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The collection of countable transitive models of ZFC (in some universe) is called the hyperverse and is very similar to the "multiverse". A typical difference between the universe and multiverse views is the attitude to the continuum hypothesis. In the universe view the continuum hypothesis is a meaningful question that is either true or false ...
Nevertheless, it is deemed unlikely that ZFC harbors an unsuspected contradiction; it is widely believed that if ZFC were inconsistent, that fact would have been uncovered by now. This much is certain — ZFC is immune to the classic paradoxes of naive set theory: Russell's paradox, the Burali-Forti paradox, and Cantor's paradox.
Although the existence of some statements independent of ZFC had already been known more than two decades prior: for example, assuming good soundness properties and the consistency ZFC, Gödel's incompleteness theorems, which were published in 1931, establish that there is a formal statement (one for each appropriate Gödel numbering scheme ...
The axiom of global choice states that there is a global choice function τ, meaning a function such that for every non-empty set z, τ(z) is an element of z.. The axiom of global choice cannot be stated directly in the language of Zermelo–Fraenkel set theory (ZF) with the axiom of choice (AC), known as ZFC, as the choice function τ is a proper class and in ZFC one cannot quantify over classes.
An initial segment of the von Neumann universe. Ordinal multiplication is reversed from our usual convention; see Ordinal arithmetic.. The cumulative hierarchy is a collection of sets V α indexed by the class of ordinal numbers; in particular, V α is the set of all sets having ranks less than α.
One common assumption is that the multiverse is a "patchwork quilt of separate universes all bound by the same laws of physics." [2] The concept of multiple universes, or a multiverse, has been discussed throughout history, including Greek philosophy. It has evolved and has been debated in various fields, including cosmology, physics, and ...
Joel David Hamkins is an American mathematician and philosopher who is the John Cardinal O'Hara Professor of Logic at the University of Notre Dame. [1] He has made contributions in mathematical and philosophical logic, set theory and philosophy of set theory (particularly the idea of the set-theoretic multiverse), in computability theory, and in group theory.
The following set theoretic statements are independent of ZFC, among others: the continuum hypothesis or CH (Gödel produced a model of ZFC in which CH is true, showing that CH cannot be disproven in ZFC; Paul Cohen later invented the method of forcing to exhibit a model of ZFC in which CH fails, showing that CH cannot be proven in ZFC. The ...