On 03/12/2024 01:28 PM, olcott wrote:
> On 3/12/2024 2:59 PM, Ross Finlayson wrote:
>> On 03/12/2024 12:32 PM, olcott wrote:
>>> On 3/12/2024 2:08 PM, Ross Finlayson wrote:
>>>> On 03/12/2024 08:52 AM, olcott wrote:
>>>>
>>>>> ∀ H ∈ Turing_Machine_Deciders
>>>>> ∃ TMD ∈ Turing_Machine_Descriptions |
>>>>> Predicted_Behavior(H, TMD) != Actual_Behavior(TMD)
>>>>>
>>>>> There is some input TMD to every H such that
>>>>> Predicted_Behavior(H, TMD) != Actual_Behavior(TMD)
>>>>>
>>>>> When we disallow decider/input pairs that are incorrect
>>>>> questions where both YES and NO are the wrong answer
>>>>> (the same way the ZFC disallowed self-referential sets) then
>>>>> pathological inputs are not allowed to come into existence.
>>>>>
>>>>> Does the barber that shaves everyone that does not shave
>>>>> themselves shave himself? is rejected as an incorrect question.
>>>>> https://en.wikipedia.org/wiki/Barber_paradox#
>>>>>
>>>>
>>>>
>>>> People learn about ZFC when their mathematical curiousity
>>>> brings them to questions of foundations.
>>>>
>>>> Then it's understood that it's an axiomatic theory,
>>>> and that axioms are rules, and in an axiomatic theory,
>>>> there result theorems derived from axioms, with axioms
>>>> themselves being considered theorems, and that no theorems
>>>> break any axioms.
>>>>
>>>> This is that axioms are somehow "true", and in the theory
>>>> they're defined to be true, meaning they're never contradicted.
>>>>
>>> That is a great insight that you and Haskell Curry and
>>> very few others agree on.
>>> https://www.liarparadox.org/Haskell_Curry_45.pdf
>>>
>>>> This is with the usual notion of contradiction and
>>>> non-contradiction, about opposition and juxtaposition,
>>>> where it's established usually that there is "true" or
>>>> there is "false" and there is no middle ground, that
>>>> a third case or tertium does not exist in the world,
>>>> "tertium non datur", the laws of excluded middle, the
>>>> principle of excluded middle, which in this axiomatic
>>>> theory, is somehow always a theorem, as it results
>>>> from the plain contemplation or consideration, that
>>>> axioms are "true", in the theory, in what is almost
>>>> always these days, a "meta-theory", that's undefined,
>>>> except that axioms are true and none of their theorems
>>>> contradict each other, saying "both true and false",
>>>> which is tertium and non datur.
>>>>
>>>>
>>>> So anyways ZFC is a theory where there's only one relation,
>>>> it's "elt". There's only one kind of object, it's "set".
>>>
>>> I have no idea what "elt" means.
>>>
>>>> For any given set P and any given set Q, either P elt Q
>>>> or Q elt P, or neither, and, not both. Then you might
>>>> wonder, "well why not both?", and it's because, one of
>>>> the axioms of ZFC is "not both".
>>>>
>>>> The axioms of ZFC either _expand_ comprehension, meaning,
>>>> "no reason why not, so, it's so", or _restrict_ comprehension,
>>>> meaning, "not: because this axiom is true in this theory,
>>>> and says no".
>>>>
>>>> This introduces the concept of "independence" of axioms,
>>>> that axioms that are independent say nothing about the
>>>> theorems of otherwise independent axioms, and that axioms
>>>> that are not independent, contradict each other, and that
>>>> restriction is defined to always win, in any case of otherwise
>>>> contradiction, when axioms aren't independent, in ZFC,
>>>> that axioms of _restriction_ of comprehension aren't
>>>> necessarily independent each other, or, the independent
>>>> axioms of _expansion_ of comprehension.
>>>>
>>>>
>>>>
>>>> So, ZFC has various axioms of restriction of comprehension,
>>> Yes, no set can be defined that contains itself.
>>>
>>>> what boil down to the "Axiom of Regularity" also known as
>>>> the "Axiom of Well-Foundedness" or "Axiom of Foundation",
>>> Yes that one
>>>
>>>> that for any two sets P elt Q or Q elt P, or neither,
>>>> but not both. This is where otherwise the axioms of
>>>> expansion of comprehension, would otherwise result,
>>>> "no reason why not", except that eventually certain
>>>> theorems _desired_, of the theory, would either not be
>>>> evident or would see contradictions.
>>>>
>>>>
>>>> So, yeah, "ZFC solution to incorrect questions: reject them",
>>>> is what's called "restriction of comprehension" and then
>>> Great !!!
>>>
>>>> what you do is get into all the various combinations of
>>>> otherwise the expansion of comprehension, then get into
>>>> why the models of the universe of the objects so related,
>>>> is a wider theory where ZFC, or ZF, set theory, is variously
>>>> considerable as either a fragment or an extension,
>>>> the universe of the objects of ZF and ZFC set theories,
>>>> in all theory in all comprehension according to what's, "true".
>>>>
>>>>
>>>> Or, you know, "not false".
