Ripheus23

Reverse forcing

In infinite set theory, forcing is a method whereby the axioms are used to arbitrarily construct a model of V such that conclusions in V can be negated in V+. Thus it can be shown that from the axioms of ZFC set theory, the basic continuum hypothesis is arbitrarily decidable over V/V+. Reverse forcing is the construction of a more limited model of V, one that limits the possible answers to questions posed in V. For example, if the axiom of replacement is not assumed, then V is limited to א_ω. But since the powerset question can still be asked (given that the powerset axiom can still be used), it follows that the powerset question can be answered under V, which means under א_ω. Accordingly, reverse forcing puts a limit on the continuum question. Combined with the formulaic requirement (that if the continuum is some specific aleph-number, its being this number is an instance of a formula), this gives us the prerequisite of the proof in the system that the continuum must be the second aleph-number.

Note that all the hyperoperator questions can be posed under omega-omega, here, but are all advergent from K ^4 K through K ^(omega-omega) K (for K > or = to aleph-zero as such).