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These sets are then taken to "be" cardinal numbers, by definition. In Zermelo–Fraenkel set theory with the axiom of choice, one way of assigning representatives to cardinal numbers is to associate each cardinal number with the least ordinal number of the same cardinality. These special ordinals are the ℵ numbers. But if the axiom of choice ...
In mathematics, a cardinal λ < Θ is a Suslin cardinal if there exists a set P ⊂ 2 ω such that P is λ-Suslin but P is not λ'-Suslin for any λ' < λ. It is named after the Russian mathematician Mikhail Yakovlevich Suslin (1894–1919).
Russian has several classes of numerals ([имена] числительные): cardinal, ordinal, collective, and also fractional constructions; also it has other types of words, relative to numbers: collective adverbial forms (вдвоём), multiplicative (двойной) and counting-system (двоичный) adjectives, some numeric ...
Existence of a cardinal number κ of a given type implies the existence of cardinals of most of the types listed above that type, and for most listed cardinal descriptions φ of lesser consistency strength, V κ satisfies "there is an unbounded class of cardinals satisfying φ".
Cardinal and Ordinal Numbers is a book on transfinite numbers, by Polish mathematician Wacław Sierpiński. It was published in 1958 by Państwowe Wydawnictwo Naukowe , as volume 34 of the series Monografie Matematyczne of the Institute of Mathematics of the Polish Academy of Sciences .
To define ℵ α for arbitrary ordinal number α, we must define the successor cardinal operation, which assigns to any cardinal number ρ the next larger well-ordered cardinal ρ + (if the axiom of choice holds, this is the (unique) next larger cardinal). We can then define the aleph numbers as follows: ℵ 0 = ω ℵ α+1 = (ℵ α) +
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A bijective function, f: X → Y, from set X to set Y demonstrates that the sets have the same cardinality, in this case equal to the cardinal number 4. Aleph-null, the smallest infinite cardinal. In mathematics, a cardinal number, or cardinal for short, is what is commonly called the number of elements of a set.