This paper is a technical continuation of ``Natural Axiom Schemata Extending ZFC. Truth in the Universe?'' In that paper we argue that $CIFS$ is a natural axiom schema for the universe of sets. In particular it is a natural closure condition on $V$ and a natural generalization of $IFS(L).$ Here we shall prove the consistency of $ZFC\ +\ CIFS$ relative to the existence of a transitive model of $ZFC$ using the compactness theorem together with a class forcing.
Consistency of partition relation for cardinal in (lambda,2^lambda)
The consistency of 2^{aleph_{0}}> aleph_{omega} + I(aleph_{2})=I(aleph_{omega})
On CH + 2^{aleph_1}-> (alpha)^2_2 for alpha < omega_2
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