Search results
Results from the WOW.Com Content Network
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 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.
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". [25]
The Multiversity featured a story arc about the DC Comics Multiverse being invaded by a race of extradimensional parasites known as The Gentry. The Gentry come from beyond the immediate DC "local" multiverse, and each member is a cultural fear or "bad idea" personified as a living, demonic entity.
The axioms of Zermelo set theory are stated for objects, some of which (but not necessarily all) are sets, and the remaining objects are urelements and not sets. Zermelo's language implicitly includes a membership relation ∈, an equality relation = (if it is not included in the underlying logic), and a unary predicate saying whether an object is a set.
Suppose, to the contrary, that there is a function, f, on the natural numbers with f(n+1) an element of f(n) for each n.Define S = {f(n): n a natural number}, the range of f, which can be seen to be a set from the axiom schema of replacement.
This set is called the minimal model of ZFC. Using the downward Löwenheim–Skolem theorem , one can show that the minimal model (if it exists) is a countable set. Of course, any consistent theory must have a model, so even within the minimal model of set theory there are sets that are models of ZF (assuming ZF is consistent).