Re: Replacement of Cardinality (real-valued)

On 08/08/2024 12:00 PM, Jim Burns wrote:

On 8/7/2024 9:03 PM, Ross Finlayson wrote:

On 08/07/2024 01:06 PM, Jim Burns wrote:

>

A theory can describe more than one model.
>
Some claims have proofs.
Those claims are true in each model.
>
Some claims are true in each model.
Those claims have proofs.
(That is a very nice result, maybe not super.obvious.)
>
True.and.false in different models
does not make a theory contradictory.
The theory is silent, not wrong.

>
What theory?

>
| The completeness theorem applies to any first-order theory:
| If T is such a theory, and
| φ is a sentence (in the same language) and
| every model of T is a model of φ,
| then there is a (first-order) proof of φ
| using the statements of T as axioms.
| One sometimes says this as
| "anything true in all models is provable".
| (This does not contradict Gödel's incompleteness theorem,
| which is about a formula φᵤ that is unprovable
| in a certain theory T
| but true in the "standard" model of the natural numbers:
| φᵤ is false in some other, "non-standard" models of T.)
>

Model theory?
That's exactly what model theory not.is.

>
| The completeness theorem makes a close link between
| model theory,
| which deals with what is true in different models, and
| proof theory,
| which studies what can be formally proven
| in particular formal systems.
>
https://en.wikipedia.org/wiki/G%C3%B6del%27s_completeness_theorem
Gödel's completeness theorem
>
>

Well, yeah, proof-theory and model-theory are equi-interpretable,
then a usual idea that foundations includes the entire space of
the integer continuum and the linear continuum, or the ubiquitous
ordinals and the hyper-dimensional holo-gram, makes that model
theory is the usual milieu with the algebraizations, arithmetizations,
and geometrizations.
Goedel's pretty simple arithmetization of an algebraization
and for Goedel's "stop hitting yourself" completion theorem,
doesn't say much.
That includes that it doesn't say much about the extra-ordinary.
Here it's better to call the extra-ordinary the extra-ordinary
instead of the non-standard, when, for example, it works out
that the "standard" model of the integers: _isn't_.
"Is it the universe of numbers or the universe of words? Yeah, it is."