Re: Replacement of Cardinality (infinite middle)

On 08/12/2024 06:06 PM, Ross Finlayson wrote:

On 08/12/2024 05:44 PM, Python wrote:

Le 13/08/2024 à 02:28, Ross Finlayson a écrit :

On 08/12/2024 04:06 PM, Jim Burns wrote:

On 8/12/2024 4:59 PM, Ross Finlayson wrote:

On 08/11/2024 09:44 PM, Jim Burns wrote:

On 8/11/2024 7:39 PM, Ross Finlayson wrote:

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Starting with a theory _without_
the constant introduced named omega,
i.e., finite sets,

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For P(z),
use a description 𝕆ᶠⁱⁿ(z) of a finite ordinal,
and ω := {z:𝕆ᶠⁱⁿ(z)} exists
>
For example, use
𝕆ᶠⁱⁿ(z)  ⇔
(z ∋ {} ∧ ∀y ∈ z+1: y≠{} ⇒ ∃x∈z: x+1=y)
∨ (z = {})
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z+1 = z∪{z}

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Then, omega, as you've defined it,

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ω := {z:𝕆ᶠⁱⁿ(z)}
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contains itself,

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I'm curious, now that you have
a beginning and an end of
the finite, or 0 and omega in ZF,

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ω is the least.upper.bound of the finites.
ω is not a finite.
ω is not the upper.end of the finites.
The upper.end of the finites doesn't exist.

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Here though

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_Where_ though?
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it's beginning ... ( ... infinitely-many ...) ... end,
where the upper.end of the finites always exists.

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For ω as I've defined it, no upper.end exists.
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for each k ∈ ω
𝕆ᶠⁱⁿ(k)
𝕆ᶠⁱⁿ(k+1)
k+1 ∈ ω
k is not the upper end of ω
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for each k ∉ ω
k is not the upper end of ω
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Then you claim to have
an axiom of restriction of comprehension of the finites

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To review:
What I claim is
⎛ ∃{}
⎜ ∀x∀y∃z=x∪{y}
⎝ and extensionality
⎛ ∃∃xx={z:P(z)}: ∀y: y ∈ {z:P(z)} ⇔ P(y)
⎝ and extensionality
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∃∃{z:P(z)} is unrestricted comprehension.
Unless we are no longer uninterested in what words mean.
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unless Russell grants you
a dispensation of Russell's retro-thesis,
and say it's always so for others, too,
congratulations,
you claim to have invented a mathematics
where you = Russell.

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Ah.
I've seen this one before.
Your tacit thesis is that
it is preferable to disagree with the Old Ones
even at the cost of being wrong.
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Well, it's a choice.
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>

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Oh, I have the entire canon here along.
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It's like yesterday, in this thread with the subject
of it talking about "infinite in the middle and always
with both ends", or, "here...", pointing out that some
modern philosophers with their Ph.D.s. resuscitate a
metaphysics that Compte and Boole and Russell and Carnap
made so nice for Marx and nihilism and extistentialism
and the sort of post-modern that begets itself.
>
>
Now, I confiscate logical positivism from Compte and
brush off Boole for De Morgan and point out Russell
and for example Whitehead suffer their own arguments
and Carnap was quite a pleasant fellow and I like Quine
yet I'm not a nominalist fictionalist. So, a stronger
logical positivism and the ontological is kept with
a strong mathematical platonism and teleological.
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Talking about "the Old Ones", you still got Zeno
shaking his head and pointing at his watch.
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Furthermore, I'm a constructivist and agree with
notions like infinite induction already and as there's
already, for example a sort of ubiquitous ordinals,
and even a sort of axiomless natural deduction seated
in reason.
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The, "material implication", or, "ex falso quodlibet",
has that material implication is neither material nor implication,
and ex falso is mistakes or lies.
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Some kinds of strong constructivists don't accept
non-constructive proofs, for example via contradiction,
de dicto, at all.
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Here though it's just "modular: always modular,
of integral wholes, infinite in the middle, modular",
just so different from "and a 1 and on down and a 2
and on down and a 3 and on down and ... an omega and
on down", or, you know, not so.
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See, "modularity" is regular, rulial, in both
increment and dispersion.
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... Which most have as properties of integers
as with regards to associates with magnitudes,
or measures.
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Heh, you brought up "The Old Ones", it's like,
what did the librarian or book-keeper say
when the paranoiac asked for self-help books,
"they're right behind you".
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So, for example, Anderson's Relevance Logic many
have as more relevant than the quasi-modal, which
is neither temporal nor modal, like De Morgan's is,
with direct implication, there are some fans of Dana Scott
and not for his coat-tailing and wall-papering,
the theory of types is often attributed to Peirce,
the completeness theorems of arithmetic often to Frege,
the extra-ordinary of set theory to Mirimanoff and also
a bit to Quine about ultimate classes, you keep the
Vienna Circle, and, I'll stick with the larger, fuller canon.
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Not that there's anything wrong with that, .....
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>

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Seriously Ross. What's the point of posting nonsense from
start to finish?
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>

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If you don't know your history
someone's bound to try and re-write it for you.
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I released a new pod-cast the other day, you could
put a voice-stress analysis on it and see if it's
perceived veracity.
https://www.youtube.com/@rossfinlayson
Wouldn't that be nice, an un-obtrusive red or green
dot on your screen indicating when someone's either
lying to your face or entirely empty?
Perhaps you might put its transcript to one of these
modern mechanical thinking apparatuses and see whether
it finds anything of value or exactly what it doesn't agree.
Though I suppose you could always down a bottle of Xanax
and some booze and not worry much either way except
for perhaps the headache the next day, or otherwise
how to reclaim the brain from a pool of re-uptake inhibitors.
On the idiot scale, damnit I'm not going to beat your wife
for you. And you can take the entire slate of rhetorical fallacies
and wash it on down with some "I told you so".