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

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,

>
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.

>
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.

>

^- Fragment

>
No.
The minimal inductive plurality is
a standard model of the integers.
>

Let's recall an example geometrically of what's
so inductively and not so in the limit.
>
Take a circle and draw a diameter, then bisect
the diameter resulting diameters of common circles,
all sharing a common diameter, vertical, say.
>
Then, notice the length of the circle, is
same, as the sum of the lengths of the half-diameter
circles, their sum.
>
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.
>
Thusly, a clear example "not.first.false" being
"ultimately.untrue".

>
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.
>
The reason that that's true is that
THE SEQUENCE OF CLAIMS is finite.
>
Whatever those CLAIMS refer to,
none of those CLAIMS are first.false.
(They're each not.first.false.)
>
Since none of those CLAIMS are first.false,
none of those CLAIMS are false.
(That sequence is finite.)
>
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,

>
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

>
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.
A case for induction for a case for induction?
What all's the "meta" theory?
So, in the limit, as it's a constant, the ratio of the
circumferences to the diameters is pi, yet, in the
limit, what's no different, than the diameter,
the ratios of those circumferences, vanishing,
equals one,so, inductive inference,
while not.first.false, is, ultimately.untrue.
Furthermore it's an example of something that
only in the infinite, changes what's a constant
measure, so it's quasi-invariant.
Then, this "yin-yang ad infinitum", is a counter-example
to naive induction, and as well, a striking example of
an infinite limit, that belies all its finite approximations.
That's also totally simply demonstrable as of
the method of exhaustion, as what it arrives to.
So, such a simple and thusly clear diagrammatical device,
helps show the failure of ill-advised inference,
and the lack of its guarantees otherwise,
and is otherwise a neat counterexample,
and helps reflects that deductive inference
is never absent from round reasoning.