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

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

On 11/15/2024 12:06 PM, Jim Burns wrote:

On 11/15/2024 1:05 PM, Ross Finlayson wrote:

Non-standard models of integers exist.

>
Yes, and,
when we discuss non.standard models,
we can assemble claim.sequences which
start with a description of a non.standard model.
And, when we do that,
augmenting true.or.not.false claims
will be true about the described non.standard models.
>
However,
non.standard models of integers are not
standard models of integers.

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

Consider Boolos's theory ST with plural quantification.
⎛ set {} exists
⎜ set x∪{y} exists
⎜ set.extensionality
⎜ plurality ⦃z:P(z)⦄ exists
⎝ plurality.extensionality
That can be augmented with
a finite sequence of claims, each claim of which
is true.or.not.first.false, and which
ends with the claim that
the standard integer.model exists, as a plurality.
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?

that Russell has fooled you with a contradictory statement,
and there are only fragments, if unbounded,
and extensions, sublime.

What statements has Russel made about ST+PQ?