Consistency of the internal forcing schema IFS

From papers

Let IFSIFS be the axiom schema asserting that for every weakly absolutely definable set XX and every partial order PP definable in L(X)L(X), if

XV[G]=XV(r.o.P)V=1,\big|\big|X^{V[G]}=X^{V}\big|\big|^{(r.o.P)^V}=1,

then there is an L(X)L(X)-generic subset GG of PP. A universe has a gap when these hypotheses hold but no such generic subset exists; IFSIFS asserts that there are no gaps. Consistency conjecture. ZFC+IFSZFC+IFS is consistent. This is a consistency claim for the proposed internal forcing schema, motivated by the idea that gaps in the existence of generics should not occur. The supplied text does not provide a relative consistency proof or a refutation.

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Sources & referencesView supporting material

Primary source

Garvin Melles, “Some Natural Internal Forcing Schemata Extending ZFC”, arXiv:math/9209209 (1992).

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