• Re: Readings in (some of the) foundations of mathematics --- tree of knowledge

    From olcott@polcott333@gmail.com to comp.theory,sci.logic,sci.lang,comp.ai.philosophy,sci.math on Thu Jun 25 11:14:50 2026
    From Newsgroup: sci.lang

    On 6/25/2026 2:21 AM, Mikko wrote:
    On 24/06/2026 23:26, olcott wrote:
    On 6/24/2026 5:00 AM, Mikko wrote:
    On 23/06/2026 17:48, olcott wrote:
    On 6/23/2026 1:06 AM, Mikko wrote:
    On 22/06/2026 15:10, olcott wrote:
    On 6/22/2026 1:49 AM, Mikko wrote:
    On 22/06/2026 02:02, olcott wrote:
    On 6/21/2026 4:08 PM, Andr|- G. Isaak wrote:
    On 2026-06-21 14:42, olcott wrote:
    On 6/21/2026 3:04 PM, Alan Mackenzie wrote:
    [ Followup-To: set ]

    In comp.theory olcott <polcott333@gmail.com> wrote:
    On 6/21/2026 6:26 AM, Alan Mackenzie wrote:
    In comp.theory olcott <polcott333@gmail.com> wrote: >>>>>>>>>>>>>> I just found the term:
    "grounding in a proof theoretic atomic base" yesterday. >>>>>>>>>>>
    You can find any number of terms.-a That doesn't mean you're >>>>>>>>>>>>> capable of
    understanding them.


    The above is the key reason why under PTS G||del 1931 >>>>>>>>>>>> incompleteness
    fails.

    I don't believe you.-a You have no respect for or
    understanding of the
    truth.-a If you really want to persuade anybody that PTS >>>>>>>>>>> somehow causes
    G||del's theorem not to hold, then cite an academic expert >>>>>>>>>>> who'll have
    some credibility.

    If they are mere gibberish words to you then you will not >>>>>>>>>>>> understand.

    You don't understand Proof-theoritic Semantics, and you >>>>>>>>>>> certainly don't
    understand G||del's Theorem, neither the theorem itself nor >>>>>>>>>>> any proof of
    it.

    It is a verified fact that G||del's G is ungrounded
    in the atomic base of PA. That you do not understand
    what: "grounded in the atomic base" means is less
    than no rebuttal at all.

    "grounded in the atomic base of PA" is an expression used only >>>>>>>>> by you, and it is one which you have never explicitly defined, >>>>>>>>> so the fault here certainly doesn't lie with Alan. It's
    certainly not a 'verified fact' when you haven't even
    adequately explained what it is that you mean.

    All of knowledge expressed in language is structured as a tree >>>>>>>> of semantic relations specified syntactically between finite
    strings.

    What makes you believe semantic relations that can be structured as >>>>>>> a tree are sufficient to contain all knowledge that is exressed in >>>>>>> some language?

    The CycL language and the Cyc Project.

    They use a tree structure for concepts. But why would one try to
    put knowledge in a tree structure?

    It must at least be a directed acyclic graph or
    the proof gets stuck in an infinite loop and never
    completes.

    How can any ordering of knowledge prevent getting stuck in a loop
    when looking for a proof?

    By looking upward in a type hierarchy.

    If you mean not looking elsewhere that may indeed prevent loops.
    In most cases that also prevents finding the proof.


    Truth Conditional Semantics (TCS) <is> incoherent
    compared to Proof Theoretic Semantics (PTS). Essentially
    PTS just coherently connects the semantic meanings
    expressed in language together into one coherent body
    of general knowledge. It does this without undecidability
    or mathematical incompleteness.
    --
    Copyright 2026 Olcott

    My 28 year goal has been to make
    "true on the basis of meaning expressed in language"
    reliably computable for the entire body of knowledge.
    The complete structure of this system is now defined.

    The entire body of knowledge expressed in language is
    comprised of two types of relations between finite strings:
    (a) *Axioms* Expressions of language that are stipulated to be true.

    My system bridges the analytic/synthetic distinction by
    expressly encoding all empirical "atomic facts" in a formal
    language such as CycL of the Cyc project.

