The core model

Weight of closed subsets topologically generating a compact group

“…Therefore, ¬( †) ⇒ ¬SCH. Dodd and Jensen [5,6] proved that the consistency of ZFC + ¬SCH implies the existence of an inner model with a measurable cardinal. Hence, to violate ( †) one needs the strength of large cardinals.…”

Scales of minimal complexity in $${K(\mathbb{R})}$$

“…7 We shall work in ZF + DC and state any additional hypotheses as we need them. This is done to keep a close watch on the use of determinacy in the proofs of our main theorems.…”

“…We initiated the mixture of two techniques to produce definable scales in K (R) beyond those in L(R): fine structure and iterated ultrapowers. After defining iterable real premice and extending the basic fine structural notions of Dodd-Jensen [7] to encompass iterable "premice above the reals," we were able to prove the following result (see [1,Theorem 4.4]). Dodd-Jensen's notion of a mouse to that of a real mouse (see [4, subsection 3.4]).…”

On the regular extension axiom and its variants

“…This large cardinal assumption might seem exaggerated, but it is known that the consistency of all uncountable cardinals being singular cannot be proved without assuming the consistency of the existence of some large cardinals. For instance, it was shown in [8] that if ℵ 1 and ℵ 2 are both singular one can obtain an inner model with a measurable cardinal. Corollary 7.1 If ZF + "All uncountable cardinals are singular" is consistent, then so are the following theories:…”