Consequences of arithmetic for set theory

In this paper, we consider certain cardinals in ZF (set theory without AC, the Axiom of Choice). In ZFC (set theory with AC), given any cardinals C and D, either C <= D or D <= C. However, in ZF this is no longer so. For a given infinite set A consider Seq(A), the set of all sequences of A without repetition. We compare |Seq(A)|, the cardinality of this set, to |P(A)|, the cardinality of the power set of A. What is provable about these two cardinals in ZF? The main result of this paper is that ZF |- for all A: |Seq(A)| not= |P(A)| and we show that this is the best possible result. Furthermore, it is provable in ZF that if B is an infinite set, then |fin(B)|<|P(B)|, even though the existence for some infinite set B^* of a function f from fin(B^*) onto P(B^*) is consistent with ZF

Relations between some cardinals in the absence of the Axiom of Choice

If we assume the axiom of choice, then every two cardinal numbers are comparable. In the absence of the axiom of choice, this is no longer so. For a few cardinalities related to an arbitrary infinite set, we will give all the possible relationships between them, where possible means that the relationship is consistent with the axioms of set theory. Further we investigate the relationships between some other cardinal numbers in specific permutation models and give some results provable without using the axiom of choice

ZFC vs. NFU

Von Beginn an ist dasjenige Gebiet der mathematischen Grundlagenforschung, das man `Mengenlehre' nennt, heiß umfehdet - und zwar nicht bloß in `mathematischer', sondern auch in philosophischer Hinsicht. Formal ist eine (axiomatische) Mengenlehre eine mathematische Theorie wie jede andere, gleichgestellt der Gruppentheorie, der Peano-Arithmetik erster Stufe oder der Theorie der partiellen Ordnungen. De facto ist sie primus inter pares, weil es möglich ist alle anderen mathematischen Theorien innderhalb der Mengenlehre zu `verhandeln'. Unmengen an verschiedenen Systemen der axiomatischen Mengenlehre gibt es mittlerweile, beinahe jeder große Logiker des zwanzigsten Jahrhunderts hat ein eigenes. Einige unterscheiden sich nur graduell, andere entstammen grundverschiedenen Denktraditionen - und dennoch beanspruchen alle, den Begriff der `Menge' angemessen zu formalisieren. Zwei solche Systeme (NFU als Repräsentant einer eher `logisch' orientierten Denktradition und ZFC als Repräsentant einer genuin `mathematischen' Tradition) sollen in dieser Arbeit gegenübergestellt werden. Dabei ist ein erstes Ziel, herauszustellen, welche Sätze in dem einen System beweisbar sind, nicht jedoch im anderen (und umgekehrt). Vor allem in der transfiniten Arithmetik wird sich zeigen, dass die Unterschiede vielfältig sind. Das zweite (und vorrangige) Ziel besteht darin, aufzuzeigen, welche konkurrierenden Intuitionen zum Begriff `Menge' die beiden Systeme motivieren, und in welchem Verhältnis diese Intuitionen zu deren formalisierten Versionen und zueinander stehen

The formal verification of the ctm approach to forcing

We discuss some highlights of our computer-verified proof of the construction, given a countable transitive set-model $M$ of $\mathit{ZFC}$, of generic extensions satisfying $\mathit{ZFC}+\neg\mathit{CH}$ and $\mathit{ZFC}+\mathit{CH}$. Moreover, let $\mathcal{R}$ be the set of instances of the Axiom of Replacement. We isolated a 21-element subset $\Omega\subseteq\mathcal{R}$ and defined $\mathcal{F}:\mathcal{R}\to\mathcal{R}$ such that for every $\Phi\subseteq\mathcal{R}$ and $M$-generic $G$, $M\models \mathit{ZC} \cup \mathcal{F}\text{``}\Phi \cup \Omega$ implies $M[G]\models \mathit{ZC} \cup \Phi \cup \{ \neg \mathit{CH} \}$, where $\mathit{ZC}$ is Zermelo set theory with Choice. To achieve this, we worked in the proof assistant Isabelle, basing our development on the Isabelle/ZF library by L. Paulson and others.Comment: 20pp + 14pp in bibliography & appendices, 2 table

All Worlds in One: Reassessing the Forest-Armstrong Argument

The Forrest-Armstrong argument, as reconfigured by David Lewis, is a reductio against an unrestricted principle of recombination. There is a gap in the argument which Lewis thought could be bridged by an appeal to recombination. After presenting the argument, I show that no plausible principle of recombination can bridge the gap. But other plausible principles of plenitude can bridge the gap, both principles of plenitude for world contents and principles of plenitude for world structures. I conclude that the Forrest-Armstrong argument, when fortified in one of these ways, demands that unrestricted recombination be rejected. The appropriate restriction comes from a consideration of what world structures are possible. I argue that, although there are too many worlds to form a set, for any world, the individuals at that world do form a set. To defend it I invoke a principle of Limitation of Size together with an iterative conception of structure

Classical Set Theory: Theory of Sets and Classes

This is a short introductory course to Set Theory, based on axioms of von Neumann--Bernays--G\"odel (briefly NBG). The text can be used as a base for a lecture course in Foundations of Mathematics, and contains a reasonable minimum which a good (post-graduate) student in Mathematics should know about foundations of this science.Comment: 162 page