The correct definition of true cardinality of a set!

[Zuhair]

Define: t.card(x)=a <=>
[a < w & x bijective to a] \/
[infinite(x) & a=w]

Where a is an ordinal and w stand for the set of all finite von Neumann ordinals.

It appears to me that this is the right definition of the true cardinality of sets!
Reason: there is nothing to force us believing in existence of uncountable sets!

Zuhair

[Jeff Barnett]

Well you certainly believe in them! The above "infinite(x)" is only
necessary if they exist. I see you believe in "w" but do not have a
power set axiom in your system. Makes your system a tad useless. Try
writing your complete set of axioms so we can see what useful theorems
you are dropping: changing a definition doesn't do much unless axioms
reference that definition. Otherwise you just define things the old way
using new names and get the same theorems with name substitution
substitutions, not new theorems.
--
Jeff Barnett

[Julio Di Egidio]

You could just say the natural numbers with a point at infinity.

Then it makes perfect sense to me, although I'd then second
Jeff Barnett's comment: could you expand a bit, indeed what
about defining the real numbers in this theory?

Julio

[Julio Di Egidio]

But even before that, you should explicitly define infinite(x).
(E.g. Dedekind-infinite with those extended natural numbers
should be suitably amended, just like you have already
amended cardinality.) Indeed, I would expect this definition
(and the consequences of defining finite vs infinite this way)
already to contain the germ of a justification for your claim
that there are no uncountable sets (in this theory)...

Julio

[Zuhair]

The strength of the background theory is not the issue here, you can go up in strength as may you! The issue is about what constitutes the *TRUE* size of a set, is it as seen internally within any model satisfying that theory, or is it what is seen externally over those models. To me, "truth" is an EXTERNAL standpoint, so what's going on internally can be false but consistent. A very clear example of that is what's going inside a countable model of ZFC, internally speaking you live in Cantor's paradice of infinities, but the truth of the matter is that all those sets are externally countable, so what was going on inside was FALSE, even though consistent.

A suitable theory to justify my definition, whould be for example: "ZFC + All models of ZFC are countable", or another kind of a theory is to have two sort objects, the first written in lower case represent sets, the second written in upper case represent classes, now write all axioms of ZFC in lower case, and add that every set is a class, add extensionality over classes, also add that for any formula \phi we have a class \{y : phi\}, and then add the axiom that ALL CLASSES are countable (written complelety in upper case). I call this theory "ZFC + All classes are countable"

Both theories justify this definition, and yet both are at least as strong as ZFC itself, actually I can take any extension of ZFC and do the same thing with it.

Now, the philosphical standpoint is that there is nothing to force us believe in existence of uncountable models of first order theories, and since set theory is phrased in first order then adhering to Occam's razor, we may content ourselves with existence of the countable ones only, i.e. postulate that all models of all first order theories are externally countable. So, the truth of the matter is that all sets are countable! Even though we can have a consistent (but false) consistent discourse about them.

[Zuhair]

Yes! This is indeed one possible scenario to go about it. It is interesting. There are known one and two place compactification of the reals, I've done 3 and 4 point compactification of the complex plane, and there is also the fully affine compactification of the complex plain. In those milieux one can define division by zero and the alike.

But, anyhow, my primary point was philosopho-logical, that is about "truth" being essentially an external standpoint, see my prior response to Jeff.

[Jeff Barnett]

I really don't think your externally/internal distinction and
correspondence to what's real and what isn't is to the point. There is
nothing real or not real about sizes of infinities in a manner of
speaking. That all hinges on what set theory you are using and what sets
that theory actually defines. Whether a set is infinite or not depends
on whether there is a 1-1 map from a set to a proper subset of itself;
and whether that map exists depends on whether your set theory contains
such a map (a set of ordered pairs that meet certain conditions).
Whether some set is countable or larger once again depends on where (in
which theory) you measure its size. Different theories are not obliged
to agree and neither necessarily represents reality.

Picking a set theory to represent reality is like picking a religion for
the same task. You can argue all day, form we vs they groups, eventually
fight wars, or ultimately waste time on USENET groups but you are
unlikely to find universal agreement. I propose that we do what most
excellent mathematicians and logicians do: remain agnostic and use
whatever system does good work for us on the current problem.

