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

On 12/4/2024 4:39 PM, Ross Finlayson wrote:

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

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

[...]

See below for

⎜ not.first.false claim [1]:

⎜ not.first.false claim [2]:

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 [predecessors],
but, from ω onward, it's negation is true.

Thank you in advance for confirming or correcting
my impression of what you mean
(something you have not yet done),
in furtherance of your
forthcoming and forthright posting history.

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.

So, not.first.false,

is an attribute of claims in
a finite sequence of claims.

So, not.first.false, is only
after some pair-wise comprehension,
because, there are ready example that
(which you have not yet done)
in "super-task comprehension",

It is not a supertask to make
finitely.many claims.
It is not a supertask to verify
the visibly not.first.false status of
finitely.many.claims.
Consider the possibility that
what you think I am saying and
what I actually am saying
are not the same.
----

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.