power of MTT that
has any additional reduction of extraneous complexity.
MTT is the only universal Tarski metalanguage that is needed to make every detail of
his proof explicit using a single language, and thus make the errors of this proof
enormously more obvious.
Re: Refuting Incompleteness and Undefinability
Posted: Sat Mar 23, 2019 6:34 pm
by Speakpigeon
PeteOlcott wrote: ↑Sat Mar 23, 2019 5:39 am
My fundamental design criteria for MTT was to make sure that its parsed expressions
decomposed into a directed acyclic graph. This would seem to be expressible in
lambda calculus.
Because MTT is an extension of First Order Logic it can express complex relationships
in a way that is much easier for the limitations of the human mind. It uses conventional
logic symbols with named predicates and functions.
In terms of human cognition MTT is the simplest possible language that can completely
express any (one or more) elements of the currently existing finite set of human
concepts. In this regard it is analogous to the slightly more complex CycL of Cycorp.
So although it is possible to magnify extraneous complexity of languages without limit
it may be impossible to derive any language with the expressive power of MTT that
has any additional reduction of extraneous complexity.
MTT is the only universal Tarski metalanguage that is needed to make every detail of
his proof explicit using a single language, and thus make the errors of this proof
enormously more obvious.
I won't pretend I really understand your paper but it broadly fits with my intuition about mathematical logic. Your result seems really important, so what could be the consequence of it?
Also, why publish on ResearchGate?
Have you received any comments on your work?
EB
Re: Refuting Incompleteness and Undefinability
Posted: Sat Mar 23, 2019 9:15 pm
by PeteOlcott
I have received many comments on my work over the years. I have finally (just barely) gained enough mathematical maturity that logicians are starting to take me seriously. I publish on Researchgate and PhilPapers. ResearchGate only permits members with actual published works.
To translate the essence of my ideas into laymen's terms all of conceptual truth is anchored in expressions of language (such as English statements and the well-formed formula of formal languages) that are defined to be true. Some mathematicians such as Haskell Curry understand that mathematical axioms are defined to be true.
The only way that we know that a {cat} is a kind of {animal} is that this is defined as a basic fact. It turns out that all of analytical truth works this same way. I refer to analytical truth because some truth depends on sensations for the sense organs and can thus not be fully expressed using words. The actual taste of strawberries is an example.
Nearly all logicians maintain two separate representations to analyze semantics. I show that this is unnecessary and and makes things much more complicated than they really need to be. We can express and analyze any and all semantics entirely syntactically without any separate assignment of meaning over and above the conventional meaning of the logical operators and the axioms specified in the formal language.
The consequence of my work would be that the Tarski Undefinability Theorem, Gödel's 1931 Incompleteness Theorem, Turing's 1931 Halting Problem proof and many other undecidable decision problems would be shown to be erroneous. The other key consequence of this work is that truth conditional semantics will finally be anchored in a formal correct notion of truth. This is a key missing piece of all artificial intelligence research.
Re: Refuting Incompleteness and Undefinability
Posted: Sun Mar 24, 2019 3:32 am
by wtf
PeteOlcott wrote: ↑Sat Mar 23, 2019 9:15 pm
The consequence of my work would be that the Tarski Undefinability Theorem, Gödel's 1931 Incompleteness Theorem, Turing's 1931 Halting Problem proof and many other undecidable decision problems would be shown to be erroneous.
Two questions if I may.
1. Turing's 1936 (not '31) proof of the unsolvability of the Halting problem is particularly simple to understand. I was wondering if you've ever sat down and shown the exact flaw in Turing's proof.
2. What do you make of all the independence results in math? Gödel and Cohen's proof of the independence of the Continuum hypothesis and the axiom of choice? All the modern independence results in the theory of large cardinals? If these results are erroneous, can you supply proofs or disproofs of CH and AC? If the axioms of set theory are complete, such proofs must exist. Do you claim such proofs must exist?
Re: Refuting Incompleteness and Undefinability
Posted: Sun Mar 24, 2019 4:05 am
by PeteOlcott
Oh, right 1936. After carefully studying the Peter Linz (simplest possible) sufficiently elaborated Halting Problem proof off and on since 2004, last December, I finally figured out the precise algorithm showing exactly how the Linz H would decide halting for the Linz (Ĥ, Ĥ) input pair.
This solution seems to be generalizable to the other conventional proofs. I will publish it as soon as I have developed the software to provide a full execution trace of this pair of virtual machines. All of the detailed design is complete, I only have to finish encoding the fully operational general purpose UTM interpreter.
