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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.
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 ...
The Infinite Monkey Theorem hypothesizes that, given a typewriter and an infinite amount of time, a monkey could in theory produce the full works of William Shakespeare. It has long been used to ...
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.
Maps are useful in presenting key facts within a geographical context and enabling a descriptive overview of a complex concept to be accessed easily and quickly. WikiProject Maps encourages the creation of free maps and their upload on Wikimedia Commons. On the project's pages can be found advice, tools, links to resources, and map conventions.
Joel David Hamkins proposes a multiverse approach to set theory and argues that "the continuum hypothesis is settled on the multiverse view by our extensive knowledge about how it behaves in the multiverse, and, as a result, it can no longer be settled in the manner formerly hoped for". [23]
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 ...
There is a free complete Boolean algebra on countably many generators. [40] There is a set that cannot be linearly ordered. There exists a model of ZF¬C in which every set in R n is measurable. Thus it is possible to exclude counterintuitive results like the Banach–Tarski paradox which are provable in ZFC.