Large cardinal

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For other uses of "Cardinal", see Cardinal (disambiguation).

In the mathematical field of

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".[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.

Most working set theorists believe that the large cardinal axioms that are currently being considered are

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

consistent
, 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

consistency strength
. That is, no exception is known to the following: Given two large cardinal axioms A1 and A2, exactly one of three things happens:

  1. Unless ZFC is inconsistent, ZFC+A1 is consistent if and only if ZFC+A2 is consistent;
  2. ZFC+A1 proves that ZFC+A2 is consistent; or
  3. ZFC+A2 proves that ZFC+A1 is consistent.

These are mutually exclusive, unless one of the theories in question is actually inconsistent.

In case 1, we say that A1 and A2 are

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.

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

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

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 model in L that believes there exists a measurable cardinal, even though L itself does not satisfy that proposition.

See also

Notes

  1. ISBN 0-19-853241-5
    .
  2. ^ Joel, Hamkins (2022-12-24). "Does anyone still seriously doubt the consistency of ZFC?". MathOverflow.

References
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