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

On 11/17/2024 10:35 AM, Jim Burns wrote:

>

On 11/16/2024 02:22 AM, Jim Burns wrote:

On 11/15/2024 9:52 PM, Ross Finlayson wrote:

On 11/15/2024 02:37 PM, Jim Burns wrote:

On 11/15/2024 4:32 PM, Ross Finlayson wrote:

>

Ah, yet according to Mirimanoff,
there do not exist standard models of integers,

>
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.

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ST+PQ does not suffer from claiming
that the set of all non.self.membered sets
is self.membered or claiming it isn't.
>
While I am at it,
ZFC does not suffer from claiming
that the set of all non.self.membered sets
is self.membered or claiming it isn't,
and
ordinal.theory=Well.Order
does not suffer from claiming
that the set of all non.self.membered sets
is self.membered or claiming it isn't.

>

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

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you have ignored Russell his paradox and so on

>
On 11/17/2024 1:52 AM, Ross Finlayson wrote:

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:

>

⎛ 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

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How about Finsler and Boffa?

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Meaning:
how about non.well.founded set theories?
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⎛ I learned a new word recently: 'sanewashing'.
⎜
⎜ For example,
⎜ it is sanewashing
⎜ when I snip context in which
⎜ you call me a liar for stating facts
⎜ accessible to Wikipedia.level research.
⎜
⎜ I don't wish to dwell on my sanewashing.
⎜ I doubt that anyone beyond you (RF) or me
⎜ would find it the least bit interesting if I did.
⎜ I only note it in passing for the benefit of
⎝ the future, when the cockroaches evolve archeologists.
>
On a lighter note,
how about Finsler or Boffa or Mirimanoff or
non.well.founded set theories?
Do they show that ST+PQ or ZFC or ordinals
suffer from Russell's {S:S∉S}?
>
No. They do not show that.
>
The less.interesting reason that they don't
is that
they are different domains of discourse.
⎛ 0 < 1/2 < 1 does not show that
⎜ there is an integer between 0 and 1
⎝ because 1/2 isn't an integer.
>
That less.interesting reason seems to
lie close to the heart of your objection.
You (RF) seem to not.believe that
things can be not.referred to.
>
In that respect,
I don't see what I can do for you.
I will continue to not.refer to
what I choose to not.refer to.
>
The more.interesting reason is that
ST+PQ and ZFC and well.ordering
do not suffer from Russell's {S:S∉S}
_by design_
>
Without looking up what Mirimanoff or
Finsler or Boffa have to say about
non.well.founded set.theories,
I am confident that their theories
do not suffer from Russell's {S:S∉S}
_by design_
>
Because they know that, otherwise,
they would be talking gibberish.
>
You (RF) seem to argue that
☠⎛ they cannot not.refer to Russell's {S:S∉S}
☠⎜ and therefore they ARE talking gibberish
☠⎝ and a standard model of the integers not.exists.
>
☠( and anyone who disagrees with that is a liar.
>
>

It is so that that is what I argue, for, yes.
Mirimanoff says so, too.
Then there's also the "not.first.false is
not necessarily not.ultimately.untrue" bit,
and the demonstration of the "yin-yang ad-infinitum"
bit, then also as with regards to doubling-spaces
and halving spaces and the continuum in mathematics.