Re: Replacement of Cardinality (real-valued)

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