Ineffable cardinal
Encyclopedia
In the mathematics
of transfinite number
s, an ineffable cardinal is a certain kind of large cardinal number, introduced by .
A cardinal number
is called almost ineffable if for every 
(where 
is the powerset of 
) with the property that 
is a subset of 
for all ordinals 
, there is a subset 
of 
having cardinal 
and homogeneous for 
, in the sense that for any 
in 
, 
.
A cardinal number
is called ineffable if for every binary-valued function 
, there is a stationary subset of 
on which 
is homogeneous: that is, either 
maps all unordered pairs of elements drawn from that subset to zero, or it maps all such unordered pairs to one.
More generally,
is called 
-ineffable (for a positive integer 
) if for every 
there is a stationary subset of 
on which 
is 
-homogeneous (takes the same value for all unordered 
-tuples drawn from the subset). Thus, it is ineffable if and only if it is 2-ineffable.
A totally ineffable cardinal is a cardinal that is
-ineffable for every 
. If 
is 
-ineffable, then the set of 
-ineffable cardinals below 
is a stationary subset of 
.
Totally ineffable cardinals are of greater consistency strength than subtle cardinal
s and of lesser consistency strength than remarkable cardinals. A list of large cardinal axioms by consistency strength is available here.
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...
of transfinite number
Transfinite number
Transfinite numbers are numbers that are "infinite" in the sense that they are larger than all finite numbers, yet not necessarily absolutely infinite. The term transfinite was coined by Georg Cantor, who wished to avoid some of the implications of the word infinite in connection with these...
s, an ineffable cardinal is a certain kind of large cardinal number, introduced by .
A cardinal number
Cardinal number
In mathematics, cardinal numbers, or cardinals for short, are a generalization of the natural numbers used to measure the cardinality of sets. The cardinality of a finite set is a natural number – the number of elements in the set. The transfinite cardinal numbers describe the sizes of infinite...
is called almost ineffable if for every (where is the powerset of ) with the property that is a subset of for all ordinals , there is a subset of having cardinal and homogeneous for , in the sense that for any in , .A cardinal number
Cardinal number
In mathematics, cardinal numbers, or cardinals for short, are a generalization of the natural numbers used to measure the cardinality of sets. The cardinality of a finite set is a natural number – the number of elements in the set. The transfinite cardinal numbers describe the sizes of infinite...
is called ineffable if for every binary-valued function , there is a stationary subset of on which is homogeneous: that is, either maps all unordered pairs of elements drawn from that subset to zero, or it maps all such unordered pairs to one.More generally,
is called -ineffable (for a positive integer ) if for every there is a stationary subset of on which is -homogeneous (takes the same value for all unordered -tuples drawn from the subset). Thus, it is ineffable if and only if it is 2-ineffable.A totally ineffable cardinal is a cardinal that is
-ineffable for every . If is -ineffable, then the set of -ineffable cardinals below is a stationary subset of .Totally ineffable cardinals are of greater consistency strength than subtle cardinal
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 for which element number δ , Aδ ⊂ δ there...
s and of lesser consistency strength than remarkable cardinals. A list of large cardinal axioms by consistency strength is available here.