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Cantor's Theorem and Cantor's Paradox - Essential Relations
Cantor's Theorem - that every set has a powerset whose power is greater than the set - and Cantor's Paradox - that there can not be a set of all ses - are intimately and essentially related. That this issue strikes the heart of mathematics remains an odd political dilemma in modern mathematics' history and practice.
Source: Robbie Lindauer, 2004-04-23
Candidate: Robert Lindauer
Cantor's Theorem that every powerset has a number greater than the set of which it is the powerset is near-axiomatic in modern mathematics. This is to say, the only thing separating it from its status as an axiom is the fact that it can be proved from some accepted axioms - namely the powerset axiom, the axiom of foundation, and the axiom of infinity. The combination of these axioms along with the definition of ">" for sets gives an adequate proof of cantor's theorem. Cantor's theorem in this form has been accepted for nearly a century and has become canonical and well accepted in mathematics despite its suprising implications.
Those suprising implications include the notion that there are inaccessible cardinals - numbers which can not be reached by counting - and that there are strongly inaccessible cardinals - numbers that can not be reached by any method at all. These two notions leave one with a strange paradox which I intend to uncover in this essay.
The Powerset Axiom and Infinity
The notion that there is a set for which the number of members is infinite is codified in the axiom of infinity which states briefly that for every set, there exists a set which is the successor of the set in cardinality or one which has for its members the set, and the set that contains the set and the empty set. To see how it works, it's relevant to pick it apart.
The empty set is {} - the group of nothings. There is precisely one empty-set. The cardinality of the empty-set is 0, the cardinality of the set of the empty-set and itself is 1, e.g. { {}, }. The cardinality of the set (1) and the set of the empty-set and itself is 2, etc. This leads to the fact that there is a set of all sets of this succession-type - namely aleph-0 - the cardinality of the first number-type.
This set is typically shown as {1, 2, 3, 4, 5, ... } or the range of primitive recursive arithmetic which states simply:
1 is a number
anything plus 1 is a number
nothing else is a number.
or
E(x): x = 1; A(x) E(y): y = x + 1
(the set-theoretical definition is meant to clarify this conception).
The powerset axiom states that there is a powerset of the set in quesiton. The set we designate w. The powerset we designate P(w). The cardinality of P(w) is greater than w. The axiomatic justification of this fact is unimpeachable - to have a greater cardinality one must have more members, and P(w) must have more members than w.
There are two exceptions to this notion. In the first place, it's not clear that in fact every set has a powerset. Upon assertaining the meaning of cardinality, it is clear that the set of cardinal numbers, for instance, does not exist, and that its powerset can not exist for this would assert the existence of a number that was simultaneously greater-than and less-than itself. This order of contradiction we'll term a "primitive contradiction" - or a contradiction of the notion of the primitive number series.
In the second place, it's possible that there is no powerset in the cases of infinite sets - while it's clear that every finite set has a powerset, it's not clear that every set has a powerset. This would require an argument establishing the existence of the powerset. Instead we are given it by axiom -that is - it is assumed.
In the case of the primitive number series its clear that there is no powerset, for assuming there were, it would mean that there was a number for which it was equal to itself plus an element not a member of itself, which is a primitive contradiction - it means that for some n, n = n + 1.
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