Re: Replacement of Cardinality (real-valued)

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

On 8/4/2024 11:46 AM, Ross Finlayson wrote:

On 08/04/2024 07:59 AM, Ross Finlayson wrote:

On 08/04/2024 07:52 AM, Ross Finlayson wrote:

On 08/04/2024 03:41 AM, FromTheRafters wrote:

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It's kind of like when people say
"hey you know the initial ordinal assignment is
what we can say 'are' cardinals",

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For each equivalence relation '#' on a class A
there is a family A/# of partitions of A
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[x] and [y] are the partitions holding x and y
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[x] = {u e A: u # x}
[y] = {u e A: u # y}
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Any element of [x] serves equally well as
a representative of the partition holding x
[x] = [y]  ⇔  x # y
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We somewhat.arbitrarily assign initial ordinal ξ
to be the One True Representative of
the partition [ξ] of sets A = |ξ|
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The initial ordinal has a certain elegance.
Being ordinals, any ordinal means an initial ordinal.
But for any set A = |ξ|, [A] = [ξ]
and A serves as a representative just as well.
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⎛ The axiom of choice is equivalent to
⎝ each partition [A] holds an ordinal.
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then it's like,
"with the Continuum Hypothesis being undecide-able and all,
then there are and aren't ordinals between
what would be those cardinals by their cardinals the ordinals",

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We have a description of sets, the ZFC axioms.
In some domains satisfying that description,
the continuum hypothesis is true.
In some domains satisfying that description,
the continuum hypothesis is false.
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Therefore,
the ZFC axioms aren't enough to decide
the continuum hypothesis.
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sort of establishing that
such a definition does and doesn't
keep itself non-contradictory,

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No, that is not established.
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A theory describes more than one model.
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Some claims have proofs.
Those claims are true in each model.
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Some claims are true in each model.
Those claims have proofs.
(That is a very nice result, maybe not super.obvious.)
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True.and.false in different models
does not make a theory contradictory.
The theory is silent, not wrong.
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What theory?
Model theory? That's exactly what model theory not.is.
... When _all_ the objects of relation are in the model,
the faithful model, witnessing all facts of the model.
I.e., either would establish something _not so_ in the
other, not just independent.
... ordinals, and their cardinals, between or ordinals,
and their cardinals, not between.
I think it's much better that ubiquitous ordinals makes
for that there aren't standard models of integers and
then that forcing, of essentially ubiquitous ordinals,
makes CH independent of ZF, and makes theories of cardinals,
for counting, and theories of ordinals, for numbering,
fundamentally different.
This is among reasons I've said for a long time or since
at least two decades that ubiquitous ordinals is a thing
and there are extra-ordinary models of infinite sets,
and "counting" and "numbering" are two different things.
... Not to be "confused".