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

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:

On 11/16/2024 08:58 AM, Ross Finlayson 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,

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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?

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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.

>

I don't say "infinity" is an axiom
primarily because
"infinity" is not an axiom of ST+PQ
ST+PQ:
⎛ set {} exists
⎜ set x∪{y} exists
⎜ set.extensionality
⎜ plurality ⦃z:P(z)⦄ exists
⎝ plurality.extensionality
>
"Infinity exists" ==
"the minimal inductive plurality exists"
is a theorem of those axioms.

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^- Fragment

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No.
The minimal inductive plurality is
a standard model of the integers.
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Let's recall an example geometrically of what's
so inductively and not so in the limit.
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Take a circle and draw a diameter, then bisect
the diameter resulting diameters of common circles,
all sharing a common diameter, vertical, say.
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Then, notice the length of the circle, is
same, as the sum of the lengths of the half-diameter
circles, their sum.
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So, repeat his dividing ad infinitum. In the limit,
the length is that of the diameter, not the perimeter,
while inductively, it's the diameter.
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Thusly, a clear example "not.first.false" being
"ultimately.untrue".

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A finite sequence of claims, each claim of which
is true.or.not.first.false  is
a finite sequence of claims, each claim of which
is true.
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The reason that that's true is that
THE SEQUENCE OF CLAIMS is finite.
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Whatever those CLAIMS refer to,
none of those CLAIMS are first.false.
(They're each not.first.false.)
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Since none of those CLAIMS are first.false,
none of those CLAIMS are false.
(That sequence is finite.)
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What those claims are ABOUT doesn't affect that.
For example,
being ABOUT an indefinite one of infinitely.many
doesn't affect that.
>
Discovering
a finite sequence of claims, each claim of which
is true.or.not.first.false
in which there IS an untrue claim
is akin to
counting the eggs in a carton and
discovering that, there, in that carton,
7 is NOT between 6 and 8.
There is a problem, but not with mathematics.
>

Then, with regards to your fragment,

>
...the minimal inductive plurality...
>

congratulations,

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Thank you.
>

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.
>

and quite fully revived Frege and given yourself
a complete theory and consistent as it may be, and
can entirely ignore all of 20'th century mathematics.
>
It's small, .... Fragment

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The minimal inductive plurality.
Big or small, that's the thing,
the whole thing, and nothing but the thing.

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

>
Tell me what you think is a lie:

>

>
Quote what I wrote which you think is a lie:
>

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Well, it's among what you clipped because it was un-answerable,

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I didn't clip any quote of me by you.
Anyway, do you need help finding earlier posts in a thread?
>
⎛ Noun
⎜ lie (plural lies)
⎜
⎜ 1. An intentionally false statement; an intentional falsehood.
⎜ 2. A statement intended to deceive, even if literally true.
⎝ 3. (by extension) Anything that misleads or disappoints.
>
https://en.wiktionary.org/wiki/lie
>
Quote what I wrote which you think is a lie.
>
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