Re: Incompleteness of Cantor's enumeration of the rational numbers (extra-standard)

On 11/16/2024 10:11 PM, Ross Finlayson wrote:

On 11/16/2024 09:59 PM, Ross Finlayson wrote:

On 11/16/2024 09:56 PM, Jim Burns wrote:

On 11/17/2024 12:30 AM, Ross Finlayson wrote:

On 11/16/2024 08:57 PM, Jim Burns wrote:

On 11/16/2024 11:35 PM, Ross Finlayson wrote:

On 11/16/2024 08:18 PM, Jim Burns wrote:

On 11/16/2024 7:17 PM, Ross Finlayson wrote:

On 11/16/2024 02:46 PM, Jim Burns wrote:

On 11/16/2024 5:31 PM, Ross Finlayson wrote:

On 11/16/2024 12:29 PM, Jim Burns wrote:

On 11/16/2024 12:07 PM, Ross Finlayson wrote:

>

you have ignored Russell his paradox and so on

>
No.
>
Selecting axioms which do not suffer from claiming
  that the set of all non.self.membered sets
  is self.membered or claiming it isn't
is not ignoring Russel,
it is responding to Russell.
>
Russell points out that
_we do not want_ to claim
that the set of all non.self.membered sets
  is self.membered or claiming it isn't.
>
We respond: Okay, we'll stop doing that.

>

Bzzt, flake-out.
It's not pretty the act of making lies.

>

Quote what I wrote which you think is a lie.

>
"We respond"

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⎛ The modern study of set theory was initiated by
⎜ Georg Cantor and Richard Dedekind in the 1870s.
⎜ However,
⎜ the discovery of paradoxes in naive set theory,
⎜ such as Russell's paradox,
⎜ led to the desire for
⎜ a more rigorous form of set theory
⎝ that was free of these paradoxes.
>
https://en.wikipedia.org/wiki/Zermelo%E2%80%93Fraenkel_set_theory
>
Got anything else?
>
>

>
How about Finsler and Boffa?
>
>

>
Of course I say "Russell's retro-thesis" is a lie itself.
>
That is to say, when, there aren't yet infinite collections gesammelt,
that's of those the whole set of those that don't contain themselves,
that naturally the infinite inductive set by expansion of comprehension
does contain itself, as plainly a fact that there are no standard
models of integers instead only extensions or fragments, thusly,
saying that it's both complete and ordinary, is a lie, it's
a falsehood, and right after saying the other way: it's a slap
in the face.
>
>

Looking back over research a few decades,
some decades ago the only surface results
for "anti-foundational set theory" were Aczel's,
who had a little industry in a modest and
non-offensive-to-some-sensibilities non-
well-founded set theory.
It took quite a bit of actual research into
the literature to find things like Finsler and
Boffa and their more-or-less completely
separate approaches to non-well-founded
set theory, a theory with one relation "elt",
as from their times.
Then, not so many years ago there was
Scott and with Lando about circle and box
modality, this being about the univalent
and illative and that not getting a really
great reception though it's sort of good,
as it's more than a "conservative" extension
of set theory.
Yet, these days at least, non-well-founded
set theory has a little Wiki page.
https://en.wikipedia.org/wiki/Non-well-founded_set_theory
It's gratifying that it mentions Mirimanoff
right away, who in the era of ZF's origination,
pretty much _the very next day_ came up
with "extra-ordinary", and it's a thing,
and should be of the surface and part-and-parcel
with these considerations, because "comprehension".
Then, when you get to Cohen forcing, who
employs ordinals and order theory to prove
the independence of CH because otherwise
it would be consistent either way and thusly
those two ways contradict each other, that
it's a little cap on ordinary, regular, well-founded
set theory, which would otherwise prove
two opposite things.
So, ubiquitous ordinals are a thing, and,
there's a model of set theory where
it's inconsistent for there to be an
ordinary, standard, well-founded,
infinite inductive set.
It's called "Russell's paradox", yet
really, it's not a lie.
Then here there's also that
"not.first.false" is not necessarily
"not.ultimately.untrue".