If there exist definite transfinite numbers, then their
reciprocals must be infinitesimals, not zero. Which is
good, as infinitesimals necessitate specific additional
laws, reciprocally making the transfinite sharper. And
one issue is immediately apparent: infinitesimals are
not compatible with the Archimedean principle. Ergo,
all infinities are countable in ordinary mathematics.

Julio

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On 04/05/2025 17:17, Ross Finlayson wrote:

On 05/04/2025 06:27 AM, Julio Di Egidio wrote:

If there exist definite transfinite numbers, then their
reciprocals must be infinitesimals, not zero.  Which is
good, as infinitesimals necessitate specific additional
laws, reciprocally making the transfinite sharper.  And
one issue is immediately apparent: infinitesimals are
not compatible with the Archimedean principle.  Ergo,
all infinities are countable in ordinary mathematics.

Pythagorean Archimedean [bollocks]

A usual account of infinity has that it's not ordinary,
rather, per Mirimanoff, extra-ordinary, then that it's
fragments or extensions, the model of integers.

Are you aware of the fact that the least upper-bound
property, which is an axiom of the standard theory of
real numbers, and a formalisation of the notion of
continuum with it, *implies the Archimedean property*?

Indeed, there is ordinary and there is extra-ordinary,
and *invalidly* then *inconsistently* mixing results
is the problem there.

(But already the prefix "extra", which is necessarily
extra-to given something, here the "ordinary", should
at least make *you* pause, and rather warn you that you
have it upside-down, what "ordinary mathematics" even
is, as per the usual inversion of all that counts.

Conversely, your balderdash, here as elsewhere, always
eventually back to your blind take-everything and fully
prosaic Platonism, remains the other side of the very
same mangled/fraudulent coin. Strictly speaking.)

Julio

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On 04/05/2025 20:24, Ross Finlayson wrote:

The incompatibility of infinitesimals with the Archimedean principle
doesn't directly imply that all infinities must be countable in standard mathematics.

I said in *ordinary* mathematics. But you won't learn.

Julio

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On 04/05/2025 21:55, Ross Finlayson wrote:

On 05/04/2025 12:24 PM, Julio Di Egidio wrote:

On 04/05/2025 20:24, Ross Finlayson wrote:

The incompatibility of infinitesimals with the Archimedean principle
doesn't directly imply that all infinities must be countable in standard >>> mathematics.

I said in *ordinary* mathematics.  But you won't learn.

Oh, I didn't write that, in that dialog with one of those
mechanical reasoners "Google Gemini", I only wrote the
paragraphs starting "Thanks GG.".

Of course, but it suits you perfectly: that *is* the measure
of the generalised insanity and inconsistency that you so
literally incarnate.

These are objects of mathematics, they exist regardless being
defined away, naive positivist.

There is nothing naive about the systematic shithole for everybody.
You are a true enemy of life and intelligence, indeed in your very
abusive inanity, the prototypical true paladin and defender of the nazi-retarded shithole for everybody that we are.

*Plonk*

Julio

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On 05/05/2025 02:06, Ross Finlayson wrote:

Simply I wouldn't say that infinitesimals and
the Archimedean principle were "incompatible",
since they're from either side of dividing/divided.

But the Archimedean principle is indeed the statement
that there are no infinitesimals, so either you are
truly incapable of any logical reasoning, or you are
rather in denial to put it charitably.

Don't worry, insofar as you express contempt and disgust of
wrong-minded oppression, I may share that sentiment.

Sentiment is the least of our problems: we are what we do,
not what we think and even less what we feel, you fucking
*idiots*!

To which you'll just retort with some more of the same
nazi-retarded mantras and propaganda, ad nauseas...

Some people on this planet are just a lost cause, many
more are rather too compromised to get it: together they
are a fucking global calamity.

I'll go puke now if you won't mind.

Julio

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On 05/05/2025 02:56, Julio Di Egidio wrote:

On 05/05/2025 02:06, Ross Finlayson wrote:

Simply I wouldn't say that infinitesimals and
the Archimedean principle were "incompatible",
since they're from either side of dividing/divided.

