18. Large cardinals

so since |B| = κ it follows that |H ξ | = κ for some ξ < λ, as desired.
(iv)⇒(i): obvious.
Now we go into the connection of weakly compact cardinals with logic, thereby justifying
the name “weakly compact”. This is optional material.
Let κ and λ be infinite cardinals. The language L κλ is an extension of ordinary first
order logic as follows. The notion of a model is unchanged. In the logic, we have a sequence
of λ distinct individual variables, and we allow quantification over any one-one sequence
of fewer than λ variables. We also allow conjunctions and disjunctions of fewer than κ
formulas. It should be clear what it means for an assignment of values to the variables to
satisfy a formula in this extended language. We say that an infinite cardinal κ is logically
weakly compact iff the following condition holds:
(*) For any language L κκ with at most κ basic symbols, if Γ is a set of sentences of the
language and if every subset of Γ of size less than κ has a model, then also Γ has a model.
Notice here the somewhat unnatural restriction that there are at most κ basic symbols.
If we drop this restriction, we obtain the notion of a strongly compact cardinal. These
cardinals are much larger than even the measurable cardinals discussed later. We will not
go into the theory of such cardinals.
Theorem 18.8. An infinite cardinal is logically weakly compact iff it is weakly compact.
Proof. Suppose that κ is logically weakly compact.
(1) κ is regular.
Suppose not; say X ⊆ κ is unbounded but |X| < κ. Take the language with individual
constants c α for α < κ and also one more individual constant d. Consider the following
set Γ of sentences in this language:
⎧
⎫
⎨ ∨ ∨ ⎬
{d ≠ c α : α < κ} ∪ (d = c α )
⎩
⎭ .
β∈X α