[FOM] the Continuum Hypothesis: where I stand.

[Adrian-Richard-David Mathias]

Harvey Friedman asked me, a few weeks ago, to contribute 
to FOM my ideas about the continuum hypothesis; having a
quiet moment, I now do so. 

I start from the belief that while certain areas of mathematics 
have, considered as systems of reasoning, a property that I shall call 
homogeneity, this property does not apply to mathematics considered 
as a whole. I doubt that I can at present define this property
exactly; any more than I can say with precision what constitutes an idea, 
even though I am prepared to say that first-year undergraduates cannot 
tolerate more than one idea per eight hours of a lecture course. 
But homogeneity would apply to an area that develops a coherent body of 
related intuitions. 

Now the continuum hypothesis, CH, postulates a particular relationship
between two objects: R, the real line, and OMEGA, the set of countable 
von Neumann ordinals. (Some might prefer me to say the set of order types 
of countable well orderings). 

I would say that CH is an inhomogeneous statement. R is the product of 
a geometric intuition: it is the completion of the rationals, thought of
as points on a line, by an appeal to notions of continuity. 
OMEGA on the other hand is not a geometric construct. It is in some sense
a completion, by appeal to (abstractly) arithmetical intuitions, of 
the counting process that begins 0, 1, 2, ...  and then realises after
omega steps that some things have been left out.  

I believe that it is this inhomogeneity that makes CH a difficult
question; it might be the same perception of inhomogeneity that has led some
scholars to maintain that CH is an impossible or an incoherent question. 
 
Woodin has drawn attention to the fact that Lusin held that the question 
of the Lebesgue measurability of the projective sets was impossibly
difficult; Lusin gave his reasons, which seem to be based on a sense 
of inhomogeneity of the question similar to the sense I have sketched
above. It is therefore very striking that the hypothesis of Projective
Determinacy solves Lusin's question. [I haven't the text of Lusin's
remarks with me.] There is a moral to be drawn from this episode. 

It is a remarkable development since Cohen showed CH to be unprovable 
that different possible values, among the alephs, for the cardinality of R
no longer seem to be equally interesting or plausible, but that attention 
has narrowed to aleph_1, aleph_2 and not-an-aleph as the three most
interesting, even plausible, possibilities. And it is a remarkable
testament to Goedel's insight that in what was possibly his last
mathematical manuscript he gave "some reasons for believing that the true 
cardinal of the continuum is aleph_2". 

For the moment, that is as far as I get. Thirty years ago I would have 
called CH completely undecidable. Now I think developments might well lead 
us to hold that it is false. 

A.R.D.Mathias, 
   Professeur de Mathématiques Pures

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