Indescribable cardinal
Large cardinal number that is hard to describe in a given language / From Wikipedia, the free encyclopedia
In set theory, a branch of mathematics, a Q-indescribable cardinal is a certain kind of large cardinal number that is hard to axiomatize in some language Q. There are many different types of indescribable cardinals corresponding to different choices of languages Q. They were introduced by Hanf & Scott (1961).
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A cardinal number is called
-indescribable if for every
proposition
, and set
with
there exists an
with
.[1] Following Lévy's hierarchy, here one looks at formulas with m-1 alternations of quantifiers with the outermost quantifier being universal.
-indescribable cardinals are defined in a similar way, but with an outermost existential quantifier. Prior to defining the structure
, one new predicate symbol is added to the language of set theory, which is interpreted as
.[2] The idea is that
cannot be distinguished (looking from below) from smaller cardinals by any formula of n+1-th order logic with m-1 alternations of quantifiers even with the advantage of an extra unary predicate symbol (for A). This implies that it is large because it means that there must be many smaller cardinals with similar properties. [citation needed]
The cardinal number is called totally indescribable if it is
-indescribable for all positive integers m and n.
If is an ordinal, the cardinal number
is called
-indescribable if for every formula
and every subset
of
such that
holds in
there is a some
such that
holds in
. If
is infinite then
-indescribable ordinals are totally indescribable, and if
is finite they are the same as
-indescribable ordinals. There is no
that is
-indescribable, nor does
-indescribability necessarily imply
-indescribability for any
, but there is an alternative notion of shrewd cardinals that makes sense when
: if
holds in
, then there are
and
such that
holds in
.[3] However, it is possible that a cardinal
is
-indescribable for
much greater than
.[1]Ch. 9, theorem 4.3