Re: Undecidability based on epistemological antinomies V2 --Mendelson--

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:
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On 4/25/2024 3:26 AM, Mikko wrote:

epistemological antinomy

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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.

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That a definition is current does not mean that is incorrect.
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...14 Every epistemological antinomy can likewise be used for a similar
undecidability proof...(Gödel 1931:43-44)
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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?
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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.
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Minimal Type Theory (YACC BNF)
https://www.researchgate.net/publication/331859461_Minimal_Type_Theory_YACC_BNF
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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.
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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(...))))
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In Prolog
?- LP = not(true(LP)).
LP = not(true(LP)).
?- unify_with_occurs_check(LP, not(true(LP))).
false.
Indicates  ~True(~True(~True(~True(...))))
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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)
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By definition epistemological antinomies cannot be true or false thus
cannot be logic sentences therefore Gödel is wrong.
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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, ...".
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...14 Every epistemological antinomy can likewise be used for a
similar undecidability proof...(Gödel 1931:43-44)
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epistemological antinomies cannot be true or false thus cannot
be propositions that must be true or false.
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Right, and Godel doesn't claim it is.
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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"
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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.
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Your skipping those steps just makes your arguement incorrect.
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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)
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Undecidable(K, ℬ) ≡ ∃ℬ ∈ K ((K ⊬ ℬ) ∧ (K ⊬ ¬ℬ))
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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.
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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.
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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).
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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.
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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,
be right, while, currently neither is complete.
(And that in a sense being incomplete is incorrect.)