18. Large cardinals

Now since κ ∈ M and (M, ∈) is a model of ZFC, Vκ
M
to V κ . Hence by (1) we get
exists, and by absoluteness it is equal
Hence
(M, ∈, U ′ ) |= ∀X ⊆ V κ ϕ V κ
(U ′ ∩ V κ ).
(M, ∈, U ′ ) |= ∃α∀X ⊆ V α ϕ V α
(U ′ ∩ V α ),
so by the elementary extension property we get
(V κ , ∈, U) |= ∃α∀X ⊆ V α ϕ V α
(U ′ ∩ V α ).
We choose such an α. Since V κ ∩On = κ, it follows that α < κ. Hence (V α , ∈, U ′ ∩V α ) |= σ,
as desired.
Measurable cardinals
Our third kind of large cardinal is the class of measurable cardinals. Although, as the
name suggests, this notion comes from measure theory, the definition and results we give
are purely set-theoretical. Moreover, similarly to weakly compact cardinals, it is not
obvious from the definition that we are dealing with large cardinals.
The definition is given in terms of the notion of an ultrafilter on a set.
• Let X be a nonempty set. A filter on X is a family F of subsets of X satisfying the
following conditions:
(i) X ∈ F.
(ii) If Y, Z ∈ F, then Y ∩ Z ∈ F.
(iii) If Y ∈ F and Y ⊆ Z ⊆ X, then Z ∈ F.
• A filter F on a set X is proper or nontrivial iff ∅ /∈ F.
• An ultrafilter on a set X is a nontrivial filter F on X such that for every Y ⊆ X, either
Y ∈ F or X\Y ∈ F.
• A family A of subsets of X has the finite intersection property, fip, iff for every finite
subset B of A we have ⋂ B ≠ ∅.
• If A is a family of subsets of X, then the filter generated by A is the set
{Y ⊆ X : ⋂ B ⊆ Y for some finite B ⊆ A }.
[Clearly this is a filter on X, and it contains A .]
Proposition 18.16. If x ∈ X, then {Y ⊆ X : x ∈ Y } is an ultrafilter on X.
An ultrafilter of the kind given in this proposition is called a principal ultrafilter. There
are nonprincipal ultrafilters on any infinite set, as we will see shortly.
Proposition 18.17. Let F be a proper filter on a set X. Then the following are equivalent:
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