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the Continuum Hypothesis is true
Mar. 11th, 2006 08:39 pmI'm normally not very interested in controversies about foundational axioms for mathematics, since they tend to concern infinities that are too abstract to mean anything to me. Would any scientists care if ZF were all they could work with?
Furthermore, I'm not a Platonist, and I don't like to talk about things in mathematics as being "true" (long ago, I added another argument to this predicate: "theory T is true" became "theory T is good model of domain D", whether D can refer to things in the physical world or to intuitions).
But yesterday, my boss, who is not an academic, let alone a logician, led me to the following insight: in a certain sense, CH is *true*. The independence result implies that one cannot construct a set S such that |Naturals| < |S| < |Reals| (if you could, this would be a proof that ~CH follows from ZFC. I wish I understood how to define this constructibility in terms of axioms).
Any sets that you stick in there are "made up" and cannot be exhibited concretely (not that the Reals is very concrete either, but adding the limits of all Cauchy sequences seems like a rather natural way to construct it).
My boss was thinking about the problem in concrete terms, something which I hadn't done in a long time.
Furthermore, I'm not a Platonist, and I don't like to talk about things in mathematics as being "true" (long ago, I added another argument to this predicate: "theory T is true" became "theory T is good model of domain D", whether D can refer to things in the physical world or to intuitions).
But yesterday, my boss, who is not an academic, let alone a logician, led me to the following insight: in a certain sense, CH is *true*. The independence result implies that one cannot construct a set S such that |Naturals| < |S| < |Reals| (if you could, this would be a proof that ~CH follows from ZFC. I wish I understood how to define this constructibility in terms of axioms).
Any sets that you stick in there are "made up" and cannot be exhibited concretely (not that the Reals is very concrete either, but adding the limits of all Cauchy sequences seems like a rather natural way to construct it).
My boss was thinking about the problem in concrete terms, something which I hadn't done in a long time.
