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I didn't. The OP did. He said that Gödel's incompleteness theorem is "erroneous." That theorem bears directly on set theory. The OP needs to either retract his claim or else take responsibility for its logical consequences.
I didn't. The OP did. He said that Gödel's incompleteness theorem is "erroneous." That theorem bears directly on set theory. The OP needs to either retract his claim or else likewise assert (and be prepared to defend) its logical consequences.
The OP's proof is also explicitly in a Curry-Howard system using type theory.
There is some further conceptual/linguistic misalignment.
Code: Select all
$toggle = true
def a
$toggle = ( not $toggle )
end
a and not a
=> true
c = a
c and not c
=> false 
OK, why choose this means of publication? I would expect most mathematicians to poo-poo this medium...PeteOlcott wrote: ↑Sat Mar 23, 2019 9:15 pm 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
PeteOlcott wrote: ↑Sat Mar 23, 2019 9:15 pm 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.
I broadly agree.PeteOlcott wrote: ↑Sat Mar 23, 2019 9:15 pm 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.
Yes, the notion of a semantic separate from the syntax is something I only discovered recently and I thought it was definitely not a good idea.PeteOlcott wrote: ↑Sat Mar 23, 2019 9:15 pm 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.
Yes. That"s what I thought reading your piece.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. 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.
Speakpigeon wrote: ↑Sun Mar 24, 2019 12:02 pm My pet peeve is the principle of explosion, which I find intuitively just moronic and I can't understand how gown-up mathematicians could countenance it.
Minimal Type Theory (formally specified at the end of my paper) is higher order logic with types, thus a kind of type theory.Logik wrote: ↑Sun Mar 24, 2019 6:58 amWhy have you chosen set theory and not type theory for your mathematical foundation?PeteOlcott wrote: ↑Sat Mar 23, 2019 9:15 pm Nearly all logicians maintain two separate representations to analyze semantics.
Apologies, that question was aimed at "wtf". I was merely using your quote as a demonstration of his own choice to represent semantics in set theory AND in English.PeteOlcott wrote: ↑Sun Mar 24, 2019 3:38 pmMinimal Type Theory (formally specified at the end of my paper) is higher order logic with types, thus a kind of type theory.Logik wrote: ↑Sun Mar 24, 2019 6:58 amWhy have you chosen set theory and not type theory for your mathematical foundation?PeteOlcott wrote: ↑Sat Mar 23, 2019 9:15 pm Nearly all logicians maintain two separate representations to analyze semantics.
More evidence for my claim that set theorists and type theorists are speaking cross-paradigms.
My last reply seems to have gotten lost. I also agree that the principle of explosion is semantically unsound.Speakpigeon wrote: ↑Sun Mar 24, 2019 12:02 pm Yes. That"s what I thought reading your piece.
I haven't yet looked at the details and I'm not a mathematician, but so far I agree with all your points.
My pet peeve is the principle of explosion, which I find intuitively just moronic and I can't understand how gown-up mathematicians could countenance it. I worked out myself two different proofs (not for publication) but your "semantic proof" about going to the grocery store is equally convincing.
I look forward to looking into the details of what you say.
EB
I have always approached these things from a philosophy of logic perspective.Speakpigeon wrote: ↑Sun Mar 24, 2019 12:02 pmOK, why choose this means of publication? I would expect most mathematicians to poo-poo this medium...PeteOlcott wrote: ↑Sat Mar 23, 2019 9:15 pm 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
OK, that may be something we could disagree about. I understand truth to be essentially another name for whatever it is we believe. We believe it's night, then we say it's true it's night. Logic then is merely the syntactic relation between our beliefs. We can't verify our beliefs anyway, but it's clearly a good idea to have "coherent" beliefs so our brain evolved to make sure we do, up to a point. Truth still as a value for each of us. There are things you can say that are true, just because there are things you know, like how you feel and whether you're hungry or pissed off. But that's not enough to discuss with each other and that's not enough even to think the world around you. So you need logic to think up the world around you, and also, secondarily, to exchange with other people "linguistically", something which has only become very important since perhaps 10,000 to 40,000 years ago. The point here is that logic is first a mental process evolved by natural selection over more than 525 million years and an intuitive capability we all have, and only secondarily a formal system, which as such is a part of nearly all human languages. As I understand it, the formal system we use is only a partial expression of the mental capability we have. Logic comes first, formalisation comes second, and then maybe not quite the thing. Still, this gives us a number of "logical truths" we all think are true, although we can choose not to, just because logical truths are formal logic, ergo part of language, and that we can invent and say pretty much anything. Our intuitive brain couldn't, though. This explains for example how mathematicians could choose, after Russell, to move away from their own logical intuition. Basically, mathematical logic is mathematics, not logic.PeteOlcott wrote: ↑Sun Mar 24, 2019 4:02 pm My last reply seems to have gotten lost. I also agree that the principle of explosion is semantically unsound.
I do not want to digress into talking about this until all of my current points are fully addressed.
Relevance logic seems to address the principle of explosion quite well.
My main point is that it impossible for truth and provability to ever diverge because truth
is ONLY verified as a formal proof to theorem consequences:
∀F ∈ Formal_Systems ∀x ∈ WFF(F) (True(F, x) ↔ (F ⊢ x))
When this above formula is accepted as the correct formalization of Truth,
then Tarski and Gödel are both refuted.
My brain says he feels good about that.
PeteOlcott wrote: ↑Sun Mar 24, 2019 5:01 pm I have always approached these things from a philosophy of logic perspective. When I finally learned enough formal logic to unequivocally express what I mean, this was rejected by logicians as just simply not the way that we do it.
PeteOlcott wrote: ↑Sun Mar 24, 2019 5:01 pm Logicians take conventional wisdom as the foundatal basis of all of their work and never question the legitimacy of this foundation.
OK, I'll look at that but the maths maybe beyond my abilities to assess.PeteOlcott wrote: ↑Sun Mar 24, 2019 5:01 pm Previously I could never find any people interested in the philosophy of logic foundations that could begin to understand any aspect of predicate logic. Now that I finally totally understand these ideas myself I can bridge the gap between the math and plain English. I added a laymen's section to the second paper of my original post.
John is John. ( A = A ⇔ True )Speakpigeon wrote: ↑Sun Mar 24, 2019 6:38 pmOK, I'll look at that but the maths maybe beyond my abilities to assess.PeteOlcott wrote: ↑Sun Mar 24, 2019 5:01 pm Previously I could never find any people interested in the philosophy of logic foundations that could begin to understand any aspect of predicate logic. Now that I finally totally understand these ideas myself I can bridge the gap between the math and plain English. I added a laymen's section to the second paper of my original post.
EB
But you don't have to use '=' for 'is'?Logik wrote:...
The verb "is" (represented by the logical symbol "=" ) is semantically overloaded. Its meaning is context-specific and inferred in the __eq__ function of the Human class.
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