On 12/04/2024 02:12 PM, Jim Burns wrote:
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:
>
[...]
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See below for
⎜ not.first.false claim [1]:
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⎜ not.first.false claim [2]:
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Yet, I think that I've always been
both forthcoming and forthright
in providing answers, and context,
in this loooong conversation [...]
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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.
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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.
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'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'.
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It is in part the absence of 'ad infinitum'
which justifies claims such as [1] and [2]
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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.
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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.
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So, not.first.false,
>
is an attribute of claims in
a finite sequence of claims.
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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",
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It is not a supertask to make
finitely.many claims.
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It is not a supertask to verify
the visibly not.first.false status of
finitely.many.claims.
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Consider the possibility that
what you think I am saying and
what I actually am saying
are not the same.
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>
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----
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.
>
>
About your posited point of detail, or question,
about this yin-yang infinitum, which is non-inductive,
and a neat also graphical example of the non-inductive,
a counter-example to the naively inductive, as with
regards to whether it's not so at some finite or
not ultimately untrue, I'd aver that it introduces
a notion of "arrival" at "the trans-finite case",
which is otherwise just another sort of stipulation,
as with regards to trans-finite induction (like
in-finite induction, if induction can be called
that, except that there's a _case_ given as via
one of deduction or raw, baseless, ungrounded stipulation,
the trans-finite induction's base-cases for each limit
ordinal >= omega and also each case for each ordinal > omega),
has that _arrival_ is what this deductive account provides.
This isn't even saying that "the point at infinity"
is a limit ordinal, i.e. with no "immediate" predecessor,
at all, with regards to its, "eventual", or "imminent",
say, predecessors.
Anyways your point stands that "not.first.false" is
not necessarily "not.ultimately.untrue", and so
does _not_ decide the outcome. Which is, ...,
not.ultimately.untrue.
Of course counting down or counting backward from
limit ordinals is pretty simple with things like
these "infinite-middle" and so on, where for example
there's wavelets as they may be that radiate from
all the 2^w points on this diameter resulting, in
the limit, and not.ultimately.untrue, the circle.
I.e., if Cantor picked up then tossed away the
notion of "counting backward", it's just as simple
to make it with "pairing", there are ways. Of course
uncountability is a profound fact, about an ordinary
infinity which may or may not exist (except as
unbounded fragments and extra-ordinary extensions).
Indeed it sort of rathers seems to demand this
the reasoning.
The il-lative, is sort of after the sub-lative,
that a whole can't have anything taken away and
remain whole, kind of the opposite of that, then
what of course demands its _own_ justification
otherwise, merely freeing deductive inference
about the spaces of things and geometrically usually
from "false not.first.false".
Also, it's like "wow, fantastic, you've reduced
your inductive cases to one inductive case,
now at worst, it will only be last.false".
Re: Incompleteness of Cantor's enumeration of the rational numbers (extra-ordinary)
Re: Incompleteness of Cantor's enumeration of the rational numbers (extra-ordinary)