I have focused all of my analytical attention (since 2004) on decision problems that are undecidable because of pathological self-reference. To the best of my knowledge this same reasoning can be extended (on the basis of truth conditional semantics) to many other types of otherwise undecidable decision problems.
Re: Refuting Incompleteness and Undefinability
Posted: Sun Mar 24, 2019 4:19 am
by wtf
PeteOlcott wrote: ↑Sun Mar 24, 2019 4:05 am
To the best of my knowledge this same reasoning can be extended ...
I just want to be clear since my background is in math and not logic.
If you say Gödel's incompleteness results are false, then are you saying the ZFC axioms of set theory are complete and consistent? That means that there must be a proof or disproof (but not both!) of each statement currently regarded as independent of ZFC. Do you make that claim?
ps -- A Google search on Peter Linz brings up a puppeteer. Can you supply a link to what you're referring to?
Re: Refuting Incompleteness and Undefinability
Posted: Sun Mar 24, 2019 5:55 am
by PeteOlcott
I would define set theory differently than the way it is defined. I don't want to digress into this.
I have not paid very much attention to things unrelated to the decision problems that I cited.
Here are the two pages of Linz that I studied off and on since 2004.
http://liarparadox.org/Peter_Linz_HP(Pages_318-319).pdf
Re: Refuting Incompleteness and Undefinability
Posted: Sun Mar 24, 2019 6:49 am
by wtf
PeteOlcott wrote: ↑Sun Mar 24, 2019 5:55 am
I would define set theory differently than the way it is defined. I don't want to digress into this.
I have not paid very much attention to things unrelated to the decision problems that I cited.
You said:
PeteOlcott wrote: ↑Sun Mar 24, 2019 5:55 am
The consequence of my work would be that the Tarski Undefinability Theorem,
Gödel's 1931 Incompleteness Theorem, Turing's 1931 Halting Problem proof and many other undecidable decision problems would be shown to be erroneous.
My emphasis.
Gödel's first incompleteness theorem says that no axiomatic theory that's powerful enough to include the arithmetic of the natural numbers can be both consistent and complete.
If that theorem is erroneous, your direct words, then there exists an axiomatic theory that can represent natural number arithmetic and is consistent and complete.
Therefore there exists a theory that definitively resolves the Continuum hypothesis, the axiom of choice, the existence of inaccessible cardinals, measurable cardinals, and all the other wild large cardinals of modern set theory.
If you assert that Gödel's first incompleteness theorem is erroneous, you must necessarily, as a logical consequence, hold that there's a complete and consistent theory that resolves CH and AC and the large cardinal axioms.
I ask you again: Do you or do you not assert that Gödel''s first incompleteness theorem is erroneous? And if so, do you assert the direct logical consequences of that claim?
I hope you (and others) can see that I am only pressing you on the logical consequences of the claim you have made.
Do you understand that in order for anyone to take your work seriously, you would inevitably have to respond to exactly the questions I'm asking? Or perhaps you went one claim too far about Gödel and would like to walk it back. Once you claim Gödel's incompleteness theorem is erroneous, you must immediately confront the logical consequences.
And what is the flaw in the proof? And what is the flaw in Turing's extremely simple original proof?
Re: Refuting Incompleteness and Undefinability
Posted: Sun Mar 24, 2019 6:58 am
by Logik
wtf wrote: ↑Sun Mar 24, 2019 6:49 am
Therefore there exists a theory that definitively resolves the Continuum hypothesis, the axiom of choice, the existence of inaccessible cardinals, measurable cardinals, and all the other wild large cardinals of modern set theory.
If you assert that Gödel's first incompleteness theorem is erroneous, you must necessarily, as a logical consequence, hold that there's a complete and consistent theory that resolves CH and AC and the large cardinal axioms.
First point of order....
To speak of Set theory and Curry-Howard systems in the same sentence is a misnomer. A confusion of paradigms.
You have made a choice (or worse - haven't!) somewhere. This is Pete's very point:
PeteOlcott wrote: ↑Sat Mar 23, 2019 9:15 pm
Nearly all logicians maintain two separate representations to analyze semantics.
Why have you chosen set theory and not type theory for your mathematical foundation?
2nd point of order.
If you are dealing with Curry-Howard systems (proofs compute) you are in a finitist paradigm. Infinities are errors ( so we can junk the Continuum hypothesis ).
The above will not halt if A is an infinite set. I am providing no citations because the above comes from my own mind.
Of course, you are free to use ZFC theory IN a Curry-Howard system, but before you can do that you need to define the algorithms (TYPES!) for the predicates "contains", "intersects", "subset" etc. etc. etc.