Totally indescribable cardinal
Encyclopedia
In mathematics
, a Q-indescribable cardinal is a certain kind of large cardinal number that is hard to describe in some language Q. There are many different types of indescribable cardinals corresponding to different choices of languages Q. They were introduced by .
A cardinal number κ is called Π-indescribable if for every Πm proposition φ, and set A ⊆ Vκ with (Vκ+n, ∈, A) ⊧ φ there exists an α < κ with (Vα+n, ∈, A ∩ Vα) ⊧ φ.
Here one looks at formulas with m-1 alternations of quantifiers with the outermost quantifier being universal.
Σ-indescribable cardinals are defined in a similar way. The idea is that κ 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.
The cardinal number κ is called totally indescribable if it is Π-indescribable for all positive integers m and n.
If α is an ordinal, the cardinal number κ is called α-indescribable if for every formula φ and every subset U of Vκ
such that φ(U) holds in Vκ+α there is a some λ<κ such that φ(U∩ Vλ) holds in Vλ+α. If α is infinite then α-indescribable ordinals are totally indescribable, and if α is finite they are the same as Π-indescribable ordinals.
Π-indescribable cardinals are the same as weakly compact cardinal
s.
A cardinal is inaccessible if and only if it is Π-indescribable for all positive integers m
Measurable cardinals are Π-indescribable, but the smallest measurable cardinal is not Σ-indescribable. However there are many totally indescribable cardinals below any measurable cardinal.
Totally indescribable cardinals remain totally indescribable in the constructible universe
.
Mathematics
Mathematics is the study of quantity, space, structure, and change. Mathematicians seek out patterns and formulate new conjectures. Mathematicians resolve the truth or falsity of conjectures by mathematical proofs, which are arguments sufficient to convince other mathematicians of their validity...
, a Q-indescribable cardinal is a certain kind of large cardinal number that is hard to describe in some language Q. There are many different types of indescribable cardinals corresponding to different choices of languages Q. They were introduced by .
A cardinal number κ is called Π-indescribable if for every Πm proposition φ, and set A ⊆ Vκ with (Vκ+n, ∈, A) ⊧ φ there exists an α < κ with (Vα+n, ∈, A ∩ Vα) ⊧ φ.
Here one looks at formulas with m-1 alternations of quantifiers with the outermost quantifier being universal.
Σ-indescribable cardinals are defined in a similar way. The idea is that κ 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.
The cardinal number κ is called totally indescribable if it is Π-indescribable for all positive integers m and n.
If α is an ordinal, the cardinal number κ is called α-indescribable if for every formula φ and every subset U of Vκ
such that φ(U) holds in Vκ+α there is a some λ<κ such that φ(U∩ Vλ) holds in Vλ+α. If α is infinite then α-indescribable ordinals are totally indescribable, and if α is finite they are the same as Π-indescribable ordinals.
Π-indescribable cardinals are the same as weakly compact cardinal
Weakly compact cardinal
In mathematics, a weakly compact cardinal is a certain kind of cardinal number introduced by ; weakly compact cardinals are large cardinals, meaning that their existence can not be proven from the standard axioms of set theory....
s.
A cardinal is inaccessible if and only if it is Π-indescribable for all positive integers m
Measurable cardinals are Π-indescribable, but the smallest measurable cardinal is not Σ-indescribable. However there are many totally indescribable cardinals below any measurable cardinal.
Totally indescribable cardinals remain totally indescribable in the constructible universe
Constructible universe
In mathematics, the constructible universe , denoted L, is a particular class of sets which can be described entirely in terms of simpler sets. It was introduced by Kurt Gödel in his 1938 paper "The Consistency of the Axiom of Choice and of the Generalized Continuum-Hypothesis"...
.