In
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 shrewd cardinal is a certain kind of
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 � ...
number introduced by , extending the definition of
indescribable cardinals.
For an
ordinal λ, a
cardinal number κ is called λ-shrewd if for every
proposition φ, and set A ⊆ V
κ with (V
κ+λ, ∈, A) ⊧ φ there exists an α, λ' < κ with (V
α+λ', ∈, A ∩ V
α) ⊧ φ. It is called shrewd if it is λ-shrewd for every λ
[M. Rathjen,]
The Art of Ordinal Analysis
. Accessed June 20 2022.(Definition 4.1) (including λ > κ).
This definition extends the concept of
indescribability to transfinite levels. A λ-shrewd cardinal is also μ-shrewd for any ordinal μ < λ.
Shrewdness was developed by
Michael Rathjen as part of his
ordinal analysis of
Π12-comprehension. It is essentially the nonrecursive analog to the
stability property for
admissible ordinals.
More generally, a cardinal number κ is called λ-Π
m-shrewd if for every Π
m proposition φ, and set A ⊆ V
κ with (V
κ+λ, ∈, A) ⊧ φ there exists an α, λ' < κ with (V
α+λ', ∈, A ∩ V
α) ⊧ φ.
Π
m is one of the levels of the
Lévy hierarchy, in short one looks at formulas with m-1 alternations of
quantifiers with the outermost quantifier being universal.
For finite ''n'', an ''n''-Π
m-shrewd cardinals is the same thing as a Π
mn-indescribable cardinal.
If κ is a
subtle cardinal In mathematics, subtle cardinals and ethereal cardinals are closely related kinds of large cardinal number.
A cardinal ''κ'' is called subtle if for every closed and unbounded ''C'' ⊂ ''κ'' and for every sequence ''A'' of length ''κ' ...
, then the set of κ-shrewd cardinals is
stationary
In addition to its common meaning, stationary may have the following specialized scientific meanings:
Mathematics
* Stationary point
* Stationary process
* Stationary state
Meteorology
* A stationary front is a weather front that is not moving ...
in κ.
Rathjen does not state how shrewd cardinals compare to
unfoldable cardinals, however.
λ-shrewdness is an improved version of λ-indescribability, as defined in Drake; this cardinal property differs in that the reflected substructure must be (V
α+λ, ∈, A ∩ V
α), making it impossible for a cardinal κ to be κ-indescribable. Also, the monotonicity property is lost: a λ-indescribable cardinal may fail to be α-indescribable for some ordinal α < λ.
References
*
*
*
Large cardinals
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