[FOM] Continuum Hypothesis

Bill Taylor W.Taylor at math.canterbury.ac.nz
Fri May 16 01:21:33 EDT 2003

sjmangin at unimelb.edu.au writes:

->1) Do you believe that the continuum hypothesis is true, or false?

Here is my somewhat vague and untutored feeling on the matter.

It is based on the half-baked idea that the only sets that "really exist"
are those that are in some way "definable", (while simultaneously realizing
that this concept itself must needs remain essentially "undefinable").

But I should warn everyone that I am a lone voice in the wilderness here!


My feeling is that prior to CH comes AC; and that AC is clearly false, 
as there are clear and simple counterexamples (e.g. non-measurable realsets).

Then looking at CH, it is either true or false, depending on exactly how
it is worded (in FOL=(eps)).  

If it says...

"there is no infinite set A c R such that neither N nor R bijects to A"

...then it is clearly true.   But if it says...

"there is a function f on domain P(R) such that f(A) is a bijection between
 either R or N or some finite set" 
 ...then it is clearly false.
 The two forms are equivalent in ZFC but not in ZF.
 When I say "clearly", OC I mean that this has only been clear 
 since CH was proved independent of ZFC.
               Bill Taylor        W.Taylor at math.canterbury.ac.nz
    Set theory is a shotgun marriage - between well-ordering and power-set.
    The two parties get along OK; but they hardly seem made for each other.

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