Large Cardinals
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In the mathematical field of
set theory Set theory is the branch of mathematical logic that studies sets, which can be informally described as collections of objects. Although objects of any kind can be collected into a set, set theory, as a branch of mathematics, is mostly conce ...
, a large cardinal property is a certain kind of property of
transfinite Transfinite may refer to: * Transfinite number, a number larger than all finite numbers, yet not absolutely infinite * Transfinite induction, an extension of mathematical induction to well-ordered sets ** Transfinite recursion Transfinite inducti ...
cardinal number In mathematics, cardinal numbers, or cardinals for short, are a generalization of the natural numbers used to measure the cardinality (size) of sets. The cardinality of a finite set is a natural number: the number of elements in the set. Th ...
s. Cardinals with such properties are, as the name suggests, generally very "large" (for example, bigger than the least α such that α=ωα). 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 other words, they can be seen, in
Dana Scott Dana Stewart Scott (born October 11, 1932) is an American logician who is the emeritus Hillman University Professor of Computer Science, Philosophy, and Mathematical Logic at Carnegie Mellon University; he is now retired and lives in Berkeley, Ca ...
's phrase, as quantifying the fact "that if you want more you have to assume more". 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 is an axiom stating that there exists a cardinal (or perhaps many of them) with some specified large cardinal property. Most working set theorists believe that the large cardinal axioms that are currently being considered are
consistent In classical deductive logic, a consistent theory is one that does not lead to a logical contradiction. The lack of contradiction can be defined in either semantic or syntactic terms. The semantic definition states that a theory is consistent i ...
with ZFC. These axioms are strong enough to imply the consistency of ZFC. This has the consequence (via Gödel's second incompleteness theorem) that their consistency with ZFC cannot be proven in ZFC (assuming ZFC is consistent). There is no generally agreed precise definition of what a large cardinal property is, though essentially everyone agrees that those in the
list of large cardinal properties This page includes a list of cardinals with large cardinal properties. It is arranged roughly in order of the consistency strength of the axiom asserting the existence of cardinals with the given property. Existence of a cardinal number κ of a g ...
are large cardinal properties.


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 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 In classical deductive logic, a consistent theory is one that does not lead to a logical contradiction. The lack of contradiction can be defined in either semantic or syntactic terms. The semantic definition states that a theory is consistent i ...
, then ZFC does ''not'' imply that any such large cardinals exist.


Hierarchy of consistency strength

A remarkable observation about large cardinal axioms is that they appear to occur in strict
linear order In mathematics, a total or linear order is a partial order in which any two elements are comparable. That is, a total order is a binary relation \leq on some set X, which satisfies the following for all a, b and c in X: # a \leq a ( reflexive ...
by
consistency strength In mathematical logic, two theories are equiconsistent if the consistency of one theory implies the consistency of the other theory, and vice versa. In this case, they are, roughly speaking, "as consistent as each other". In general, it is not po ...
. That is, no exception is known to the following: Given two large cardinal axioms ''A''1 and ''A''2, exactly one of three things happens: #Unless ZFC is inconsistent, ZFC+''A''1 is consistent if and only if ZFC+''A''2 is consistent; #ZFC+''A''1 proves that ZFC+''A''2 is consistent; or #ZFC+''A''2 proves that ZFC+''A''1 is consistent. These are mutually exclusive, unless one of the theories in question is actually inconsistent. In case 1, we say that ''A''1 and ''A''2 are
equiconsistent In mathematical logic, two theories are equiconsistent if the consistency of one theory implies the consistency of the other theory, and vice versa. In this case, they are, roughly speaking, "as consistent as each other". In general, it is not p ...
. In case 2, we say that ''A''1 is consistency-wise stronger than ''A''2 (vice versa for case 3). If ''A''2 is stronger than ''A''1, then ZFC+''A''1 cannot prove ZFC+''A''2 is consistent, even with the additional hypothesis that ZFC+''A''1 is itself consistent (provided of course that it really is). This follows from Gödel's second incompleteness theorem. 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 Saharon Shelah ( he, שהרן שלח; born July 3, 1945) is an Israeli mathematician. He is a professor of mathematics at the Hebrew University of Jerusalem and Rutgers University in New Jersey. Biography Shelah was born in Jerusalem on July 3, ...
has asked, " there some theorem explaining this, or is our vision just more uniform than we realize?" Woodin, however, deduces this from the Ω-conjecture, the main unsolved problem of his
Ω-logic In set theory, Ω-logic is an infinitary logic and deductive system proposed by as part of an attempt to generalize the theory of determinacy of pointclasses to cover the structure H_. Just as the axiom of projective determinacy yields a canonic ...
. It is also noteworthy that many combinatorial statements are exactly equiconsistent with some large cardinal rather than, say, being intermediate between them. 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 In mathematics, a cardinal number κ is called huge if there exists an elementary embedding ''j'' : ''V'' → ''M'' from ''V'' into a transitive inner model ''M'' with critical point (set theory), critical point κ and :^M \subset M.\! Here, ''&al ...
is much stronger, in terms of consistency strength, than the existence of a
supercompact cardinal In set theory, a supercompact cardinal is a type of large cardinal. They display a variety of reflection properties. Formal definition If ''λ'' is any ordinal, ''κ'' is ''λ''-supercompact means that there exists an elementary ...
, but assuming both exist, the first huge is smaller than the first supercompact.


