A Galvin-Hajnal theorem for generalized cardinal characteristics

We prove that a variety of generalized cardinal characteristics, including meeting numbers, the reaping number, and the dominating number, satisfy an analogue of the Galvin-Hajnal theorem, and hence also of Silver's theorem, at singular cardinals of uncountable cofinality.Comment: 18 page

Funkce kontinua na singulárních kardinálech

Bakalářská práce se zabývá chováním funkce kontinua na singulárních kardinálech v teorii ZFC. Práce je rozdělena na dvě části. První část se soustředí na Silverovu větu a rozebírá dva různé důkazy této věty, původní Silverův a čistě kombinatorický důkaz dle Baumgartnera a Přikrého. Druhá část je věnována hypotéze singulárních kardinálů, která ovlivňuje chování funkce kontinua. V práci je ukázáno, za předpokladu velkých kardinálů, že hypotéza singulárních kardinálů je nedokazatelná nad teorií ZFC. Pomocí Eastonova a Přikrého forcingu je nalezen model ZFC, ve kterém hypotéza singulárních kardinálů neplatí.Bachelor thesis studies the behaviour of the continuum function on singular cardinals in theory ZFC. The work is divided into two parts. The focus of the first part is on the Silver's Theorem and it analyzes two different proofs of this Theorem, Silver's original proof and the second, purely combinatorial, proof by Baumgartner and Prikry. The second part is devoted to the Singular Cardinal Hypothesis, which influences the behaviour of the continuum function. In the thesis it is shown that, in the presence of large cardinals, Singular Cardinal Hypothesis is not provable in ZFC. Using Easton and Prikry forcing a model is found where the Singular Cardinal Hypothesis does not hold.Department of LogicKatedra logikyFaculty of ArtsFilozofická fakult

The power function

The axioms of ZFC provide very little information about the possible values of the power function (i.e. the map K---->2ᴷ). In this dissertation, we examine various theorems concerning the behaviour of the power function inside the formal system ZFC , and we :;hall be p:trticul:trly interested in results which provide eonstraints on the possible values of the power function. Thus most of the results presented here will be consistency results. A theorem of Easton (Theorem 2.3.1) shows that, when restricted to regular cardinals, the power function may take on any reasonable value, and thus a considerable part of this thesis is concerned with the power function on singular cardinals. We also examine the influence of various strong axioms of infinity, and their generalization to smaller cardinals, on the possible behaviour of the power function

The power function

The axioms of ZFC provide very little information about the possible values of the power function (i.e. the map K---->2ᴷ). In this dissertation, we examine various theorems concerning the behaviour of the power function inside the formal system ZFC , and we :;hall be p:trticul:trly interested in results which provide eonstraints on the possible values of the power function. Thus most of the results presented here will be consistency results. A theorem of Easton (Theorem 2.3.1) shows that, when restricted to regular cardinals, the power function may take on any reasonable value, and thus a considerable part of this thesis is concerned with the power function on singular cardinals. We also examine the influence of various strong axioms of infinity, and their generalization to smaller cardinals, on the possible behaviour of the power function