[FOM] Continuum Hypothesis
Harvey Friedman
friedman at math.ohio-state.edu
Sun May 18 14:34:21 EDT 2003
Reply to Spector 11:56PM 5/18/03.
>On Saturday, May 17, 2003, at 09:35 PM, Harvey Friedman wrote:
>>There is no consensus among logicians, and apparently
>>not even among set theorists, that anyone has presented
>>reasonably compelling axioms that decide CH, or might
>>decide CH.
>
>AD implies CH, in the form that every uncountable set
>of reals can be placed in 1-1 correspondence with the
>set of all reals.
>
>Moreover, the proof uses a method that Cantor would
>appreciate, in that every uncountable set of reals
>contains a perfect subset.
>
>I think AD can be characterized as "reasonably compelling".
>
AD is not set theory, in the sense that the people who work on CH
mean by "set theory". I.e., any conceivable arrangement of sets that
is not too big forms a set. This is because AD implies the negation
of the axiom of choice, and conceivably one can pick an element of
out each set in a set of pairwise disjoint nonempty sets.
AD is an interesting statement in another theory, call it smet
theory. Altough this is besides the point, smet theory has never been
discussed in any conceptually clear way: certainly not in any
sufficiently conceptually clear way that would make AD compelling.
There is a strong consensus - not unanimous, as shown by Spector -
among set theorists, concerning the previous two paragraphs.
There are several wise set theorists on this FOM list who could comment on CH.
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