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)
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 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)
We review different conceptions of the set-theoretic multiverse and evaluate their features and strengths. In Section 1, we set the stage by briefly discussing the opposition between the 'universe view' and the 'multiverse view'. Furthermore, we propose to classify multiverse conceptions in terms of their adherence to some form of mathematical realism. 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)