[FOM] Mathias on the Continuum Hypothesis.

Harvey Friedman friedman at math.ohio-state.edu
Thu Jun 26 18:44:09 EDT 2003

Reply to Mathias 8:39PM 6/25/03.
>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.

Obviously we would like to hear a bit more about "homogeneity". For 
instance, you could tell us about some other inhomogeneities, and 
also some paradigms of homogeneities. What if you mix integers and 
real numbers? Rationals and real numbers? Pointwise convergence and 
linear functions?

>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).

Of course, you are not mentioning the general concept of function here.

>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.

They are both completions in some sesne.

>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.

You are not emphasizing the general notion of function here.

>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.

Due to Mycielski, I think. But what do you make of Projective 
Determinacy? Are you recommending that it be adopted as an axiom for 
mathematics? If so, why, and if not, why not?

>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.

Is this focus on these three  permanent or just what we have now?

>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".

Is his ideas there, which are reputed to be technically incorrect, 
related to any subsequent work?

>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.

Is there any reason to regard CH as false short of accepting current 
points of view that fix the value? I.e., is it easier to argue 
against CH than to argue what cardinalt 2^omega is?

Harvey Friedman

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