[FOM] CH and vagueness

tf at maths.cam.ac.uk tf at maths.cam.ac.uk
Fri Aug 22 12:37:40 EDT 2014

There may be something to be gained in this discussion by putting oneself 
in the shoes of a good student who is just being exposed to these ideas. To 
put it in a hand-wavy fashion (i crave your indulgence) the real line is a 
continuous/analogue object and the set of countable ordinals is a 
discrete/digital object - in the terms probably used by undergraduates. It 
is not at all clear to them that these quantities are even *commensurable* 
(to use a good, old word) *at all*. (And even to get to that stage they 
have to grasp the idea of the set of all countable ordinals, and that 
requires some confidence and some sophistication. The same problem appears 
in embryonic form when one tries to persuade them that there are precisely 
$2^{\aleph_0}$ reals - i.e., that there is a bijection between the reals 
and the power set of the naturals. For various intelligible reasons this 
does not seem to them a natural thing to do, and altho' there are cute 
bijections they are not at all easy (even for good students) to find - they 
are so unnatural.

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