But the Archimedean principle is indeed the statement
that there are no infinitesimals, so either you are
truly incapable of any logical reasoning, or you are
rather in denial to put it charitably.

Don't worry, insofar as you express contempt and disgust of
wrong-minded oppression, I may share that sentiment.

Sentiment is the least of our problems: we are what we do,
not what we think and even less what we feel, you fucking
*idiots*!

Indeed the physicalists... but only when it's in support
of some material abuse, otherwise only fallacious idealism.

It's fucking embarrassing if nothing at all.

To which you'll just retort with some more of the same
nazi-retarded mantras and propaganda, ad nauseas...

Some people on this planet are just a lost cause, many
more are rather too compromised to get it: together they
are a fucking global calamity.

I'll go puke now if you won't mind.

Julio

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On 05/05/2025 06:41, Ross Finlayson wrote:

The idea though is that n/d makes standard infinitesimals
even as if only in the, unbounded, the Archimedean.

...and war is peace, and freedom is getting a job...

Then, if the un-countable, is not a constructivist result,
is a point of contention, as that its proofs employ contradiction,
and some constructivists have that's at odds with constructivism.

You are full of shit. 1) Diagonal arguments can be proved
constructively: ex falso quodlibet is not the same as
reductio ad absurdum: unless one deems ordinary induction
extra-ordinary, but that would be plain stupid. OTOH,
2) the connection from binary sequences or subsets of the
natural numbers to real numbers is not immediate and not
granted, and not any more granted is uncountability.

Speaking of which, what do you even think this thread is
about?? THERE IS NO SUCH THING AS THE EXTRA-ORDINARY,
in ordinary and/or concrete (foundational!) mathematics:
so in any mathematics! That's eventually my thesis.

Now try and give me a counter-example... that does not
rely on the real numbers being uncountable.

Julio

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On 06/05/2025 03:03, Ross Finlayson wrote:

On 05/05/2025 04:57 PM, Julio Di Egidio wrote:

1) Diagonal arguments can be proved constructively
2) the connection to uncountability is not immediate

Au contraire, uncountability arguments are non-constructive

The new world: cannot reason in plain English.

Get the fuck out of here, get a real job for a change.

*Plonk*

Julio

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On 06/05/2025 01:57, Julio Di Egidio wrote:

Speaking of which, what do you even think this thread is
about??  THERE IS NO SUCH THING AS THE EXTRA-ORDINARY,
in ordinary and/or concrete (foundational!) mathematics:
so in any mathematics!  That's eventually my thesis.

The underlying presumption being that a mathematics that
cannot be instantiated is at best an exercise in futility.

Now try and give me a counter-example... that does not
rely on [i.e. assume] the real numbers being uncountable.

You cannot, can you. LOL

Julio

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On 06/05/2025 16:31, Julio Di Egidio wrote:

On 06/05/2025 14:10, Julio Di Egidio wrote:

Or, as we say around here, `lim_{n->oo} n = oo`,
aka "the point at infinity".

Aka "(infinity is) the point at infinity".

(It's like close encounters of the third kind:
I am not a mathematician but I keep seeing that
mountain... I should read Feferman.)

Julio

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On 06/05/2025 14:10, Julio Di Egidio wrote:

On 06/05/2025 01:57, Julio Di Egidio wrote:

Speaking of which, what do you even think this thread is
about??  THERE IS NO SUCH THING AS THE EXTRA-ORDINARY,
in ordinary and/or concrete (foundational!) mathematics:
so in any mathematics!  That's eventually my thesis.

The underlying presumption being that a mathematics that
cannot be instantiated is at best an exercise in futility.

Now try and give me a counter-example... that does not
rely on [i.e. assume] the real numbers being uncountable.

