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 concer ...

, an extender is a system of

ultrafilter In the mathematical field of order theory, an ultrafilter on a given partially ordered set (or "poset") P is a certain subset of P, namely a maximal filter on P; that is, a proper filter on P that cannot be enlarged to a bigger proper filter o ...

s which represents an

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 ofte ...

witnessing

large cardinal 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 ...

properties. A nonprincipal ultrafilter is the most basic case of an extender. A (κ, λ)-extender can be defined as an elementary embedding of some model

M

of ZFC⁻ (ZFC minus the power set axiom) having critical point κ ε ''M'', and which maps κ to an ordinal at least equal to λ. It can also be defined as a collection of ultrafilters, one for each

n

tuple In mathematics, a tuple is a finite ordered list (sequence) of elements. An -tuple is a sequence (or ordered list) of elements, where is a non-negative integer. There is only one 0-tuple, referred to as ''the empty tuple''. An -tuple is defi ...

drawn from λ.

Formal definition of an extender

Let κ and λ be cardinals with κ≤λ. Then, a set

E = \

is called a (κ,λ)-extender if the following properties are satisfied: # each

E_a

is a κ-complete nonprincipal ultrafilter on kappa;sup><ω and furthermore ## at least one

E_a

is not κ⁺-complete, ## for each

\alpha \in \kappa,

at least one

E_a

contains the set

\.

# (Coherence) The

E_a

are coherent (so that the ultrapowers Ult(''V'',''E_a'') form a directed system). # (Normality) If

f

is such that

\ \in E_a,

then for some

b \supseteq a,\ \ \in E_b.

# (Wellfoundedness) The limit ultrapower Ult(''V'',''E'') is wellfounded (where Ult(''V'',''E'') is the

direct limit In mathematics, a direct limit is a way to construct a (typically large) object from many (typically smaller) objects that are put together in a specific way. These objects may be groups, rings, vector spaces or in general objects from any cat ...

of the ultrapowers Ult(''V'',''E_a'')). By coherence, one means that if

a

and

b

are finite subsets of λ such that

b

is a superset of

a,

then if

X

is an element of the ultrafilter

E_b

and one chooses the right way to project

X

down to a set of sequences of length

, a, ,

then

X

is an element of

E_a.

More formally, for

b = \,

where

\alpha_1 < \dots < \alpha_n < \lambda,

and

a = \,

where

m \leq n

and for

j \leq m

the

i_j

are pairwise distinct and at most

n,

we define the projection

\pi_ : \ \mapsto \\ (\xi_1 < \dots < \xi_n).

Then

E_a

and

E_b

cohere if

X \in E_a \iff \ \in E_b.

Defining an extender from an elementary embedding

Given an elementary embedding

j : V \to M,

which maps the set-theoretic universe

V

into a transitive

inner model In set theory, a branch of mathematical logic, an inner model for a theory ''T'' is a substructure of a model ''M'' of a set theory that is both a model for ''T'' and contains all the ordinals of ''M''. Definition Let L = \langle \in \rangle b ...

M,

with critical point κ, and a cardinal λ, κ≤λ≤''j''(κ), one defines

E = \

as follows:

\text a \in

lambda Lambda (}, ''lám(b)da'') is the 11th letter of the Greek alphabet, representing the voiced alveolar lateral approximant . In the system of Greek numerals, lambda has a value of 30. Lambda is derived from the Phoenician Lamed . Lambda gave ris ...

, X \subseteq

kappa Kappa (uppercase Κ, lowercase κ or cursive ; el, κάππα, ''káppa'') is the 10th letter of the Greek alphabet, representing the voiceless velar plosive sound in Ancient and Modern Greek. In the system of Greek numerals, has a value ...

: \quad X \in E_a \iff a \in j(X). One can then show that

E

has all the properties stated above in the definition and therefore is a (κ,λ)-extender.

References

* * {{settheory-stub Inner model theory Mathematical logic Model theory Large cardinals Set theory