In the mathematical field of set theory, a large cardinal property is a property of cardinal numbers, such that the existence of such a cardinal provably cannot be proved within the standard axiomatic set theory ZFC, if one assumes ZFC itself is consistent. Furthermore, unlike the case of the continuum hypothesis, it is (provably) not possible to show that any large cardinal axiom is even consistent with ZFC, from the assumption that ZFC is consistent. There is a rough convention that results provable from ZFC alone may be stated without hypotheses, but that if the proof requires other assumptions (such as the existence of large cardinals), these should be stated. Whether this is simply a linguistic convention, or something more, is a controversial point among distinct philosophical schools (see Motivations and epistemic status below). Mathematics is often defined as the study of topics such as quantity, structure, space, and change. ... Set theory is the mathematical theory of sets, which represent collections of abstract objects. ... In linguistics, cardinal numbers is the name given to number words that are used for quantity (one, two, three), as opposed to ordinal numbers, words that are used for order (first, second, third). ... Set theory is a branch of mathematics created principally by the German mathematician Georg Cantor at the end of the 19th century. ... The Zermelo-Fraenkel axioms of set theory (ZF) are the standard axioms of axiomatic set theory on which, together with the axiom of choice, all of ordinary mathematics is based in modern formulations. ... In mathematics, the continuum hypothesis is a hypothesis about the possible sizes of infinite sets. ...

A large cardinal axiom is an axiom stating that there exists a cardinal (or perhaps many of them) with some specified large cardinal property.

There is no generally agreed precise definition of what a large cardinal property is, though essentially everyone agrees that those listed at List of large cardinal properties are large cardinal properties.

Hierarchy of consistency strength

A remarkable observation about large cardinal axioms is that they appear to occur in strict linear order by consistency strength. That is, no exception is known to the following: Given two large cardinal axioms A1 and A2, one of three (mutually exclusive) things happens: In mathematics, a total order or linear order on a set X is any binary relation on X that is antisymmetric, transitive, and total. ...

1. ZFC proves that (ZFC+A1 is consistent if and only if ZFC+A2 is consistent),
2. ZFC+A1 proves that ZFC+A2 is consistent,
3. ZFC+A2 proves that ZFC+A1 is consistent.

In case 1 we say that A1 and A2 are equiconsistent. In case 2, we say that A1 is consistency-wise stronger than A2 (vice versa for case 3). If A2 is stronger than A1, then ZFC+A1 cannot prove A2 is consistent, even with the additional hypothesis that ZFC+A1 is itself consistent (provided of course that it really is). This follows from Gödel's second incompleteness theorem. In mathematical logic, Gödels incompleteness theorems are two celebrated theorems proved by Kurt Gödel in 1931. ...

The observation that large cardinal axioms are linearly ordered by consistency strength is just that, an observation, not a theorem. (Without an accepted definition of large cardinal property, it is not subject to proof in the ordinary sense). Also, it is not known in every case which of the three cases holds.

It should also be noted that the order of consistency strength is not necessarily the same as the order of the size of the smallest witness to a large cardinal axiom. For example, the existence of a huge cardinal is much stronger, in terms of consistency strength, than the existence of a supercompact cardinal, but assuming both exist, the first huge is smaller than the first supercompact. In mathematics, a cardinal number κ is called huge iff there exists an elementary embedding j : V → M from V into a transitive inner model M with critical point κ and j(κ)M ⊆ M. Categories: Math stubs ... In mathematics, a cardinal number κ is called supercompact iff for all ordinal numbers α there exists an elementary embedding j : V → M from V into a transitive inner model M with critical point κ and αM ⊆ Categories: Mathematical logic stubs | Set theory ...

