For a list of examples, see list of large cardinal properties. See also Large cardinal property This page is a list of some types of cardinals; it is arranged roughly in order of the consistency strength of the axiom asserting

In the mathematical field of set theory, a large cardinal property is a certain kind of property of transfinite cardinal numbers. This article describes cardinal numbers in mathematics For cardinals in linguistics see Names of numbers in English. Cardinals with such properties are, as the name suggests, generally very "large" (for example, bigger than aleph zero, bigger than the cardinality of the continuum, etc. In Mathematics, the cardinality of the continuum, sometimes also called the power of the continuum, is the size ( Cardinality) of the set of ). The proposition that such cardinals exist cannot be proved in the most common axiomatization of set theory, namely ZFC, and such propositions can be viewed as ways of measuring how "much", beyond ZFC, one needs to assume to be able to prove certain desired results. In Mathematics, an axiomatic system is any set of Axioms from which some or all axioms can be used in conjunction to logically derive Theorems Zermelo–Fraenkel set theory with the axiom of choice, commonly abbreviated ZFC, is the standard form of Axiomatic set theory and as such is the most common In other words, they can be seen, in Dana Scott's phrase, as quantifying the fact "that if you want more you have to assume more". Dana Stewart Scott (born 1932 is the emeritus Hillman University Professor of Computer Science, Philosophy, and Mathematical Logic at Carnegie [1]

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).

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. See also Large cardinal property This page is a list of some types of cardinals; it is arranged roughly in order of the consistency strength of the axiom asserting

## Partial definition

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 ZFC and it has been proven that if ZFC is consistent, then ZFC + "no such cardinal exists" is consistent. Zermelo–Fraenkel set theory with the axiom of choice, commonly abbreviated ZFC, is the standard form of Axiomatic set theory and as such is the most common

## Hierarchy of consistency strength

A remarkable observation about large cardinal axioms is that they appear to occur in strict linear order by consistency strength. In Mathematics and Set theory, a total order, linear order, simple order, or (non-strict ordering is a Binary relation In Mathematics, specifically in Mathematical logic, formal theories are studied as Mathematical objects Since some theories are powerful enough to model That is, no exception is known to the following: Given two large cardinal axioms A1 and A2, one of three (mutually exclusive) things happens:

1. ZFC proves "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 Mathematics, specifically in Mathematical logic, formal theories are studied as Mathematical objects Since some theories are powerful enough to model 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, Gödel's incompleteness theorems, proved by Kurt Gödel in 1931 are two Theorems stating inherent limitations of all but the most

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. Saharon Shelah has asked, "[i]s there some theorem explaining this, or is our vision just more uniform than we realize?". Saharon Shelah (שהרן שלח born July 3, 1945 in Jerusalem) is an Israeli Mathematician.

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 &kappa is called huge If and only if there exists an Elementary embedding j: V &rarr In Set theory, a supercompact cardinal a type of Large cardinal.

## 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. In Set theory and related branches of Mathematics, the von Neumann universe, or von Neumann hierarchy of sets, denoted V, is the class Transfinite induction is an extension of Mathematical induction to well-ordered sets, for instance to sets of ordinals or cardinals. In Mathematics, given a set S, the power set (or powerset) of S, written \mathcal{P}(S P ( S) 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. In Set theory, an uncountable regular cardinal number is called weakly inaccessible if it is a Weak limit cardinal, and strongly inaccessible In Mathematical logic, the universe of a structure (or model) is its domain. 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 Mathematics, a measurable cardinal is a certain kind of Large cardinal number Gödel universe redirects here For Kurt Gödel 's cosmological solution to the Einstein field equations, see Gödel metric.

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. The Cabal was or perhaps is a grouping of set theorists in Southern California particularly at UCLA and Caltech, perhaps also at UC Irvine. 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). In the mathematical field of Set theory, Martin's axiom, named after Donald A The axiom of constructibility is a possible Axiom for Set theory in mathematics that asserts that every set is constructible. The hardcore realists in this group would state, more simply, that large cardinal axioms are true. The philosophy of mathematics is the branch of Philosophy that studies the philosophical assumptions foundations and implications of Mathematics.

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. The philosophy of mathematics is the branch of Philosophy that studies the philosophical assumptions foundations and implications of Mathematics. 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. In Set theory, a set (or class) A is transitive, if whenever x ∈ A, and y ∈ x, then

## Notes

1. ^ Bell, J. L. (1985). Boolean-Valued Models and Independence Proofs in Set Theory. Oxford University Press, viii. ISBN 0198532415.

## 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 0-444-10535-2.
• Jech, Thomas (2002). Thomas J Jech (Tomáš Jech ˈtɔmaːʃ ˈjɛx born January 29, 1944 in Prague) is a set theorist who was at Penn State for more Set theory, third millennium edition (revised and expanded). Springer. ISBN 3-540-44085-2.
• Maddy, Penelope (1988). Penelope Maddy is a UCI Distinguished Professor of Logic and Philosophy of Science and of Mathematics at the University of California Irvine "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.
• Shelah, Saharon (2002). Saharon Shelah (שהרן שלח born July 3, 1945 in Jerusalem) is an Israeli Mathematician. The Future of Set Theory.
• 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.

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