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An axiom to settle the continuum hypothesis ?


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#1 Adib

Adib

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Posted 26 July 2008 - 10:07 PM

An axiom to settle the continuum hypothesis ?

(Logic Colloquium 2004 contributed abstract)
Paul Cohen used a set of generic reals to prove the consistency of the negation of the continuum hypothesis with other axioms. It is my opinion that such sets do not really exist for a Platonist. My opinion is that the continuum hypothesis is true.

Here is a tentative axiom from me to try to prove it. Axiom : An infinite subset of the power set of N has a bijection either with a countable union of (pair wise disjoint) sets of n elements or with a countable Cartesian products of (pair wise disjoint) sets of n elements.

Mr Andreas Blass proved that this axiom is equivalent to the continuum hypothesis. So, the axiom is consistent with the other usual axioms and independent from them, from the works of Kurt Godel and Paul Cohen, respectively. Mr Andreas Blass used the assumption that the Cartesian product is not the empty set but he did not use the axiom of choice.

The question which remains is : is it a good axiom ? My opinion is that it is realistic for a Platonist. But may be the axiom is not simple enough and may be a simpler one could be found.
Adib Ben Jebara.