[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.
------------------------------------------------------------------------------
More information about the FOM
mailing list