Crítica, Revista Hispanoamericana de Filosofía, Volume 21, number 62, agosto 1989
La hipótesis generalizada del continuo (HGC) y su relación con el axioma de elección (AE)
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José Alfredo Amor
Facultad de Ciencias
Universidad Nacional Autónoma de México


Abstract: The so called Generalized Continuum Hypothesis (GCH) is the sentence: "If A is an infinile set whose cardinal number is K and 2K denotes the cardinal number of the set P(A) of subsets of A (the power set of A), and K + denotes the succesor cardinal of K, then 2K = K +".
The Continuum Hypothesis (CH) asserts the particular case K = o. It is clear that GCH implies CH.
Another equivalent version of GCH, is the sentence: 'Any subset of the set of subsets of a given infinite set is or of cardinality less or equal than the cardinality of the given set, or of the cardinality of all the set of subsets".
Gödel in 1939, and Cohen in 1963, settled the relative consistency of the Axiom of Choice (AC) and of its negation not-AC, respectively, with respecllo the Zermelo-Fraenkel set theory (ZF). On the other hand, Gödel in 1939, and Cohen in 1963 settled too, the relative consistency of GCH , CH and of its negations not-GCH, not-CH, respectively, with respect to the Zermelo-Fraenkel set theory with the Axiom of Choice (ZF + AC or ZFC). From these results we know that GCH and AC are undecidable sentences in ZF set theory and indeed, the most famous undecidable sentences in ZF; but, which is the relation between them?
From the above results, in the theory ZF + AC is not demonstrated GCH; it is clear then that AC doesn't imply GCH in ZF theory, Bul does GCH implies AC in ZF theory? The answer is yes! or equivalently, there is no model of ZF +GCH + not-AC.
A very easy proof can be given if we have an adecuate definition of cardinal number of a set, that doesn't depend of AC but depending from the Regularity Axiom, which asserls that aIl sets have a range, which is an ordinal number associated with its constructive complexity. We define the cardinal number of A, denoted |A|, as foIlows:

|A|= {

The least ordinal number equipotent with A, if A is well orderable
The set of all sets equipotent with A and of minimum range, in other case.

It is clear that without AC, may be not ordinal cardinals and all cardinals are ordinal cardinals if all sets are well orderable (AC). Now we formulate:

GCH*: For all ordinal cardinal I<, 2K = I< +

In the paper is demonstrated that this formulation GCH* is implied by the traditional one, and indeed equivalent to it.
Lemma, The power set of any well orderable set is well orderable if and only if AC.
This is one of the many equivalents of AC in ZF,due lo Rubin, 1960.
Proposition. In ZF is a theorem: GCH* implies AC.
Supose GCH*. Let A be a well orderable set; then |A| = K an ordinal cardinal, so A is equipotent with K and then P~A) is equipotent with P(K); therefore |P(A)I|= |P(K)| = 2K = K+. But then |P(A)|= K+ and P(A) 'is equipotent with K+ and K+ is an ordinal cardinal; therefore P(A) is well orderable with the well order induced by means of the bijection, from the well order of K+.
Corolary: In ZF are theorems: GCH impIies AC and GCH is equivalent to GCH*.
We see from this proof, that GCH asserts that the cardinal number of the power set of a well orderable set A is an ordinal, which is equivalent to AC, but GCH asserts also that that ordinal cardinal is |A|+ , the ordinal cardinal succesor of the ordinal cardinal of the well orderable set A.
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