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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 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.
A necessary condition for a property of cardinal numbers to be a large cardinal property is that the existence of such a cardinal is not known to be inconsistent with ZF and that such a cardinal Κ would be an uncountable initial ordinal for which L Κ is a model of ZFC. If ZFC is consistent, then ZFC does not imply that any such large ...
The notion that online gaming could help players develop charitable habits seemed bold when the anti-poverty nonprofit Comic Relief US tested its own multiverse on the popular world-building app ...
The continuum hypothesis is a statement in the language of ZFC that is not provable within ZFC, so ZFC is not complete. In this case, there is no obvious candidate for a new axiom that resolves the issue. The theory of first-order Peano arithmetic seems consistent. Assuming this is indeed the case, note that it has an infinite but recursively ...
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]
ZF stands for Zermelo–Fraenkel set theory, and DC for the axiom of dependent choice.. Solovay's theorem is as follows. Assuming the existence of an inaccessible cardinal, there is an inner model of ZF + DC of a suitable forcing extension V[G] such that every set of reals is Lebesgue measurable, has the perfect set property, and has the Baire property.