Large cardinal
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In the mathematical field of
set theory Set theory is the branch of mathematical logic that studies Set (mathematics), 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 mathema ...
, a large cardinal property is a certain kind of property of transfinite
cardinal number In mathematics, a cardinal number, or cardinal for short, is what is commonly called the number of elements of a set. In the case of a finite set, its cardinal number, or cardinality is therefore a natural number. For dealing with the cas ...
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'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 deductive logic, a consistent theory is one that does not lead to a logical contradiction. A theory T is consistent if there is no formula \varphi such that both \varphi and its negation \lnot\varphi are elements of the set of consequences ...
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 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 deductive logic, a consistent theory is one that does not lead to a logical contradiction. A theory T is consistent if there is no formula \varphi such that both \varphi and its negation \lnot\varphi are elements of the set of consequences ...
, 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 by consistency strength. 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: #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 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 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. 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 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.


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 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 po ...
operation, which collects together all
subset In mathematics, a 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 a ...
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 , . Models can be divided int ...
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, a cardinal number is a strongly inaccessible cardinal if it is uncountable, regular, and a strong limit cardinal. A cardinal is a weakly inaccessible cardinal if it is uncountable, regular, and a weak limit cardinal. Since abou ...
, 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. It comprises all of existence, any fundamental interaction, physical process and physical constant, and therefore all forms of matter and energy, and the structures they form, from s ...
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). 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''. 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 set In set theory, a branch of mathematics, a set A is called transitive if either of the following equivalent conditions holds: * 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. S ...
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


Notes


References

* * * * * * * * *


External links


"Large Cardinals and Determinacy"
at the
Stanford Encyclopedia of Philosophy The ''Stanford Encyclopedia of Philosophy'' (''SEP'') is a freely available online philosophy resource published and maintained by Stanford University, encompassing both an online encyclopedia of philosophy and peer-reviewed original publication ...
{{Mathematical logic Axioms of set theory