Re: Replacement of Cardinality (infinite middle)

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:

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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?