I wish you good luck converting true believers to your viewpoint.
--
Jeff Barnett

[Julio Di Egidio]

On Saturday, 17 December 2022 at 07:06:52 UTC+1, Zuhair wrote:
> On Friday, December 16, 2022 at 10:16:16 PM UTC+3, ju...@diegidio.name wrote:
> > On Friday, 16 December 2022 at 19:40:12 UTC+1, Julio Di Egidio wrote:
> > > On Friday, 16 December 2022 at 16:45:22 UTC+1, Zuhair wrote:
> > > > Define: t.card(x)=a <=>
> > > > [a < w & x bijective to a] \/
> > > > [infinite(x) & a=w]
> > > >
> > > > Where a is an ordinal and w stand for the set of all finite von Neumann ordinals.
> > > >
> > > > It appears to me that this is the right definition of the true cardinality of sets!
> > > > Reason: there is nothing to force us believing in existence of uncountable sets!
> > >
> > > You could just say the natural numbers with a point at infinity.
> > >
> > > Then it makes perfect sense to me, although I'd then second
> > > Jeff Barnett's comment: could you expand a bit, indeed what
> > > about defining the real numbers in this theory?
> >
> > But even before that, you should explicitly define infinite(x).
> > (E.g. Dedekind-infinite with those extended natural numbers
> > should be suitably amended, just like you have already
> > amended cardinality.) Indeed, I would expect this definition
> > (and the consequences of defining finite vs infinite this way)
> > already to contain the germ of a justification for your claim
> > that there are no uncountable sets (in this theory)...
>

> Yes! This is indeed one possible scenario to go about it. It is interesting. There are known one and two place compactification of the reals, I've done 3 and 4 point compactification of the complex plane, and there is also the fully affine compactification of the complex plain. In those milieux one can define division by zero and the alike.
>
> But, anyhow, my primary point was philosopho-logical, that is about "truth" being essentially an external standpoint, see my prior response to Jeff.

I understand that, which is why you *should*, not just
could, say what I said, instead of mentioning concrete
realizations. Indeed, you rather keep misspeaking of
philosophy/logic and missing/mangling all the critical
points:

- Your external/internal is altogether fallacious.
"Truth" does *not* belong to mathematics: GIT docet!
And only along that line, and by not missing the point that
G is *true hence unprovable*, and the point that if we
formalize GIT's meta-mathematics we just end up with a
higher-level G, only then you can get to say that some
mathematics and indeed the whole model-theoretical
approach, are literally *false*, which, to reiterate, is not
and cannot be by purely mathematical argument.

- Your "philosophical reason" is ludicrous:
Philosophy/logic proceeds by *necessity*, not an
Occam's razor. Indeed you altogether miss the point
that *standard* mathematics (ZF/ZFC, but this really
has origins in PM) is actually *broken* re anything
infinite, not just useless: logically invalid and then
properly false! And that is a *cogent* motivation.

Which is overall a pity, since your mathematical
work could otherwise be precious, but you sort of
meticulously undermine it yourself nor you ever
complement it with anything actually concrete,
e.g. a definition of the bloody real numbers...

Was this thread of yours just a troll? Reasons
eliminated, any crank idiot could have come up
with it, just add a point at infinity...

Julio

[Ross A. Finlayson]

Is it Nelson and IST?

Cohen, Feferman, Kunen, makes for that pure set theory is extra-ordinary.

Ubiquitous ordinals is usual for that the model of naturals is
always a fragment or extension.

Axiomless natural deduction arrives at this, it helps that it
includes both the classical and modern canon.

Of course it'll require a paradox and uncountability slate,
and continuity from zero'eth principles.

....

[Julio Di Egidio]

On Sunday, 18 December 2022 at 22:18:09 UTC+1, Ross A. Finlayson wrote:

> [Obscenities and stupidities]

I write yet another seminal post, indeed exposing the century long
fraud that this retarded empire is, and you just cannot let that go,
can you, you other insane pieces of retarded nazi shit.

Get extinguished already, you and the whole insane bandwagon.

*Spammer fraud alert*

Julio

[Ross Finlayson]

Set theory is our modern mathematics' canonical foundations.

Descriptive set theory, ....