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 branch of

mathematics Mathematics is an area of knowledge that includes the topics of numbers, formulas and related structures, shapes and the spaces in which they are contained, and quantities and their changes. These topics are represented in modern mathematics ...

, a rank-into-rank embedding is a

large cardinal property In the mathematical field of set theory, a large cardinal property is a certain kind of property of transfinite cardinal numbers. Cardinals with such properties are, as the name suggests, generally very "large" (for example, bigger than the least � ...

defined by one of the following four

axiom An axiom, postulate, or assumption is a statement that is taken to be true, to serve as a premise or starting point for further reasoning and arguments. The word comes from the Ancient Greek word (), meaning 'that which is thought worthy or f ...

s given in order of increasing consistency strength. (A set of rank < λ is one of the elements of the set V_λ of the von Neumann hierarchy.) *Axiom I3: There is a nontrivial

elementary embedding In model theory, a branch of mathematical logic, two structures ''M'' and ''N'' of the same signature ''σ'' are called elementarily equivalent if they satisfy the same first-order ''σ''-sentences. If ''N'' is a substructure of ''M'', one often ...

of V_λ into itself. *Axiom I2: There is a nontrivial elementary embedding of V into a transitive class M that includes V_λ where λ is the first fixed point above the critical point. *Axiom I1: There is a nontrivial elementary embedding of V_λ+1 into itself. *Axiom I0: There is a nontrivial elementary embedding of L(V_λ+1) into itself with critical point below λ. These are essentially the strongest known large cardinal axioms not known to be inconsistent in ZFC; the axiom for

Reinhardt cardinal In set theory, a branch of mathematics, a Reinhardt cardinal is a kind of large cardinal. Reinhardt cardinals are considered under ZF (Zermelo–Fraenkel set theory without the Axiom of Choice), because they are inconsistent with ZFC (ZF with the A ...

s is stronger, but is not consistent with the

axiom of choice In mathematics, the axiom of choice, or AC, is an axiom of set theory equivalent to the statement that ''a Cartesian product of a collection of non-empty sets is non-empty''. Informally put, the axiom of choice says that given any collectio ...

. If j is the elementary embedding mentioned in one of these axioms and κ is its critical point, then λ is the limit of

j^n(\kappa)

as n goes to ω. More generally, if the

holds, it is provable that if there is a nontrivial elementary embedding of V_α into itself then α is either a

limit ordinal In set theory, a limit ordinal is an ordinal number that is neither zero nor a successor ordinal. Alternatively, an ordinal λ is a limit ordinal if there is an ordinal less than λ, and whenever β is an ordinal less than λ, then there exists an ...

cofinality In mathematics, especially in order theory, the cofinality cf(''A'') of a partially ordered set ''A'' is the least of the cardinalities of the cofinal subsets of ''A''. This definition of cofinality relies on the axiom of choice, as it uses the ...

ω or the successor of such an ordinal. The axioms I0, I1, I2, and I3 were at first suspected to be inconsistent (in ZFC) as it was thought possible that

Kunen's inconsistency theorem In set theory, a branch of mathematics, Kunen's inconsistency theorem, proved by , shows that several plausible large cardinal axioms are inconsistent with the axiom of choice. Some consequences of Kunen's theorem (or its proof) are: *There is no ...

that

s are inconsistent with the axiom of choice could be extended to them, but this has not yet happened and they are now usually believed to be consistent. Every I0 cardinal κ (speaking here of the critical point of ''j'') is an I1 cardinal. Every I1 cardinal κ (sometimes called ω-huge cardinals) is an I2 cardinal and has a stationary set of I2 cardinals below it. Every I2 cardinal κ is an I3 cardinal and has a stationary set of I3 cardinals below it. Every I3 cardinal κ has another I3 cardinal ''above'' it and is an ''n''-

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 ...

for every ''n''<ω. Axiom I1 implies that V_λ+1 (equivalently, H(λ⁺)) does not satisfy V= HOD. There is no set S⊂λ definable in V_λ+1 (even from parameters V_λ and ordinals <λ⁺) with S cofinal in λ and , S, <λ, that is, no such S witnesses that λ is singular. And similarly for Axiom I0 and ordinal definability in L(V_λ+1) (even from parameters in V_λ). However globally, and even in V_λ,Consistency of V = HOD With the Wholeness Axiom, Paul Corazza, Archive for Mathematical Logic, No. 39, 2000. V=HOD is relatively consistent with Axiom I1. Notice that I0 is sometimes strengthened further by adding an "Icarus set", so that it would be *Axiom Icarus set: There is a nontrivial elementary embedding of L(V_λ+1, Icarus) into itself with the critical point below λ. The Icarus set should be in V_λ+2 − L(V_λ+1) but chosen to avoid creating an inconsistency. So for example, it cannot encode a well-ordering of V_λ+1. See section 10 of Dimonte for more details.

Notes

References

*. * * . *. * {{citation, last1=Solovay, first1=Robert M., authorlink2=William Nelson Reinhardt, first2=William N. , last2=Reinhardt, first3= Akihiro , last3=Kanamori, year=1978, title=Strong axioms of infinity and elementary embeddings, journal=Annals of Mathematical Logic, volume=13, issue=1, pages=73–116, authorlink=Robert M. Solovay, doi=10.1016/0003-4843(78)90031-1, doi-access=free. Large cardinals Determinacy