Mathematics with the potential or actual infinite or without emptiness

[I.V. Serov]

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Dennis Hamilton writes 
(https://cs.nyu.edu/pipermail/fom/2023-February/023742.html):

Now it is certainly a fair point that obtaining zero as the limit as n 
goes to infinity of 2^-n is the limit of an infinite sequence.

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Antonio Drago replies (in 
https://cs.nyu.edu/pipermail/fom/2023-February/023768.html):

Not true. Any approximation does not equate the zero, always a distance 
remains; or evenly: a segment, representing an approximation, is defined 
by two extreme points; it cannot be reduced to one point only; this is 
an old criticism to epsilon-delta technique …

… illusion that leads to believe that the approximations at last crash 
to zero. It is just an appeal to actual infinity that justifies this 
crashing as effective. Really, it is true in a world of mathematical 
ideas only according to the “realist [Platonic] attitude”. The entire 
undergraduated teaching of mathematics is based on the actual infinity.

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Does zero exist at all, does it not?

Potentially it does not exist as it can never be reached as in the 
example of 2^-n.
Limits are always potential unless they are actually achieved.

For an actualist, zero may exist: compare (2^+n) to 1./(2^+n) and try to 
justify the difference.
When zero is to exist than infinity is to exist too and vice versa.
Note, that zero appears here to be a kind of actually infinite number, 
i.e., not finite.

The difficulty with the actual well-founded infinity is its 
uncountability, i.e., non-constructiveness.
Counting is at the heart of all mathematicians' life; even non-counters 
count non-countable infinities.

Look from another perspective: does the empty set exist? or, otherwise, 
is every set to have at least one element, which is also a set?

A non-well-founded universe Y with elements y can be defined in the Von 
Neumann way:

(A y): y+1 = y V {y};

There 1 is a natural unit and a V b denotes the union of the sets a and 
b.

Similarly, consider the genuine Zermelo-like set-up:

(A y): y+1 = {y}.

Here, the universe Y in a posterior state is the singleton of the prior 
state of the universe Y.

Note, that Russell's paradox is not solved with the above definitions, 
as it is unclear whether the universe Y is its own element.

To address the paradox, consider to think about the following circular 
axioms, where elements x of the set X are called *gets*:

(A x): x = x V {x}; genuine Von Neumann way;
(A x): x = {x}; genuine Zermelo way.
(A x): Other approaches are possible; for example, each *get* can be 
considered to be a doubleton (pair).

These look more like equations to be solved.
Achilles and the tortoise meet each other when the equations are solved.

The relations of x to other elements of X are described elsewhere.

In any case, there is a predecessor for every state of the universe X;
Neither natural unit 1, nor zero are members of the universe X.
Similarly, for the universe Y.

The empty set does not have a predecessor and therefore is so extremely 
different from any other member of the universe X,
so that the empty set is not a member of the universe X. Similarly, for 
the universe Y.

Is it otherwise possible to ever construct an infinity without already 
being an infinity?
Potentialists stay finite and only dream of infinities.

Well-grounded actualists discern infinities,
although they do not envision their ultimate construction and therefore 
experience vertigo of uncountability,
while assuring themselves and others that we all are in paradise.

When a mathematician seeks generalized countability and infinity in her 
universe X or Y,
she has to be non-well-founded - say genuine - she has to exclude zero 
from the universe.

The generalized countability is out there, since every genuine number x, 
an element of X, has an immediate predecessor and an immediate 
successor.

Herein lies the catch, the connection between potentially infinite and 
the genuine:
when the predecessor for any state of the genuine universe exists only 
potentially, then one gets a finitist world!

The generalized countability is out there in both cases!

Recall the sine whose derivative is cosine; whose derivative is minus 
sine.
Negate existence of an *end* and you move from finitism to potential 
infinitism;
Negate then existence of a *beginning* and you move from potential 
infinitism to actual non-well-founded infinitism;
The latter can be well-ordered with any arbitrary (!) element as 
neutral;
and this well-ordering has all the features of finitism or countable 
infinitism when viewed from a potentialist point of view.

Either one assumes the universe to be actually infinite with 
predecessors in any state, or it is finite in any state and only 
potentially infinite, i.e., actually finite;
the only other option being that it is not constructively defined, which 
is a serious issue for each and every constructivist; and there are very 
many out here.

One may object: recall the transfinite induction.
What about, for example, the first limit ordinal w? It looks like an 
actually infinite number and it has no immediate predecessor.
The ordinal w is the least upper bound of the natural numbers. Is there 
an upper bound of the natural numbers somewhere out there?
Ordinals do not offer a solution, for this very reason: there are limit 
ordinals, which do not have predecessors making generalized counting 
impossible.
Or is there a way to arrive at the ordinal w from a natural n without 
taking a limit?
Even then, imagine we are at the limit w; is there anything just before 
it? No? How is it possible that nothing is there?
Or maybe all the integers are there before it? May be ... Why does one 
want to start counting from w again in a like-wise manner: w, w+1, w+2, 
...?
Because all these w, w+1, w+2, ... are in paradise? Where are the 
natural numbers then? Down at earth 0, 0+1, 0+2, ...?
Is there any difference between w and 0? In any case, let us connect the 
two worlds so that it becomes one, the good one.

What does one need therefore?
one needs love of the parents x and (S x) and the offspring relation (I 
x) in X;
one needs the dichotomy, and
one needs the dialectics.

Please be so kind as to take non-well-foundedness seriously when 
thinking about foundations of mathematics.
Even the quantum spacetime is ready for this: non-well-foundedness is 
fundamental, it is genuine.

After all, we are not only living in a new century, we are living in a 
new millennium, so do dare.

I.V. Serov