Mechanising Set Theory: Cardinal Arithmetic and the Axiom of Choice
نویسنده
چکیده
منابع مشابه
Mechanizing Set Theory: Cardinal Arithmetic and the Axiom of Choice
Fairly deep results of Zermelo-Frænkel (ZF) set theory have been mechanized using the proof assistant Isabelle. The results concern cardinal arithmetic and the Axiom of Choice (AC). A key result about cardinal multiplication is κ⊗ κ = κ, where κ is any infinite cardinal. Proving this result required developing theories of orders, order-isomorphisms, order types, ordinal arithmetic, cardinals, e...
متن کاملSome remarks on cardinal arithmetic without choice
One important consequence of the Axiom of Choice is the absorption law of cardinal arithmetic. It states that for any cardinals m and n, if m 6 n and n is infinite, then m+ n = n and if m 6= 0, m · n = n. In this paper, we investigate some conditions that make this property hold as well as an instance when such a property cannot be proved in the absence of the Axiom of Choice. We further find s...
متن کاملRelations between Some Cardinals in the Absence of the Axiom of Choice Dedicated to the Memory of Prof. Hans Läuchli
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 betwe...
متن کاملFraenkel’s axiom of restriction: Axiom choice, intended models and categoricity
A recent debate has focused on different methodological principles underlying the practice of axiom choice in mathematics (cf. Feferman et al., 2000; Maddy, 1997; Easwaran, 2008). The general aim of these contributions can be described as twofold: first to clarify the spectrum of informal justification strategies retraceable in the history of mathematical axiomatics. Second, to evaluate and to ...
متن کاملInconsistency of the Zermelo-Fraenkel set theory with the axiom of choice and its effects on the computational complexity
This paper exposes a contradiction in the Zermelo-Fraenkel set theory with the axiom of choice (ZFC). While Gödel’s incompleteness theorems state that a consistent system cannot prove its consistency, they do not eliminate proofs using a stronger system or methods that are outside the scope of the system. The paper shows that the cardinalities of infinite sets are uncontrollable and contradicto...
متن کامل