Motivations and epistemic status

Large cardinals are understood in the context of the von Neumann universe V, which is built up by transfinitely iterating the powerset operation, which collects together all subsets of a given set. Typically, models in which large cardinal axioms fail can be seen in some natural way as submodels of those in which the axioms hold. For example, if there is an inaccessible cardinal, then "cutting the universe off" at the height of the first such cardinal yields a universe in which there is no inaccessible cardinal. Or if there is a measurable cardinal, then iterating the definable powerset operation rather than the full one yields Gödel's constructible universe, L, which does not satisfy the statement "there is a measurable cardinal" (even though it contains the measurable cardinal as an ordinal). In axiomatic set theory and related branches of mathematics, the Von Neumann universe, or Von Neumann hierarchy of sets is the class of all sets, divided into a transfinite hierarchy of individual sets. ... Transfinite induction is the proof technique of mathematical induction when applied to (large) well-ordered sets, for instance to sets of ordinals or cardinals, or even to the class of all ordinals. ... In mathematics, given a set S, the power set of S, written P(S) or 2S, is the set of all subsets of S. In formal language, the existence of power set of any set is presupposed by the axiom of power set. ... A is a subset of B, and B is a superset of A. In mathematics, especially in set theory, a set A is a subset of a set B, if A is contained inside B. Every set is a subset of itself. ... In mathematics, a cardinal number k > (aleph-null) is called weakly inaccessible, or just inaccessible, if the following two conditions hold. ... Bold textcopper mellon pressure washer. ... In mathematics, the constructible universe (or Gödels constructible universe), denoted , is a particular class of sets which can be described entirely in terms of simpler sets. ...

Thus, from a certain point of view held by many set theorists (especially those inspired by the tradition of the Cabal), large cardinal axioms "say" that we are considering all the sets we're "supposed" to be considering, whereas their negations are "restrictive" and say that we're considering only some of those sets. Moreover the consequences of large cardinal axioms seem to fall into natural patterns (see Maddy, "Believing the Axioms, II"). For these reasons, such set theorists tend to consider large cardinal axioms to have a preferred status among extensions of ZFC, one not shared by axioms of less clear motivation (such as Martin's axiom) or others that they consider intuitively unlikely (such as V = L). The hardcore realists in this group would state, more simply, that large cardinal axioms are true. The Cabal was, or perhaps is, a grouping of set theorists in Southern California, particularly at UCLA and Caltech, perhaps also at UC Irvine. ... In axiomatic set theory, Martins axiom is a statement which is independent of the usual axioms of ZFC Set Theory. ... Philosophy of mathematics is that branch of philosophy which attempts to answer questions such as: why is mathematics useful in describing nature?, in which sense(s), if any, do mathematical entities such as numbers exist? and why and how are mathematical statements true?. Various approaches to answering these questions will...

This point of view is by no means universal among set theorists. Some formalists would assert that standard set theory is by definition the study of the consequences of ZFC, and while they might not be opposed in principle to studying the consequences of other systems, they see no reason to single out large cardinals as preferred. There are also realists who deny that ontological maximalism is a proper motivation, and even believe that large cardinal axioms are false. And finally, there are some who deny that the negations of large cardinal axioms are restrictive, pointing out that (for example) there can be a transitive submodel of L that believes there exists a measurable cardinal, even though L itself does not satisfy that proposition. Philosophy of mathematics is that branch of philosophy which attempts to answer questions such as: why is mathematics useful in describing nature?, in which sense(s), if any, do mathematical entities such as numbers exist? and why and how are mathematical statements true?. Various approaches to answering these questions will... In set theory, a set (or class) A is transitive, if whenever x ∈ A, and y ∈ x, then y ∈ A, or, equivalently, whenever x ∈ A, and x is not an urelement, then x is a subset of A. In fact, x can be a urelement and the implication x ∈ A...

References

• Drake, F. R. (1974). Set Theory: An Introduction to Large Cardinals (Studies in Logic and the Foundations of Mathematics ; V. 76), Elsevier Science Ltd. ISBN 0444105352.
• Jech, Thomas (2002). Set theory, third millennium edition (revised and expanded), Springer. ISBN 3540440852.
• Maddy, Penelope (1988). Believing the Axioms, I, Journal of Symbolic Logic, 53(2): 481-511.
• Maddy, Penelope (1988). Believing the Axioms, II. Journal of Symbolic Logic 53 (3): 736-764.
• Solovay, Robert M., William N. Reinhardt, and Akihiro Kanamori (1978). Strong axioms of infinity and elementary embeddings Annals of Mathematical Logic 13 (1): 73-116.