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Subject: Re: Undecidability based on epistemological antinomies V2
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From: Ross Finlayson <ross.a.finlayson@gmail.com>
Date: Fri, 26 Apr 2024 09:38:33 -0700
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On 04/26/2024 08:28 AM, olcott wrote:
> On 4/26/2024 3:42 AM, Mikko wrote:
>> On 2024-04-25 14:27:23 +0000, olcott said:
>>
>>> On 4/25/2024 3:26 AM, Mikko wrote:
>>>> epistemological antinomy
>>>
>>> It <is> part of the current (thus incorrect) definition
>>> of undecidability because expressions of language that
>>> are neither true nor false (epistemological antinomies)
>>> do prove undecidability even though these expressions
>>> are not truth bearers thus not propositions.
>>
>> That a definition is current does not mean that is incorrect.
>>
>
> ...14 Every epistemological antinomy can likewise be used for a similar
> undecidability proof...(Gödel 1931:43-44)
>
>> An epistemological antinomy can only be an undecidable sentence
>> if it can be a sentence. What epistemological antinomies you
>> can find that can be expressed in, say, first order goup theory
>> or first order arithmetic or first order set tehory?
>>
>
> It only matters that they can be expressed in some formal system.
> If they cannot be expressed in any formal system then Gödel is
> wrong for a different reason.
>
> Minimal Type Theory (YACC BNF)
> https://www.researchgate.net/publication/331859461_Minimal_Type_Theory_YACC_BNF
>
>
> I created MTT so that self-reference could be correctly represented
> it is conventional to represent self-reference incorrectly. MTT uses
> adapted FOL to express arbitrary orders of logic. When MTT expressions
> are translated into directed graphs a cycle in the graph proves that
> the expression is erroneous.
>
> Here is the Liar Paradox in MTT: LP := ~True(LP)
> 00 root (1)
> 01 ~    (2)
> 02 True (0) // cycle
> Same as ~True(~True(~True(~True(...))))
>
> In Prolog
> ?- LP = not(true(LP)).
> LP = not(true(LP)).
> ?- unify_with_occurs_check(LP, not(true(LP))).
> false.
> Indicates  ~True(~True(~True(~True(...))))
>
> In mathematical logic, a sentence (or closed formula)[1] of a predicate
> logic is a Boolean-valued well-formed formula with no free variables. A
> sentence can be viewed as expressing a proposition, something that must
> be true or false.
> https://en.wikipedia.org/wiki/Sentence_(mathematical_logic)
>
> By definition epistemological antinomies cannot be true or false thus
> cannot be logic sentences therefore Gödel is wrong.
>

Actually what results is that Goedel refers to a particular kind
of enforced, opinionated, retro-Russell ordinarity, that sees it
so that "logical paradox" of quantifier ambiguity or quantifier
impredicativity, resulting one of these one-way opinions, stipulations,
assumptions, non-logical axioms of restriction of comprehension,
makes it sort of like so for Goedel as "completeness, you know,
yet, incompleteness, ...".

Where the entire thing arrives as extra-ordinary, not necessarily
with any restrictions of comprehension or including the entire
universe of truth, dually null/universal, then Goedel's result
is not so, so, the only way Goedel's result is not so, is this way.

So, if you think that Goedel's results are not so, then, either
you are in a fragment where other usual things are not so,
other usual completions of things, or, you are in an extension,
either way not the "Standard Model" you expect, that in the
completeness of the replete regularity and completeness of
things, and in their consistency as "infra-consistency",
it's the one theory like so.

Otherwise about universals and particulars or the "upper
ontology" or "the Sowa debates" or these kinds of things,
results that while it's reasonable to want to have a
certum of verum, it's sort of not a thing necessarily
the ordinary way.

Some years ago somebody had the great idea that while
standard ordinary axiomatic set theory for descriptive
set theory which is our standard modern mathematics today
had pair-wise union, it didn't have infinite union, so
what they had in mind was to add a univalency axiom so
that what results was something like "the strength of ZFC
plus two large cardinals", ..., the illative or univalency,
vis-a-vis something like "projective determinacy in New
Foundations", "New Foundations with Ur-elements", "New
Foundations with Universes", you know, like Quine's Atoms
or Ultimate Classes, "ordinary ordinals", "Nelson, who
showed Internal Set Theory co-consistent with ZFC for
some usual results in standard infinitesimals, looking
at a reason why ZFC was inconsistent", that being another
kind of thing, these kinds of things.

Anyways then it was pretty easy to find a bunch of results
that had tacked on basically two sorts of regularity fighting
each other instead of resolving them as somehow replete together
from underneath, so, all sorts usual theorem-finders only
could come back with canceling each other out.

It seems much easier for the continuity and infinity and
being standard and all for real analytical character to
have it like so the line-reals, field-reals, signal-reals,
about Standard (Sparse), Square, and Signal Cantor space,
and not being Cartesian the Equivalency Function, then a
lot of the reasons why the univalent or illative were
desired because they result direct strokes for standard
real analysis and the line and path integral, all encumbered
these days in a hand-wavy language of complete metrizing ultrafilters,
all get resolved from the get-go and then mathematics and its
logic is quite a bit better and less limited as what's a fragment
or limited and hazardous as what's a false axiom of restriction or
hazardous as ambiguous under quantification.

It's a thing, ..., "ubiquitous ordinals", in a "Comenius language".

.... Numbers and words, with geometry arising naturally.

.... "Real" numbers.