Kristian Berry
  • Member for 1 year, 11 months
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I am not a moth.

So in my system of set theory, there is an argument not directly for CH, but for the slightly different proposition that j(CH) = (ω-1)/ω. (Here, j(S) is the justification function on sentences S.) Or, conversely, that the unjustification function on CH yields the justification value (degree) 1/ω, i.e. CH is unjustified to that extent. The argument starts from forcing as an S-justifier.† Then it is supposed that the length of a forcing coincides with its degree of justification: the longer the forcing, the less justified it is. This is not quite obvious, so it is noted that for reasons of cofinality, 20 does not equal ℵω. Accordingly, we can say that j("20 = ℵω") = 0. With the modal argument for 2n < ℵω in place, we suppose then that the justification sequence for S reporting the cardinality of the Continuum proceeds from 0 = 0/ω = (ω-ω)/ω, so that when we abstract over this, we arrive at j("20 = ℵn") = (ω - n)/ω. Then j("20 = ℵ1") = (ω-1)/ω. QED

Here, we understand forcing as a special kind of disjunction elimination, with the CH issue occurring relative to an infinite disjunction. Namely, modulo what is provable in ZFC, the only upper bound on the cardinality of the Continuum is i = the first strongly inaccessible uncountable cardinal, so we have an i-long disjunction's worth of options. In other words, ZFC has the resources to manifest this disjunction, but not to perform disjunction elimination on it in a satisfactory manner. Nevertheless, to force this or that disjunct to be true is imagined to in a sense be a real operation we can perform on set-theoretic sentential space, with an attendant question of abstract justification in place.

Also, let שn be the sequence cofinal with ℵ0, modulo W when W = the smallest worldly cardinal. This gives us that שω = W, which is an aesthetically nice representation of the fact.

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