Tag Archives: Concept of the continuum

Re: Paper and slides on indefiniteness of CH

Dear Sy,

There is no retreat from my view that the concept of the continuum (qua the set of arbitrary subsets of the natural numbers) is an inherently vague or indefinite one, since any attempt to make it definite (e.g. via L or an L-like inner model) runs counter to what it is supposed to be about. I talk here about the concept of the continuum, not the supposed continuum itself, as a confirmed anti-platonist.  Mathematics in my view is about intersubjectively shared (human) conceptions of idealized structures, not any supposed such structures in and of themselves.  See my article “Conceptions of the continuum” (Intellectica 51 (2009), 169-189).

I can’t have claimed that I have established that CH is neither a definite mathematical problem nor a definite logical problem, since one can’t say precisely what such problems are in either case.  Rather, as workers in mathematics and logic, we generally know one when we see one.  So, the Goldbach conjecture and the Riemann Hypothesis (not “Reimann” as has appeared elsewhere in this exchange) are definite mathematical problems.  And the decidability of the first order theory of the reals with exponentiation is a definite logical problem.  (Logical problems make use of the concept of formal language and are relative to models or axioms.) Even though CH has the appearance of a definite mathematical problem, it has ceased to be one for all intents and purposes because it was long recognized that only logical considerations could be brought to bear to settle it, if at all.  So then what would make it a definite logical problem? Something as definite as: CH is true in L.  I can’t exclude that some time in the future, some model or axiom system will be produced that will be as canonical in nature for some concept of set as L is for the concept of hereditarily predicatively definable set.  But I’m not holding my breath either.

I don’t know whether your concept of set-theoretical truth can be assimilated to Maddy’s A-realism, but in either case I see it as trying to have your platonist cake without eating it.  It allows you to accept CH v not-CH, but so what?

Best,
Sol