> a letter that Leibniz wrote a couple of months before his death in which
> he states without euphemisms "I told them that I [= Leibniz] did not
> believe at all in the existence of magnitudes actually infinite or actually
> infinitesimal..."
Thank you for the direct Leibniz' citation (today, to my regret, I have no
possibility to attend a library - because of the Russian government's "jokes").
If my memory serves (what follows, I cites by my very old outlines), Cantor
himself writes in his "To Study of Transfinity" [in Russian: "K ucheniju o
Transfinitnom"]: "… in my monograph "Grundlagen einer allgemeinen
Mannigfaltigkeitslehre" (Leipzig, 1883), I decline … the Leibniz authority who
turned out very inconsequent in this question [AZ: on the actual infinity]
I think the fact of Leibniz's inconsequence,
> Leibniz's opinions/beliefs (?) about the infinite seemed to vary according
> to 'pompa & circunstancia',
has two reasons:
1) As (later) Gauss, Leibniz did not want to waste his time for empty
discussions with his own "Beothyers" [sorry, I have no possibility to specify
this name by some linguistic dictionary], and
2) Leibniz, I suppose, knew well that all algorithms (and their results) of his
differential and integral calculuses are not to depend on the senses (actual or
potential) that a front-end, outer user put in the symbol "dx".
I think the same situation takes place today as to Cantor's symbol W (omega)
As is known, Cantor spent a lot of time and words to justify the actual
infinity of all finite natural numbers series:
1, 2, 3, …, n, … (1)
And after that only, he constructs his famous series of transfinite integers
1,2,…,W,W+1,W+2,…W2, W2+1,W2+2,…,W^2,…,W^3,…W^W,…W^W^W,…e0,… (2)
under the strict assumption on the actuality of the common series of all finite
natural numbers (1).
Apropos of the actuality or the potentiality of (1), all mathematicians are
divided into two absolutely not-friendly camps during more than 100 years.
But as Hilbert said, if, say, X is a symbolic construction of a formal system,
say, S, then nothing will change if you will understand X as "a table, a
chair, or a beer mug". But the "nothing" - from the point of view of a
feasibility of algorithms and their results in the framework of the given
system S, of course.
So, now I say: the series (1) can be completed never and it will never
contain all its elements because it is the potentially infinite series. BUT!
Let us denote such the evolving, being constructed, potentially infinite (i.e.
finite for any fixed moment of time) series (1), say, by symbol X, - as we
usually do when a living object is denoted by a "symbol", say, SIMPSON, and
such the object is SIMPSON in its birth moment, in its child birth moments,
and, alas, even long after his death.
In such the case, IF I call X an integer, THEN I turn out in the frame of
usual Peano's (not Cantor's) Arithmetic:
If X is an integer, then X+1 is an integer too; if X+1 is an integer, then X+2
is an integer too; and so on. So, I obtain the potentially infinite series
X, X+1, X+2, … , X+n, … (1-1)
Now, according to Hilbert, I can imagine X = "a table", X = "a chair" or
X = "beer mug" and so on. Nothing will change in the Peano-like series (1-1).
But consider two more interesting cases: A) if I imagine X = "0" then the
series (1-1) is identical to the Peano series (1), B) if I imagine X= "W"
then the series (1-1) is identical (up to the interpretation of the special
symbol "…") to the transfinite semi-interval [W,W2) of the Cantor series (2).
Now, I denote the series (1-1) by a new symbol, say, X2 and call it an integer.
In such case:
If X2 is an integer, then X2+1 is an integer too; if X2 + 1 is an integer,
then X2+2 is an integer too; and so on. So, I obtain the potentially infinite
series
X2, X2 + 1, X2 + 2, … , X2 + n, … (1-2)
I can continue this process of constructing Peano's-like series (1-1), (1-2),
(1-3), … up to any infinity.
Then I paste together in one line (concatenate) all these series and obtain the
Cantor's-like series:
1,2,…,X, X+1, X+2,…, X2, X2+1, X2+2,…,X^2,…,X^3,…X^X,…X^X^X,…, f0. (3)
If somebody has a wish, the construction of the last series (3) can be continue
further over the symbol f0 (the obvious analog for Cantor's symbol
e0=W^W^W^W^... ).
Remark 1. Of course, we can introduce any axioms for formal symbols of the
notation (3), say, of such the kind: 1+X=X=/=X+1, X+X^X = X^X =/= X^X+X and so
on.
Remark 2. Of course, if I use only common names (as finite sequences of symbols
in a finite alphabet), the "number" f0 will be not a first unattainable (in
Cantor's sense) "number" of the series (3) because an amount of such names is
simply not sufficient. In order to overcome the similar difficulty in his
theory of transfinite ordinal integers, Cantor accomplished the ingenious
invention - his famous genetic algebraic polynomial algorithm for generating
any amount of naturally "well-ordered" names (thanks to the well-known genetic,
algebraic and polynomial properties of his algorithm ) - the second class
"numbers", in his terminology:
W, Wn, W^m + Wn, (W^W)l + W^m+Wn, and so on,
where the symbols l,m,n, and so on are common finite natural numbers of the
Peano series (1).
Summarizing,
1) I repeat once more: if the infinity of the Godlike (by L.Kronecker) Natural
Numbers Series (1) we consider as the potential infinity, it will change
nothing in classical G.Cantor's Theory of transfinite ordinal integers.
2) I hope Gregory Moore will agree that LEIBNIZ belongs to my list of the
message:
>On Sat, 12 Sep 1998, Alexander Zenkin wrote:
(*
As is known, Aristotle was the first person who explicitly and definitely
postulated: "Infinitum actu non datur". I think it would be interesting
(for different goals :-) ) to list all his famous like-minded persons.
Today, I have the following very approximate beginning of such the list:
ARISTOTLE, LEIBNIZ, GAUSS, CAUCHY, POINCARE, BROUWER, WEYL, LUZIN, ...
*)
Some other aspect of the problem is considered in the "epochal" Joke No.7 :
"WHETHER GOD EXISTS IN THE TRANSFINITE PARADISE OF GEORG CANTOR?" :-)
For more information you can visit my (completely redrafted at 23 September'98)
WEB-Homepage: http://www.com2com.ru/alexzen
Regards,
Alexander Zenkin
P.S. :-( ! Dear Julio Gonzalez Cabillon and Gregory Moore, as to this answer,
excuse me please for some delay: alas, in Russia of today I have in general
"The End of <my> Science" (not in John Horgan sense, but in the direct,
physical one).