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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 11:19:16 -0700
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On 04/26/2024 10:58 AM, Richard Damon wrote:
> On 4/26/24 1:15 PM, olcott wrote:
>> On 4/26/2024 11:38 AM, Ross Finlayson wrote:
>>> 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, ...".
>>>
>>
>> ...14 Every epistemological antinomy can likewise be used for a
>> similar undecidability proof...(Gödel 1931:43-44)
>>
>> epistemological antinomies cannot be true or false thus cannot
>> be propositions that must be true or false.
>>
>
> Right, and Godel doesn't claim it is.
>
> Your problem is you just don't understand what he is saying here,
> because you don't understand the meaning of the phase "can likewise be
> used for a similar undecidability proof"
>
> The Epistemological Antinomy is NOT used as the final proposition that
> needs to be proven, but provides the structural form, to construct
> ANOTHER syntactically similar, but semantically different statement,
> that is then used to build the Primitive Recursive Relationship based on
> it, that forms the statement G.
>
> Your skipping those steps just makes your arguement incorrect.
>
>
>> An undecidable sentence of a theory K is a closed wf ℬ of K such that
>> neither ℬ nor ¬ℬ is a theorem of K, that is, such that not-⊢K ℬ and
>> not-⊢K ¬ℬ. (Mendelson: 2015:208)
>>
>> Undecidable(K, ℬ) ≡ ∃ℬ ∈ K ((K ⊬ ℬ) ∧ (K ⊬ ¬ℬ))
>>
>> To hazard a guess about what you mean, or to precisely state exactly
>> what I mean there is no such ℬ in K because such a ℬ in K could not
>> be a proposition of K.
>>
>
> Right, and since Godel's G isn't the Epistemological Antinomy you think
> he is using, your argument just fails. G is, in fact, a statement that
> MUST have a truth value, as it is about the lack of existance of a
> finite number that matches a property, and such a number MUST either
> exist or not. If it exists, that existance proves G wrong via a finite
> sequence of steps to evaluate that property, and if the number doesn't
> exist, it is proven by the infinite number of steps of testing every one
> of the countably infinte numbers and testing them with a finite number
> of steps, and seeing that none of the satisfy the relationship.
>
> But, if the number doesn't exist, that method can NOT be used as a
> "Proof", as a proof must use a finite number of steps, and you can't
> individually check an infinite number of values in a finite proof. (You
> would need to find a induction or recussion method to reduce the steps
> to something finite, which might not exist).
>
> The fact that in the META theory, we can show that no such number can
> exist, using facts not present in the theory, so that proof doesn't move
> down to there, so we can demonstrate that G must be true, and by
> knowledge also in the meta-theory, we can show that no proof can exist
> in the theory to show it.
>
> Of course, since the concept of what a meta-theory seems to be foreign
> to you, as even what a "Formal System" is, this is beyond your
> understanding.

What if you take turns, ....

Meeting in the middle, "middle of nowhere".

The "inductive impasse" again helps reflect that there are
certain propositions that are true, yet, inductive inference
will never arrive at them, and not just because they're
non-constructive, yet because they reflect the essential
impasses that a line can not be made points,
and points, can not be made a line,
yet a line is as anywhere a points
and points are in as on a line.

So, making for an Aristotle's continuum and an Archimedean field,
is for both the standard infinitely-divisible Aristotle's and
Archimedes', and, a, "standard", infinitely-divided, Aristotle's
contiguous and Archimedes' not-a-field, Aristotle's other, usual,
arrived-at, model of a continuum or continuous domain,
in mathematics, an Archimedean ring of a bounded interval,
with rather-restricted transfer principle,
like Bishop and Cheng arrived at, con-structively.

Other usual examples include in logic: quantification
over unbounded domains, equality in unbounded domains,
membership in unbounded domains, not-membership in
unbounded domains, discernibility in unbounded domains,
and so on, predicativity and quantifier unambiguity
in unbounded domains.

What I'm saying is that both of you can, in a sense,
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