Motivations and epistemic status

Large cardinals are understood in the context of the
von Neumann universe In set theory and related branches of mathematics, the von Neumann universe, or von Neumann hierarchy of sets, denoted by ''V'', is the class of hereditary well-founded sets. This collection, which is formalized by Zermelo–Fraenkel set theory (Z ...
V, which is built up by transfinitely iterating the
powerset In mathematics, the power set (or powerset) of a set is the set of all subsets of , including the empty set and itself. In axiomatic set theory (as developed, for example, in the ZFC axioms), the existence of the power set of any set is postu ...
operation, which collects together all
subset In mathematics, Set (mathematics), set ''A'' is a subset of a set ''B'' if all Element (mathematics), elements of ''A'' are also elements of ''B''; ''B'' is then a superset of ''A''. It is possible for ''A'' and ''B'' to be equal; if they are ...
s of a given set. Typically,
models A model is an informative representation of an object, person or system. The term originally denoted the plans of a building in late 16th-century English, and derived via French and Italian ultimately from Latin ''modulus'', a measure. Models c ...
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 In set theory, an uncountable cardinal is inaccessible if it cannot be obtained from smaller cardinals by the usual operations of cardinal arithmetic. More precisely, a cardinal is strongly inaccessible if it is uncountable, it is not a sum of ...
, then "cutting the universe off" at the height of the first such cardinal yields a
universe The universe is all of space and time and their contents, including planets, stars, galaxies, and all other forms of matter and energy. The Big Bang theory is the prevailing cosmological description of the development of the universe. Acc ...
in which there is no inaccessible cardinal. Or if there is a
measurable cardinal In mathematics, a measurable cardinal is a certain kind of large cardinal number. In order to define the concept, one introduces a two-valued measure on a cardinal , or more generally on any set. For a cardinal , it can be described as a subdivisio ...
, then iterating the ''definable'' powerset operation rather than the full one yields
Gödel's constructible universe In mathematics, in set theory, the constructible universe (or Gödel's constructible universe), denoted by , is a particular class of sets that can be described entirely in terms of simpler sets. is the union of the constructible hierarchy . It ...
, L, which does not satisfy the statement "there is a measurable cardinal" (even though it contains the measurable cardinal as an ordinal). Thus, from a certain point of view held by many set theorists (especially those inspired by the tradition of the
Cabal A cabal is a group of people who are united in some close design, usually to promote their private views or interests in an ideology, a state, or another community, often by intrigue and usually unbeknownst to those who are outside their group. Th ...
), 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 In the mathematical field of set theory, Martin's axiom, introduced by Donald A. Martin and Robert M. Solovay, is a statement that is independent of the usual axioms of ZFC set theory. It is implied by the continuum hypothesis, but it is consist ...
) 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''. 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 In philosophy, ontological maximalism is a preference for largest possible universe, i.e. anything which could exist does exist. See also *Ontology *Maximalism *Large cardinal property *Continuum hypothesis *Mathematical universe hypothesis In ...
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 set In set theory, a branch of mathematics, a set A is called transitive if either of the following equivalent conditions hold: * whenever x \in A, and y \in x, then y \in A. * whenever x \in A, and x is not an urelement, then x is a subset of A. Simil ...
model in L that believes there exists a measurable cardinal, even though L itself does not satisfy that proposition.


See also

*
List of large cardinal properties This page includes a list of cardinals with large cardinal properties. It is arranged roughly in order of the consistency strength of the axiom asserting the existence of cardinals with the given property. Existence of a cardinal number κ of a g ...


Notes


References

* * * * * * * * *


External links


"Large Cardinals and Determinacy"
at the
Stanford Encyclopedia of Philosophy The ''Stanford Encyclopedia of Philosophy'' (''SEP'') combines an online encyclopedia of philosophy with peer-reviewed publication of original papers in philosophy, freely accessible to Internet users. It is maintained by Stanford University. Eac ...
{{Mathematical logic Axioms of set theory