Georg Cantor, article:
"On different points of view on the actually infinite"
In mathematics, the general (continuous function) is connected
to the particular limit (limit of an infinitesimal quantity)
a) Infinite large is not available to increase.
b) Infinite small is conceived as "transfinite" -
"corresponding to the smallest super-finite number."
c) The actual physical connection between them is unclear.
-------
1. Infinitesimal quantities are Planck's energy quantums of action h and Boltzmann's k-molar particles.
2- An infinitely large continuum is a cosmic vacuum, which is ignored by science.
3- The interactions between the cosmic vacuum and infinitesimal quantities (h and k) is unknown.
========..

That looks wrong. Transfinite set theory is the formalisation of number systems which include numbers bigger than any finite number. I.e. infinitely big.

A number that is infinitely small (i.e. smaller in magnitude than any finite number) is not transfinite, but infinitesimal.

In the normally defined and studied objects and structures that make use of these concepts (for example cardinals, ordinals, surreals etc.) there is no largest transfinite number and no smallest infinitesimal. That is to say (a) and (b) are not statements made by mathematics or (to the best of my memory) Georg Cantor.

I do notice that (b) seems to contain a self-contradiction: The smallest super-finite number is not "infinite[ly] small", it is bigger than any finite number and thus "infinitely large".

Planck's energy, while small is not infinitesimally so. It corresponds to a finite number, not a number smaller than any finite number.

Thommo wrote:
Planck's energy, while small is not infinitesimally so. It corresponds to a finite number, not a number smaller than any finite number.

Planck's 'h' energy is physical constant, a physical micro small limit of energy
and therefore can be called 'Infinitesimal quantity"

It would be just as useful to call it “eensy weensy”, as to use your definition of a number being “micro small”. Wouldn’t “nano small” be smaller? “Pico small” would be smaller yet.

"ronto small" and "quecto small".

quecto small x 0.5!

All big numbers have smaller numbers
To count and then divide them
And all small numbers have smaller still
And so ad infinitum

Of course, it could be what doctors call “Very, very small. So small it couldn’t possibly make off with a whole leg.”

Chapman’s Law again.

THWOTH wrote:"ronto small" and "quecto small".

quecto small x 0.5!

All big numbers have smaller numbers
To count and then divide them
And all small numbers have smaller still
And so ad infinitum

The number π=3,141592 . . . ---->> . . . to infinite decimals (∞) you cannot divide

socratus wrote:

THWOTH wrote:"ronto small" and "quecto small".

quecto small x 0.5!

All big numbers have smaller numbers
To count and then divide them
And all small numbers have smaller still
And so ad infinitum

The number π=3,141592 . . . ---->> . . . to infinite decimals (∞) you cannot divide

So what? Are you confusing irrational with infinitesimal?

socratus wrote:Planck's 'h' energy is physical constant, a physical micro small limit of energy
and therefore can be called 'Infinitesimal quantity"

Mathematically, that is simply incorrect.

Planck's constant is approximately 6.62607015 x 10-34, which means it has 33 leading 0s after the decimal point.

An infinitesimal is defined as being a number that is strictly greater than 0, but strictly less than any finite number greater than 0. Which is any number that has a decimal representation with a non-zero figure literally anywhere in its decimal expansion. So 34 leading 0s then a significant figure is "infinitely" further away from an infinitesimal than 0 is. So is 340 leading zeroes, or 340 million leading 0s.

Conceptually you can think of infinitesimals as expanding on decimal representations by filling out numbers that consist of infinitely many leading 0s after the decimal point, then a significant figure after that.

Very small =/= infinitesimal in maths or physics, only in colloquial English.

Cantor's set theory is extremely pedantic and precise. You simply can't use that kind of equivocation with it.

Comment
"Georg Cantor's article "On different points of view on the actually infinite"
raises intriguing points about the nature of the infinite in mathematics.
While the connection between the general (continuous function) and the
particular limit (limit of an infinitesimal quantity) is a fundamental concept
in calculus, the actual physical connection between them remains elusive.
Cantor's notions of the infinite large being unavailable for increase and
the infinite small being conceived as "transfinite" are thought-provoking.
In the realm of physics, the infinitesimal quantities are represented
by Planck's constant (h) and Boltzmann's constant (k), which play crucial roles
in quantum mechanics and thermodynamics, respectively. On the other hand,
the infinitely large continuum is often associated with the cosmic vacuum,
a concept that is largely ignored in science. The interplay between the mathematical
and physical interpretations of the infinite is a fascinating area of exploration
that continues to challenge our understanding of the universe."
/by Ali Nodeh/
---

Physics has its limits: Planck's constant (h), Boltzmann's constant (k),
Einstein's (c), Sommerfeld's (α=e^2/ℏc≈1/137) . . .
From these limits everything was created according to quantum laws.
=====

socratus wrote:Cantor's notions of the infinite large being unavailable for increase and
the infinite small being conceived as "transfinite" are thought-provoking.

Again: Cantor did not call "infinite small" things transfinite.

E.g.:
https://www.jamesrmeyer.com/infinite/or ... ransfinite

The absurdity of the notion of such transfinite numbers is that it deliberately assumes that there can be an ordering of integers where the first transfinite ordinal integer comes “after” the “end” of the finite integers, that is, the claim is that there is a sequence of integers and transfinite integers where the transfinite ones come “after” all the finite ones.

By convention the smallest transfinite ordinal is called ω and the smallest transfinite cardinal is called ℵ0. The smallest transfinite number is the smallest number such that it is greater than any finite number, in both of those numbering systems.

You can also find examples of how Cantor himself used the term in places like p103 of this translation of one of his original papers:
https://ia801308.us.archive.org/33/item ... 00cant.pdf