Radek Honzik

Easton's theorem and large cardinals

Abstract The continuum function α ↦ 2 α on regular cardinals is known to have great freedom. Let us say that F is an Easton function iff for regular cardinals α and β , cf ( F ( α ) ) > α and α β → F ( α ) ≤ F ( β ) . The classic example of an Easton function is the continuum function α ↦ 2 α on regular cardinals. If GCH holds then any Easton function is the continuum function on regular cardinals of some cofinality-preserving extension V [ G ] ; we say that F is realised in V [ G ] . However if we also wish to preserve measurable cardinals, new restrictions must be put on F . We say that κ is F ( κ ) -hypermeasurable iff there is an elementary embedding j : V → M with critical point κ such that H ( F ( κ ) ) V ⊆ M ; j will be called a witnessing embedding. We will show that if GCH holds then for any Easton function F there is a cofinality-preserving generic extension V [ G ] such that if κ , closed under F , is F ( κ ) -hypermeasurable in V and there is a witnessing embedding j such that j ( F ) ( κ ) ≥ F ( κ ) , then κ will remain measurable in V [ G ] .

Multiverse Conceptions in Set Theory

We review different conceptions of the set-theoretic multiverse and evaluate their features and strengths. In Sect. 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 Sect. 2, we use this classification to review four major conceptions. Finally, in Sect. 3, we focus on the distinction between actualism and potentialism with regard to the universe of sets, then we discuss the Zermelian view, featuring a ‘vertical’ multiverse, and give special attention to this multiverse conception in light of the hyperuniverse programme introduced in Arrigoni and Friedman (Bull Symb Logic 19(1):77–96, 2013). We argue that the distinctive feature of the multiverse conception chosen for the hyperuniverse programme is its utility for finding new candidates for axioms of set theory.

Fusion and large cardinal preservation

Abstract 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. Dobrinen and Friedman (2010) [3] , Friedman and Halilovic (2011) [5] , Friedman and Honzik (2008) [6] , Friedman and Magidor (2009) [8] , Friedman and Zdomskyy (2010) [10] , Honzik (2010) [12] ).

On strong forms of reflection in set theory: On strong forms of reflection

In this paper we review the most common forms of reflection and introduce a new form which we call sharpgenerated reflection. We argue that sharp-generated reflection is the strongest form of reflection which can be regarded as a natural generalization of the Lévy reflection theorem. As an application we formulate the principle sharp-maximality with the corresponding hypothesis IMH#. IMH# is an analogue of the IMH (Inner Model Hypothesis, introduced in [3]) which is compatible with the existence of large cardinals.

On strong forms of reflection in set theory

In this paper we review the most common forms of reflection and introduce a new form which we call sharp-generated reflection. We argue that sharp-generated reflection is the strongest form of reflection which can be regarded as a natural generalization of the Levy reflection theorem. As an application we formulate the principle sharp-maximality with the corresponding hypothesis IMH # . IMH # is an analogue of the IMH (Inner Model Hypothesis, introduced in Friedman (Bull Symb Log 12(4):591–600, 2006)) which is compatible with the existence of large cardinals.

Global singularization and the failure of SCH

We say thatis µ-hypermeasurable (or µ-strong) for a cardinal µ ≥ � + if there is an embedding j : V → M with critical pointsuch that H(µ) V is included in M and j(�) > 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 satisfying some mild restrictions, then there exists a cardinal-preserving forcing extension V ∗ where F is realised on all V -regular cardinals and moreover: all F(�)-hypermeasurable cardinals �, where F(�) > � + , with a witnessing embedding j such that either j(F)(�) = � + or j(F)(�) ≥ F(�), are turned into singular strong limit cardinals with cofinality !. This provides some partial information about the possible structure of a continuum function with respect to singular cardinals with countable cofinality. As a corollary, this shows that the continuum function on a singular strong limit cardinalof cofinality ! is virtually independent of the behaviour of the continuum function below �, at least for continuum functions which are simple in that 2 � ∈ {� + ,� ++ } for every cardinalbelow � (in this case every � ++ -hypermeasurable cardinal in the ground model is witnessed either by a j with either j(F)(�) ≥ F(�) or j(F)(�) = � + ).

The tree property at the ℵ 2 n 's and the failure of SCH at ℵ ω

Abstract We show – starting from a hypermeasurable-type large cardinal assumption – that one can force a model where 2 ℵ ω = ℵ ω + 2 , ℵ ω is a strong limit cardinal, and the tree property holds at all ℵ 2 n , for n > 0 . This provides a partial answer to the question whether the failure of SCH at ℵ ω is consistent with many cardinals below ℵ ω having the tree property.

A Lifting Argument for the Generalized Grigorieff Forcing

In this short paper, we describe another class of forcing notions which preserve measurability of a large cardinal k from the optimal hypothesis, while adding new unbounded subsets to k. In some ways these forcings are closer to the Cohen-type forcings — e.g. we show that they are not minimal — however, they share some properties with tree-like forcings. We show that they admit fusion-type arguments which allow for a uniform lifting argument.

Definability of Satisfaction in Outer Models

Let M be a transitive model of ZFC. We say that a transitive model of ZFC, N, is an outer model of M if M ⊆ N and ORD ∩ M = ORD ∩ N. The outer model theory of M is the collection of all formulas with parameters from M which hold in all outer models of M (which exist in a universe in which M is countable; this is independent of the choice of such a universe). Satisfaction defined with respect to outer models can be seen as a useful strengthening of first-order logic. Starting from an inaccessible cardinal κ, we show that it is consistent to have a transitive model M of ZFC of size κ in which the outer model theory is lightface definable, and moreover M satisfies V = HOD. The proof combines the infinitary logic L ∞,ω , Barwise’s results on admissible sets, and a new forcing iteration of length strictly less than κ+ which manipulates the continuum function on certain regular cardinals below κ. In the Appendix, we review some unpublished results of Mack Stanley which are directly related to our topic.

Large cardinals and their effect on the continuum function on regular cardinals

In this survey paper, we will summarise some of the more and less known results on the generalisation of the Easton theorem in the context of large cardinals. In particular, we will consider inaccessible, Mahlo, weakly compact, Ramsey, measurable, strong, Woodin, and supercompact cardinals. The paper concludes with a result from the opposite end of the spectrum: namely, how to kill all large cardinals in the universe.