FOM: Leibniz and the actual infinity

[Alexander Zenkin]

On Wed, 23 Sep 1998, Moshe' Machover wrote (in particular):

(*
2. The issue of the reality or otherwise of infinite/infinitesimal *magnitudes*
is not quite the same (at least it was not the same for Leibniz) as the issue
of the existence of actual infinity. The former concerns mathemtics; the latter
concerns physical reality.
*)

I completely agree with this Moshe' Machover remark. Add only that " the latter
concerns <AZ: not only> physical reality " but also philosophical, psychlogical
and similar "realities".

One small remark about the infinity in Mathematics and Physics.
The most fundamental infinity in Mathematics is, of couse, the infinity ( not
for a while yet  without distinguishing the actual/potential one) of the common
natural numbers series:
     0, 1, 2, 3, …, n , …     (1)
Hehel called the infinty of the series (1) as the (maybe other synonim?) nasty
infinity because of its anti-dialectical nature. The last means, from the
mathematical point of view,  that any natural number n possesses the same,
unchangeable, constant property with respect to the mathematical operation
"+1", i.e. the adding "+1" to any n does not generate a new quality for the
next number n+1 as to an applicability of the operation "+1" to the number n+1.
It holds for any n.
It is obvious that all said above holds as well for the infinity process of the
bisecsion of a segment, say, L0 = [0,1]:
      L0, L1, L2, L3, …, Ln, …     (2)
 where for any n    Ln = 1/(2^n).
      In Physics, the universal process of a divition ("bisecsion") of the
matter has a form, say:

[Galaxy]-1, [Sun System]-2, [Earth]-3, [tectonic (maybe other synonim?) plates
system]-4, [substance]-5, [molecule]-6, [atom]-7, [elementary particle]-8,
…      (3)

     As is easy to see, the last process is an essentailly dialectical one:
every division step generates objects, entities with cardinally new properties.
So, I can state that for any finite n the mathematical processes (1) and (2)
are differ cardinally from the physical process (3).
But what we shall have under a passage to the limit: n = = > oo ?
     I can state that for the series (1) we shall get (see below) … Cantor's
least transfinite integer W (omega), for the series (2) we shall get a common
geometrical point (or a real number  (-  [0,1] ) - both as the Leibniz's
(Hilbert's too?) "ideal elements" which are " useful fictions, that can be used
to shorten [arguments] and to speak more generally (pour parler
universellement)". What shall we get in the case of physical process (3) ? -I
think, a physical "ideal elements", i.e. a "point" with the cardinally new
physical property "not to be" a physical object but which is " useful fiction,
that can be used to shorten [arguments] and to speak more generally (pour
parler universellement)".

     Now, why the Cantor "omega" W and a common geometrical point, say, x (-
[0,1] have the same ontological status of "ideal elements" by my opinion?
     As is easy to see, there is the following trivial isomorphism between the
Peano axiomatics for the natural number series (1) and a "theory" for the
bisection process (3) of the segmen L0 = [0,1]:

1. Axiom 1. There is an "undefindable" symbol:   0  <= =>  L0;
2. The main unary operation:      "+1" <= => "/2";
3. Axiom 2:  if n then n+1 <= =>  if Ln then L_(n+1) = Ln/2;
4. Axiom 3.  It does not exist an other n  <= => It does not exist an other Ln;

5. Axiom 4.  IF n=m then (n+1)=(m+1) <= =>  IF Ln=Lm THEN Ln/2 = Lm/2
6. Axiom 5.  For any n: n+1 =/= 0  <= => For any Ln: Ln/2 =/= L0.
7. Axiom 6 (Math. induction). _A P [[P(0)& _A n [P(n) = => P(n+1)]] = => _A n
P(n)] <= =>      _A P [[P(L0)& _A n [P(Ln) = => P(L_(n+1)]] = => _A n
P(Ln)].{here _A is "for any"}

Remark (by S.Kleene): Peano uses 7. Axiom 6 instead of  4. Axiom 3.

Using this isomorphism, we obtain the absolutely regorous mirror-like
(symmetrical in John Barwise sense) proofs of the following conditional
statements.

THEOREM 1. IF the infinite process (2) generates [in any ontological sense !]
an individual geometrical point (real number) THEN the infinite process (1)
generates [in the same ontological sense !] the Cantor "omega".

I do not make more precise this ontological sense (say, it is not of interest
for me), BUT IF I say "A" THEN I must say the mirror-like "B".

THEOREM 2. IF Cantor's "omega" W is the least transfinite integer THEN a
"length" Lw of the corresponding geometrical point is the largest
transfinite-small real number (or infinitesimal, by Leibniz, or the
non-standard real number of the form: 0,000…1_w).

THEOREM 3. IF the construction W+1 has an ontological  sense THEN the
construction 0,000…01 has THE SAME ontological SENSE and is a non-standart real
number of a more high order of  [term ? ] smallness ("infintesimal"-ness ?).

It is obvious, IF Cantor go into the transfinite-large area:   W, W+1, W+2,
W+3, … , THEN we has a (mirror-like) right to go into the non-standard
transfinite-small area:   0,000..1_w; 0,000…01_(w+1); 0,000…001_(w+2); and so
on.

Concluding, I can only cite [believe quite close to the original, instead of my
translation from the Russian] the following fabulous, amaze Prophecy:

“Transfinite integers themselves are, in a certain sense, new irrationalities.
Indeed, in my opinion, the method for the definition of finite irrational
numbers is quite analogous, I can say, is the same one as my method for
introducing  transfinite integers. It can be certainly said: transfinite
integers stand and fall together with finite irrational numbers.”
- Georg Cantor ("At study about transfinity") .

Some more info about said above is in papers that are accessible at my new
WEB-Homepage:

http://www.com2com.ru/alexzen

Sincerely yours,

A.Z.

P.S. I am not sure I have attached the Joke-7 file to my on Wed, 23 Sep 1998
message. If so, I shall try to send it to those who took an interest. Dear
Moshe' Machover, inform me please about receiving that file.


[ Moderator's note: I removed the Joke-7 MS-Word attachment from
Dr. Zenkin's 23 Sep 1998 posting.  The reason is that all FOM postings
are supposed to consist solely of plain ASCII text, as explained in
fom.info.  
-- Steve Simpson ]