Re: Incompleteness of Cantor's enumeration of the rational numbers (extra-ordinary, not.ultimately.untrue)

On 12/04/2024 11:37 AM, Jim Burns wrote:

On 12/3/2024 8:09 PM, Ross Finlayson wrote:

On 12/03/2024 04:16 PM, Jim Burns wrote:

>

[...]

>
it was very brave of you when you admitted that
"not.first.false"
is not so much justifying itself and
not un-justifying itself,

>
Not.first.false claims are justified when
they are in finite sequences of such claims,
in which each claim is true.or.not.first.false.
>
You (RF) occasionally attribute to me (JB)
the most surprising things.
Not only do I not recognize them as mine,
but they are so distant from mine that
I can't guess what you've misunderstood.
>

with regards to the "yin-yang ad infinitum",
which inductively is a constant
yet in its completion is different,

>
This description of "yin-yang ad infinitum"
suggests to me that
you are describing what the claims _are about_
whereas 'not.first.false' describes
the claims _themselves_
>

with regards to the "yin-yang ad infinitum",
which inductively is a constant
yet in its completion is different,

>
Consider this finite sequence of claims
⎛⎛ By 'ordinals', we mean those which
⎜⎜ have only sets.with.minimums and {}
⎜⎝ ('well.ordered')
⎜
⎜⎛ By 'natural numbers', we mean those which
⎜⎜ have a successor,
⎜⎜ are a successor or 0,  and
⎜⎝ are an ordinal.
⎜
⎜ (not.first.false claim)
⎜
⎜ (not.first.false claim)
⎜
⎜ (not.first.false claim)
⎜
⎜ ...
⎜
⎜ not.first.false claim [1]:
⎜⎛ Each non.zero natural number
⎜⎜ has,
⎜⎜ for it and for each of its non.zero priors,
⎜⎝ an immediate ordinal.predecessor.
⎜
⎜ not.first.false claim [2]:
⎜⎛ The first transfinite ordinal, which we name 'ω',
⎜⎜ and each of its ordinal.followers
⎜⎜ does not have,
⎜⎜ for it and for each of its non.zero priors,
⎜⎜ an immediate ordinal.predecessor.
⎜⎜ That is,
⎜⎜ there is a non.zero prior without
⎝⎝ an immediate ordinal.predecessor.
>
----

Yet, I think that I've always been
both forthcoming and forthright
in providing answers, and context, in
this loooong conversation [...]

>
Please continue being forthcoming and forthright
by confirming or correcting my impression that
"yin-yang ad infinitum"
refers to how, up to ω, claim [1] is true,
about immediate precessors,
but, from ω onward, it's negation is true.
>
The thing is,
'not.first.false' is not used to describe ordinals,
in the way that 'yin.yang.ad.infinitum'
is used to describe ordinals.
>
'Not.first.false' is used to describe
_claims about ordinals_ of which we are
  here only concerned with finitely.many claims.
There is no 'ad infinitum' for 'not.first.false'.
>
It is in part the absence of 'ad infinitum'
which justifies claims such as [1] and [2]
>
A linearly.ordered _finite_ set must be well.ordered.
If all claims are true.or.not.first.false,
there is no first false claim.
Because well.ordered,
if there is no first false,
then there is no false,
and all those not.first.false claims are justified.
>
The natural numbers are not finitely.many.
But that isn't a problem for this argument,
because it isn't the finiteness of the _numbers_
which it depends upon,
but the finiteness of the claim.sequence.
>
>

Oh, well, tertium non datur or PEM, principle
of excluded middle, LEM, law of excluded middle,
TND, no third ground, have that inductive accounts
may _not_ bring their own completeness, if what
you'd rather is an inductive account that no
inductive account is not.ultimately.untrue.
Which is well known since antiquity and also
as to why the only thing left is for metaphysics
and foundations to arrive at a theory as with
regards to not making an inductive account
after its implied un-founded (and often un-stated)
assumptions - we should figure out here for whom
it's called that any theory with finitely many
axioms is having a neat, and brief, formal counterexample,
and it's the same for each of them, though you
can point at Goedel with regards to complete/not-complete
these what are _partial_, at best, inductive accounts.
(At best, ....)
So, regularity is still a thing of course, and
monotonic entailment, and for a theory with a modality,
yet anything mathematical introduces itself to a
great relevant concern called "the domain of discourse"
or "the universe of mathematical objects", and then
there's Zeno again "hey, how about a brief discourse
on metaphysics?"
Also a physics, ..., and "the theory".
So, not.first.false, is only after some pair-wise
comprehension, because, there are ready example
that in "super-task comprehension", or what's
called the illative when it's correct and completes
and what's usually called "undefined" when it doesn't,
lazy positivists, it's that not.first.false is either
not.ultimately.untrue, or it's not.
And sometimes (some times, a modal account), ..., it's not.
Of course there are then _reasons_ _why_
what is so is so.