Re: ZFC solution to incorrect questions: reject them

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.
--
Copyright 2024 Olcott "Talent hits a target no one else can hit; Genius
hits a target no one else can see." Arthur Schopenhauer