>>>>
>>>>
>>>>
>>>> So of course there are names for all these things and
>>>> studies of all these things and criteria for all these
>>>> things, what basically results for "Set Theory" what's
>>>> called "Class/Set Distinction", a sort of, meta-theory,
>>> NBG set theory
>>> https://www.britannica.com/science/proper-class
>>>
>>>> about set theory, where "elt" has a sort of complement
>>>> "members" reflects "elt's sets are contained in sets"
>>>> while "members' classes contain classes", that also the
>>>> Class/Set distinction reflects objects as of the,
>>>> "Inconsistent Multiplicities", of set theory, that
>>>> one can relate to the, "Indeterminate Forms", of
>>>> mathematics, that variously kind of do or don't have
>>>> structure, "models" in the "model theory", where a
>>>> theory has a model and a model has a theory is the meta-theory,
>>>> helping explain why the wider world of theory knows that
>>>> ZFC, or ZF, set theory, is a fragment of the universe of
>>>> the objects of ZF set theory, which is its model in
>>>> the wider model theory,
>>>>
>>>> Theory of ZF, of course, doesn't actually acknowledge,
>>>> "a universe of objects of ZF, the domain of discourse"
>>>> not so much as it's axiomatic "there is no universe of
>>>> objects in ZF set theory", but that it's a theorem that's
>>>> a consequence of restriction of comprehension, "foundational
>>>> well-foundedness", which follows from otherwise a very
>>>> useful result called "uncountability", .
>>>>
>>>> Now, "Regularity" means "the Rulial", it rules or defines
>>>> a rule, so other theories otherwise about sets can have
>>>> their own sorts rulial definitions, just saying that the
>>>> theory where Well-Foundedness is rulial, just indicates
>>>> that this is moreso "ZF's axiom that establishes ZF's
>>>> main restriction of comprehension as ruliality, AoR
>>>> the Axiom of Regularity, is particular to ZF, and it's
>>>> called Well-Foundedness or Foundation, to reflect that
>>>> theories without it are called Non-Well-Founded or
>>>> sometimes Anti-Well-Founded, with regards to the regular
>>> Yes I get that and have known about it for some years.
>>>
>>>> or rulial the ruliality of what may be other theories,
>>>> of sets, which are defined by one relation, elt".
>>>>
>>>>
>>>>
>>>> So anyways, there are others.
>>>
>>> *Your knowledge of these things seem truly superb*
>>> I had forgotten many of the details that you referenced.
>>>
>>
>> Well, yeah, my mathematical curiousity brought me
>> to questions of foundations.
>>
> I love foundations because I noticed errors in the understanding
> of the notion of true_on_the_basis_of_semantic_meaning(x) back
> in 2004.
>
>> "Question" is a word, and it's kind of loaded. For
>> example, the German language has two different words
>> for "question-able", "fraglich", as, dubious, and,
>> "question-providing", "fragwuerdig", as, profound.
>>
>> I.e., the profound, opens new questions, vis-a-vis
>> the interrogable, which may or may not.
>>
>> I'm reading about this in Steiner on Heidegger as
>> of from old Roger Bacon. (I've sort of got a
>> trio of anti-Plato's in Wittgenstein, Nietzsche,
>> and Heidegger as various rejections of logical
>> positivism in the accommodation of logical positivism
> The coherent notion of true_on_the_basis_of_semantic_meaning(x)
> seems to affirm logical positivism over Gödel.
>
>> after their compensation in search of teleology
>> after the breakwater of ontology, that of course
>> they're each strong Platonists in otherwise a
>> world of mundane, subjective, inauthentic,
>> Existentialists, the adrift logical positivists.
>> Of course that's for a strong logical positivism
>> overall, with re-attaching the silver thread, or cord.)
>>
>>
>>
>> The universe of logical and mathematical objects
>> is its own, intuitively structured, thing. All
>
> Yes, hence true_on_the_basis_of_semantic_meaning(x) does
> not apply to reality only mental models of reality.
>
>> matters of relation are assumed to exist in it,
>> both the concrete as realizable mathematically,
>> and of course all suchly matters of mathematical
>> or logical consistency, and inconsistency, so effable,
>> or ineffable.
>>
> Perhaps with true_on_the_basis_of_semantic_meaning(x)
> it becomes effable.
>
>> Then, humans or objectively thinkers, or course have
>> limited or finite means, and, communication has its
>> own finite yet open means.
>>
> Yet with Montague semantics can be formalized to eliminate
> all ambiguity.
>
>>
>>
>> I thank you for your compliments, affinity,
>> then would suggest that you look to the fuller
>> complements, complementarity, as what such
>> notions of the fuller dialectic, arrive largely
>> as fundamentally from "complementary duals",
>> that not only is something filled, also filling.
>>
> I can only understand those aspects of my ideas that you
> directly referenced that I acknowledged that I understood.
> This was much more tan I expected to understand. I had
> forgotten some of the key details that you referenced.
>
>> I.e., "is there an axiom?" "Inverse".
>>
>> Or, you know, yes and no.
>>
>>
>>
>> You can find some hundred hours or readings
>> and lectures on my youtube channel as of
>> like https youtube /@rossfinlayson .
>>
>> Or, you know, the 10,000's posts to sci.math.
>