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NNTP-Posting-Date: Sat, 16 Nov 2024 04:08:21 +0000
Subject: Re: Incompleteness of Cantor's enumeration of the rational numbers
 (extra-standard)
Newsgroups: sci.math
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From: Ross Finlayson <ross.a.finlayson@gmail.com>
Date: Fri, 15 Nov 2024 20:08:27 -0800
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On 11/15/2024 06:52 PM, Ross Finlayson wrote:
> On 11/15/2024 02:37 PM, Jim Burns wrote:
>> If it is true that
>> our domain of discourse is a model of ST+PQ
>> then it is true that
>> our domain of discourse holds a standard integer.model.
>>
>> What is Mirimanoff's argument that
>> it doesn't exist?
>
> Mirimanoff's? Russell's Paradox.
>
>
> That "If it is true that our domain of discourse
> is a model of ST+PQ then it is true that our
> domain of discourse holds a standard integer.model"
> is a pretty long axiom - why not just say
> "infinity", that's the usual approach.
>
>
>
>


People want to learn philosophy and theory,
then it's like "well you have to study Kant,
he was considered the best". Then it's like,
"well what's his point?", and there's, "well,
he's got that there's in a thing in itself, and
there's a thing bigger than itself". "Well that
doesn't seem according to those words ..."
and it's like "yeah some people don't get it".

It's fair to say for some people "there's no
infinity", and that's sane, "logically consistent",
yet, it's also small. It's finger counting, and
anybody can do it, and it's sane, and it's small.

Then, we know since ancient times that there's
either Archimedean "infinitely-many and no
infinitely-grand", and that's sane, and then
there's Democritus, "infinitely-small and
makes one whole", that's sane, and then
there's Euclid, "it's axiomatized if necessary
between any two points is a shortest, straightest
line", that's sane.

Then it's like, "well I put together Euclid and
Democritus and Archimedes, now there's
infinitely-grand to be infinitely-small",
then it's like "well, now you see how Kant
is stood up as a great philosopher".

Then it's like "you point at Duns Scotus,
he read enough Aristotle to put together
Aristotle on Archimedes and Democritus
and Euclid because of Zeno, so, infinity is
in".

How can that all be sane? Well, it's simple,
it's called "resolving the paradoxes of logic",
and it involves a dialectic from either side
of "so" and "not so", "finite and infinite",
why there's a theory and philosophy at
all, that's sane.

Then, for people who haven't gone through
all that, then it's like, "well, there's Kant",
"according to Kant's idealism there's a sane
infinity even if it's non-standard". I.e. they
lean on that in case not having all together
made a theory for themselves from the axiomless
natural deduction how it's arrived at.

Then there's that, and it's like, "well how's that",
then it's like, "well, you see there's Hegel, and,
he refers to Kant, then points out there's nothing,
then he gives it a proper name, Nothing, and
then it's a reflection on points and their spaces of
geometry as continuous and infinite and words
and their spaces of algebra as finite and discrete
and then that's enough sane for pretty much all of it".