Mathematics with the potential or actual infinite or without emptiness
I.V. Serov
i.v.serov at chf.nu
Sat Feb 11 11:51:56 EST 2023
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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
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