• Ann. Pure Appl. Logic
• 2013
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. [4, 5, 6, 8,(More)
• Ann. Pure Appl. Logic
• 2008
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)
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 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 אω, אω strong limit, while there is a lightface definable wellorder of H(אω+1) (“definable failure” of the singular cardinal hypothesis at(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 [7], we will show that if V satisfies GCH and F is an Easton function from the regular cardinals into cardinals(More)
• Ann. Pure Appl. Logic
• 2012
The equiconsistency of a measurable cardinal with Mitchell order o(κ) = κ++ with a measurable cardinal such that 2κ = κ++ follows from the results by W. Mitchell [13] and M. Gitik [7]. These results were later generalized to measurable cardinals with 2κ larger than κ++ (see [8]). In [5], we formulated and proved Easton’s theorem [4] in a large cardinal(More)
The paper reviews special Aronszajn trees, both at ω1 and κ + for an uncountable regular κ. It provides a comprehensive classification of the trees and discusses the existence of these trees under different set-theoretical assumptions. The paper provides details and proofs for many folklore results which circulate (often without a proper proof) in the(More)
• 2013
Suppose κ is λ-supercompact witnessed by an elementary embedding j : V →M with critical point κ, and further suppose that F is a function from the class of regular cardinals to the class of cardinals satisfying the requirements of Easton’s theorem: (1) ∀α α < cf(F (α)) and (2) α < β =⇒ F (α) ≤ F (β). In this article we address the question: assuming GCH,(More)
• J. Symb. Log.
• 2016
Let M be a transitive model of ZFC. We say that a transitive model of ZFC, N , is an outer model ofM ifM ⊆ N and ORD∩M = ORD∩N . The outer model theory ofM is the collection of all formulas with parameters from M which hold in all outer models ofM (which exist in a universe in which M is countable; this is independent of the choice of such a universe).(More)