    (b) *Inference Rules* Expressions of language that are semantically
    entailed syntactically from (a) and/or (b).
    --- Synchronet 3.22a-Linux NewsLink 1.2
  • From Mikko@mikko.levanto@iki.fi to comp.theory,sci.logic,sci.lang,comp.ai.philosophy,sci.math on Fri Jun 26 09:39:59 2026
    From Newsgroup: sci.lang

    On 25/06/2026 19:14, olcott wrote:
    On 6/25/2026 2:21 AM, Mikko wrote:
    On 24/06/2026 23:26, olcott wrote:
    On 6/24/2026 5:00 AM, Mikko wrote:
    On 23/06/2026 17:48, olcott wrote:
    On 6/23/2026 1:06 AM, Mikko wrote:
    On 22/06/2026 15:10, olcott wrote:
    On 6/22/2026 1:49 AM, Mikko wrote:
    On 22/06/2026 02:02, olcott wrote:
    On 6/21/2026 4:08 PM, Andr|- G. Isaak wrote:
    On 2026-06-21 14:42, olcott wrote:
    On 6/21/2026 3:04 PM, Alan Mackenzie wrote:
    [ Followup-To: set ]

    In comp.theory olcott <polcott333@gmail.com> wrote:
    On 6/21/2026 6:26 AM, Alan Mackenzie wrote:
    In comp.theory olcott <polcott333@gmail.com> wrote: >>>>>>>>>>>>>>> I just found the term:
    "grounding in a proof theoretic atomic base" yesterday. >>>>>>>>>>>>
    You can find any number of terms.-a That doesn't mean >>>>>>>>>>>>>> you're capable of
    understanding them.


    The above is the key reason why under PTS G||del 1931 >>>>>>>>>>>>> incompleteness
    fails.

    I don't believe you.-a You have no respect for or
    understanding of the
    truth.-a If you really want to persuade anybody that PTS >>>>>>>>>>>> somehow causes
    G||del's theorem not to hold, then cite an academic expert >>>>>>>>>>>> who'll have
    some credibility.

    If they are mere gibberish words to you then you will not >>>>>>>>>>>>> understand.

    You don't understand Proof-theoritic Semantics, and you >>>>>>>>>>>> certainly don't
    understand G||del's Theorem, neither the theorem itself nor >>>>>>>>>>>> any proof of
    it.

    It is a verified fact that G||del's G is ungrounded
    in the atomic base of PA. That you do not understand
    what: "grounded in the atomic base" means is less
    than no rebuttal at all.

    "grounded in the atomic base of PA" is an expression used only >>>>>>>>>> by you, and it is one which you have never explicitly defined, >>>>>>>>>> so the fault here certainly doesn't lie with Alan. It's
    certainly not a 'verified fact' when you haven't even
    adequately explained what it is that you mean.

    All of knowledge expressed in language is structured as a tree >>>>>>>>> of semantic relations specified syntactically between finite >>>>>>>>> strings.

    What makes you believe semantic relations that can be structured as >>>>>>>> a tree are sufficient to contain all knowledge that is exressed in >>>>>>>> some language?

    The CycL language and the Cyc Project.

    They use a tree structure for concepts. But why would one try to
    put knowledge in a tree structure?

    It must at least be a directed acyclic graph or
    the proof gets stuck in an infinite loop and never
    completes.

    How can any ordering of knowledge prevent getting stuck in a loop
    when looking for a proof?

    By looking upward in a type hierarchy.

    If you mean not looking elsewhere that may indeed prevent loops.
    In most cases that also prevents finding the proof.

    Truth Conditional Semantics (TCS) <is> incoherent
    compared to Proof Theoretic Semantics (PTS). Essentially
    PTS just coherently connects the semantic meanings
    expressed in language together into one coherent body
    of general knowledge. It does this without undecidability
    or mathematical incompleteness.

    Looking for a proof does not need any semantics so it is not obvious
    how switching to another semantics could improve it.
    --
    Mikko
    --- Synchronet 3.22a-Linux NewsLink 1.2