[FOM] As to strict definitions of potential and actual infinities.
Alexander Zenkin
alexzen at com2com.ru
Tue Dec 31 04:45:15 EST 2002
As far back as about two millenniums ago Aristotle first
explicitly distinguished and described two opposite types of infinite -
potential infinite and actual infinite. From that time long,
ineffective, mainly speculative discussions are carried on whether the
actual infinite exists, in what sense it can exist, whether an infinite
set can be considered as a completed, individual entity, whether the
generality quantifier in "for all n P(n), n (- N" makes the infinite
set, N={1,2,3,...}, actual, whether it is possible in general to use
actually infinite, completed sets in mathematics, etc.
Cantor was first who in 70s of XIX century explicitly introduced
the actual infinity in mathematics.
Today every meta-mathematician and set theorist knows well that
Cantor's 'naive' set theory as well as all modern axiomatic set theories
are based on the actual infinity concept. Indeed if, for example, a set
of real numbers as well as these numbers themselves as infinite, say
binary, sequences are potential in Aristotle's sense then the Cantor's
theorem on the uncountability of continuum becomes unprovable and
therefore any distinguishing of infinite sets by their cardinalities as
well as transfinite cardinal and ordinal 'numbers' themselves lose any
sense and attraction.
However hitherto there are not strict definitions of the concepts
of the potential and actual infinities. This fact generated a widespread
opinion that "the ideas of potential vs. actual infinity are vague,
fuzzy, speculative, that the idea of actual infinity makes sense only
within a platonistic conception of mathematics, and therefore all such
the ideas are at the informal, philosophical level and can't be a part
of the axiomatic set theories."
However I believe that strict mathematical definitions of the
concepts of the potential and actual infinities are possible. If so,
such the definitions could help to understand better a lot of basic
problems of the modern Foundations of Mathematics, Logic, and Philosophy
of Infinity.
I offer below one of possible versions of such the definitions.
Any remarks, opinions and suggestions are welcome.
Happy New Year-2003 to all.
Alexander Zenkin
^^^^^^^^^^^^^^^^^^^^^^^^^^^^^^^^^^^^^^^^^^^^^^^^^^^^^^^^^^^
Prof. Alexander A. Zenkin
Doctor of Physical and Mathematical Sciences
Leading Research Scientist
Department of Artificial Intelligence Problems
Computing Center of the Russian Academy of Sciences
Vavilov st. 40,
117967 Moscow GSP-1,
Russia
e-mail: alexzen at com2com.ru
URL: http://www.com2com.ru/alexzen/
^^^^^^^^^^^^^^^^^^^^^^^^^^^^^^^^^^^^^^^^^^^^^^^^^^^^^^^^^^^
A STRICT DEFINITION OF THE CONCEPTS
OF POTENTIAL AND ACTUAL INFINITIES.
DEFINITIONS OF THE CONCEPT OF POTENTIAL INFINITE (PI).
D1) ARISTOTLES' DEFINITION (AZ: the insertions in brackets below
are of mine): " ... the infinite exists through one thing [n+1] taken
after [>] another [n], what is taken being always finite [n < oo], but
ever other and other [n -- > oo]".
D2) AXIOMATIC DEFINITION.
(P1) There exists a 'thing' '0' (since any well-ordered
finite sequence of 'things' has a first 'thing'; we denote this first
'thing', say, as '0').
(P2) [if n is a 'thing' (a natural) then n+1 is a 'thing' (a
natural) as well] & [n < n+1].
(P3) There are not 'things' (natural numbers) that are
different from those defined by (P1)&(P2).
As is easy to see, it is a strict, formal, axiomatic, inductive
definition of the common series of the common finite natural numbers:
1, 2, 3, ..., n, ...
(*)
and the points (P1)-(P3) are the first three axioms of Peano's
system.
D3) THE MAIN MATHEMATICAL PROPERTY of the PI-series (*).
THEOREM. There does not exist a maximal, last element in the
series (*).
PROOF. Assume that n* is a last element of (*). Since n* is
natural then n*+1 is natural too and n*+1 > n*. So, n* is not a last
element of (*). Contradiction. QED.
ALGORITHMICAL DEFINITION.
The process of the series (*) construction can be presented by the
computer program:
BEGIN INTEGER i; LABEL L; i:=0; L: i:=i+1; PRINT(i); GOTO L END
(**)
D4) The step-by-step PI-process (**) of the series (*)
construction does not ever arrive at its 'STOP' ('HALTING') state.
And THEREFORE
D5) the PI-process (**) can't produce an INDIVIDUAL mathematical
object as its FINAL RESULT, e.g., according to Cantor's 'naive' and
vague 'definition', as "a completed, not changeable, but definite and
invariable in all its parts" non-finite number, sequence, set, and the
like.
DEFINITION OF THE CONCEPT OF ACTUAL INFINITE (AI).
I) CANTOR'S DEFINITION (almost verbatim): "it is well known that
the number of finite natural numbers in the series (*) is infinite, and
therefore there is no maximal, last number in (*) ...; however
contradictory it might seem [it is really contradictory very much, as
Cantor himself was well understanding! - AZ], there is in fact no
absurdity ["The essence of pure mathematics is its freedom"! - So any
fantasies are admissible! - AZ] in denoting the series (*) as a whole
using a name (or symbol), say, 'omega', calling the name 'omega' an
integer and then going on to count:
'omega', 'omega'+1, 'omega'+2, 'omega'+3, . . . ,"
(C)
and one can add - in the complete conformity with ... the
Aristotle-Peano's axiom: "if a 'thing' is [called - AZ] integer then the
'thing'+1 is integer too" (for any 'thing' independently of a "real
nature" of the 'thing' and what we think of the 'thing').
So, Cantor accepts the Aristotle-Peano's axioms D2 and the main
mathematical property D3, but rejects their algorithmic consequences
D4-D5, i.e., he accepts the negations of D4 and D5 in the following
form.
ALGORITHMICAL DEFINITION of the AI-notion.
D3) THE MAIN MATHEMATICAL PROPERTY of the series (*).
THEOREM. There does not exist a last element in the series (*).
D'5) the process (**) produces (AS ITS FINAL RESULT) a sequence
(*) as an INDIVIDUAL 'mathematical' object - the famous Cantor's minimal
transfinite ordinal type 'omega' that is a "completed, not changeable,
but definite and invariable in all its parts" entity.
And THEREFORE
D'4) The step-by-step process (**) of the series (*) construction
arrives at its 'STOP' ('HALTING') state.
REMARK 1. The point D'4 is a necessary feature of the actual
infinity definition, since otherwise the process (**) must continue
ever, e.g., in the time when Cantor claims the series (*) an integer
'omega' and constructs his 'transfinite' series (C). That is if D'4 does
not hold then Cantor's 'omega' can't be a "completed, not changeable,
but definite and invariable in all its parts" individual mathematical
object.
REMARK 2. I used above the original Cantor's definition of the
actual infinity because today there is not a better definition in all
modern 'non-naive' axiomatic set theories.
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