You cannot, can you.  LOL

Here is a very interesting one [*], M. Rathjen,
"Proof Theory of Constructive Systems: Inductive Types
and Univalence" (2018), <https://arxiv.org/abs/1610.02191>

For example, among other things:

<< Martin-Löf type theory appears to capture the abstract notion
of an inductively defined type very well via its W-type. There are,
however, intuitionistic theories of inductive definitions that at
first glance appear to be just slight extensions of Feferman's
explicit mathematics (see Feferman's quote from Sect. 1) but have
turned out to be much stronger than anything considered in ML type
theory. They are obtained from `T^i_0` by the augmentation of a
monotone fixed point principle which asserts that every monotone
operation on classifications (Feferman's notion of set) possesses
a least fixed point. To be more precise, there are two versions of
this principle. `MID` merely postulates the existence of a least
solution, whereas `UMID` provides a uniform version of this axiom
by adjoining a new functional constant to the language, ensuring
that a fixed point is uniformly presentable as a function of the
monotone operation. >>

Or, as we say around here, `lim_{n->oo} n = oo`,
aka "the point at infinity".

Julio

[*] Found via this SE answer by L. Pujet which provides
more introduction: "Proof-theoretic comparison table?" <https://proofassistants.stackexchange.com/a/1210>

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On 09/05/2025 02:26, Ross Finlayson wrote:

On 05/08/2025 12:45 PM, Chris M. Thomasson wrote:

On 5/6/2025 1:59 PM, FromTheRafters wrote:

Chris M. Thomasson explained :

On 5/6/2025 7:48 AM, Julio Di Egidio wrote:

On 06/05/2025 16:31, Julio Di Egidio wrote:

On 06/05/2025 14:10, Julio Di Egidio wrote:

Or, as we say around here, `lim_{n->oo} n = oo`,
aka "the point at infinity".

Aka "(infinity is) the point at infinity".

Wrt projection we can create a finite point in space and just call it
a point at infinity. However it is not infinity?... ;^)

That depends upon the space selected. :)

It's kind of like choosing a "really" large number wrt the naturals.
Say, okay, for our purposes, this is infinity, even though its not?

https://www.youtube.com/watch?v=9r-HbQZDkU0&list=PLb7rLSBiE7F4_E-POURNmVLwp-dyzjYr-&index=29

"Logos 2000: natural infinities"

Before I call you sons of bitches names, can I
please ask you:

On 06/05/2025 16:31, I have posted a message with
links to articles that starts with "Here is a very
interesting one"?

Can you see that message at all?

Julio

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On 09/05/2025 19:10, Ross Finlayson wrote:

On 05/09/2025 06:21 AM, Julio Di Egidio wrote:

On 09/05/2025 02:26, Ross Finlayson wrote:

On 05/08/2025 12:45 PM, Chris M. Thomasson wrote:

On 5/6/2025 1:59 PM, FromTheRafters wrote:

Chris M. Thomasson explained :

On 5/6/2025 7:48 AM, Julio Di Egidio wrote:

On 06/05/2025 16:31, Julio Di Egidio wrote:

On 06/05/2025 14:10, Julio Di Egidio wrote:

Or, as we say around here, `lim_{n->oo} n = oo`,
aka "the point at infinity".

Aka "(infinity is) the point at infinity".

Wrt projection we can create a finite point in space and just call it >>>>>> a point at infinity. However it is not infinity?... ;^)

That depends upon the space selected. :)

It's kind of like choosing a "really" large number wrt the naturals.
Say, okay, for our purposes, this is infinity, even though its not?

https://www.youtube.com/watch?v=9r-HbQZDkU0&list=PLb7rLSBiE7F4_E-POURNmVLwp-dyzjYr-&index=29


"Logos 2000: natural infinities"

Before I call you sons of bitches names, can I
please ask you:

On 06/05/2025 16:31, I have posted a message with
links to articles that starts with "Here is a very
interesting one"?

Can you see that message at all?

It was about "fixed-point", with regards to fixed-point theorems,
it's a sort of usual thing then with regards to "point at infinity".

It seemed not saying much, ....

It was about foundations, anyway thanks for confirming
you guys do not even have that excuse.

Julio

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