[FOM] [HM] Infinity and the "Noble Lie"
Alexander Zenkin
alexzen at com2com.ru
Sun Dec 25 23:52:16 EST 2005
James Landau wrote at the [HM]-list (on 19 Dec 2005 5:58 ):
<begin quote>
I believe there is a way around having to use what Alexander Zenkin
describes as
"Bourbaki formulate this "tenet of the "ZFC religion" as follows:
AXIOM OF INFINITY. There exists an infinite set."
(Sun, 11 Dec 2005 20:11:41 +0300 From: "Alexander Zenkin" <
alexzen at com2com.ru >)
1) Let us start with the positive integers. Without using the (as yet
undefined) word "set" let us define certain binary operations on the
integers, of which one-to-one correspondence is the most important.
We now define "countably infinite set" as "anything which can be
placed in one-to-one correspondence with the positive integers".
2) What about "uncountably infinite sets"? Cantor's diagonal proof is
not necessary, as we can proceed via measure theory. Any countably
infinite set has measure zero. There exist certain sets that have a
measure greater than zero, e.g. any interval on the real number line. We
now define any set with a positive measure as an "uncountably infinite
set." Have I cheated?
<end quote>
Yes, you have cheated: the famous Cantor set ('Cantor's dust') is an
"uncountably infinite set" that has the Lebesgue measure of zero, and
therefore it is a counter-example to your 'definition' of uncountable
sets. Of course, you may ("The essence of mathematics lies in its
freedom") call any set with a positive measure as an "uncountably
infinite set", but I doubt very much that you can prove that the
cardinality of your "uncountably infinite set" is greater than the
cardinality of the set of positive integers, i.e. prove that there is
not a 1-1-correspondence between these sets, not using Cantor's diagonal
method.
James Landau concludes:
<begin quote>
What I think I have accomplished is to defend Cantor and other set
theorists of the charge that they failed to explicitly state certain
axioms. Rather they have assumed or implied certain axioms, including
"the positive integers exist" and "the real number line exists", that
their critics such as Brouwer (cited above) also implicitly accept.
<end quote>
You certainly proved that in modern Axiomatic Set Theory (further -AST)
there are implicit ('assumed or implied') statements that are in reality
necessary conditions of some AST-proofs. But you could not prove that
the explication of such hidden statements is forbidden in AST. Moreover
the history of mathematics itself is a history of explications, more
clear formulations, and formalizations of primarily informal, fuzzy, and
hidden statements some of which later became axioms.
The explication of any hidden statement which is used as a necessary
condition of a mathematical proof can't do harm to mathematics, but
sometimes can help to avoid a trouble in the case when the addition of
the hidden statement to an axiom system of a given area leads to a
contradiction.
So, we have the following situation.
The paradigmatic AST-Theorem on the uncountability of continuum is
proved using algorithmically some consequences of the Cantor axiom ("all
infinite sets are actually infinite"), since if they use the Aristotle
axiom ("the infinite exists potentially"), the AST-Theorem becomes
unprovable (together with all Cantor's 'Study on Transfinitum'). However
modern AST outlaws the Cantor axiom and forbids to mention it at all.
I see no mathematical reasons against the addition of the Cantor
axiom to ZFC-system. By the way, a strict definition of the actual
infinity notion, used in Cantor's axiom formulation, is given first in
the papers shown in my previous message to FOM- and HM-lists.
I (together with Kronecker, Hermit, Poincare, Weyl, Brouwer,
etc., etc., etc.) have a firm belief (together with a strict proof of
the belief) that nobody will be able ever to defend Cantor's theory of
transfinite ordinals and cardinals.
Merry Christmas and Happy New Year !
Alexander Zenkin
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