From:  Jim Burns <james.g.burns@att.net>
Date:  07 Sep 2024 03:40:06 Hong Kong Time
Newsgroup:  news.alt119.net/sci.logic
Subject:  

Re: Replacement of Cardinality (infinite middle)

NNTP-Posting-Host:  null

On 9/5/2024 4:14 PM, Ross Finlayson wrote:
> On 09/05/2024 12:57 PM, Ross Finlayson wrote:
>> On 09/03/2024 01:50 PM, Jim Burns wrote:

>>> [...]
>> [...]
>
> Back in the 80's and 90's
> it was Nelson's Internal Set Theory
> where it was figured that
> the avenue toward true non-standard real analysis
> was to result.

This "true non-standard real analysis" must concern
something other than
the Dedekind.complete ordered field.

> I.e.,
> not-a-real-functions with real analytical character,
> like Dirac's delta function or
> here for example
> the Natural/Unit Equivalency Function,
> it is expected that
> "foundations" _does_ formalize them, and that
> what doesn't, simply, isn't,
> respectively.

You (RF) may be tired of nuance by now,
but
I think we need to distinguish between
what _simply_ isn't and
what _a specific foundation_ won't say is.

Consider Boolos's ST as a toy foundation.
⎛ ∃{}
⎜ ∃z = x∪{y}
⎝ extensionality

ST supports the existence of each finite ordinal
via a finite not.first.false claim.sequence.

ST does not support the existence of
a set of all finite ordinals.
At least, I don't see how it could.
ST doesn't support its non.existence, either.
At least, I don't see how it could.

An ordinal which has itself as an element
simply isn't.
That depends pretty much completely on
_what ordinal are_  well.ordered.

Getting around that prohibition would
require ordinals which were something else.
But that's not actually getting around it.
That's only playing a game similar to
"if we rename 2 as 3, then 1+1=3"

> Then this "infinite middle" is just about
> the simplest "non-Archimedean" that there is,
> and in fact even simpler, than for example
> axiomatizing "0" and "omega"

"omega" must be
something other than
the first transfinite ordinal.

> axiomatizing "0" and "omega"
> with an infinite-middle pretty much
> exactly like ZF does,
> except symmmetric about the middle
> instead of non-inductive yet declared fiat
> (stipulated).

1+1=3?