The continuum function F on regular cardinals is known to have great freedom; if α, β are regular cardinals, then F needs only obey the following two restrictions: (1) cf(F (α)) > α, (2) α < β → F (α) ≤ F (β). However, if we wish to preserve measurable cardinals in the generic extension, new restrictions must be put on F. We say that κ is F… (More)
We show first that it is consistent that κ is a measurable cardinal where the GCH fails, while there is a lightface definable wellorder of H(κ +). Then with further forcing we show that it is consistent that GCH fails at ℵ ω , ℵ ω strong limit, while there is a lightface definable wellorder of H(ℵ ω+1) (" definable failure " of the singular cardinal… (More)
Let κ < λ be regular cardinals. We say that an embedding j : V → M with critical point κ is λ-tall if λ < j(κ) and M is closed under κ-sequences in V. Silver showed that GCH can fail at a measurable cardinal κ, starting with κ being κ ++-supercompact. Later, Woodin improved this result, starting from the optimal hypothesis of a κ ++-tall measurable cardinal… (More)
We say that κ is µ-hypermeasurable (or µ-strong) for a cardinal µ ≥ κ + if there is an embedding j : V → M with critical point κ such that H(µ) V is included in M and j(κ) > µ. Such j is called a witnessing embedding. Building on the results in , we will show that if V satisfies GCH and F is an Easton function from the regular cardinals into cardinals… (More)
In this paper we introduce some fusion properties of forcing notions which guarantee that an iteration with supports of size ≤ κ not only does not collapse κ + but also preserves the strength of κ (after a suitable preparatory forcing). This provides a general theory covering the known cases of tree iterations which preserve large cardinals (cf.
We wish to review some conceptions of the set-theoretic multiverse and evaluate their strength. In §1, we introduce the uni-verse/multiverse dichotomy and discuss its significance. In §2, we discuss three alternative conceptions. Finally, in §3, we present our own conception as integral to the Hyperuniverse Programme launched by Friedman and Arrigoni in… (More)
The equiconsistency of a measurable cardinal with Mitchell order o(κ) = κ ++ with a measurable cardinal such that 2 κ = κ ++ follows from the results by W. Mitchell  and M. Gi-tik . These results were later generalized to measurable cardinals with 2 κ larger than κ ++ (see ). In , we formulated and proved Easton's theorem  in a large… (More)