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

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.
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.
Is it possible that you (RF)
have misunderstood Mirimanoff?

That
 "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"
is a pretty long axiom - why not just say
"infinity", that's the usual approach.

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.
I said, in such a long.winded manner, that
  "infinity exists" is theorem of ST+PQ
because,
although I know that claim is well.justified,
it sounded as though this person Mirimanoff
has shown that I am mistaken about that.
I wanted to put my justification out there
for his argument to attack.