Aleph Alpha GmbH is a German artificial intelligence (AI) startup company founded by Jonas Andrulis and Samuel Weinbach, both of whom have professional...

{\displaystyle \aleph _{\alpha }^{\aleph _{\beta }}\leq \aleph _{\alpha }^{\aleph _{\alpha }}\leq (2^{\aleph _{\alpha }})^{\aleph _{\alpha }}=2^{\aleph _{\alpha }\cdot...

  {\displaystyle 0,1,2,3,\ldots ,n,\ldots ;\aleph _{0},\aleph _{1},\aleph _{2},\ldots ,\aleph _{\alpha },\ldots .\ } This sequence starts with the natural...

derived from the Phoenician letter aleph , which is the West Semitic word for "ox". Letters that arose from alpha include the Latin letter A and the Cyrillic...

ℵ 0 , {\displaystyle \beth _{0}=\aleph _{0},} ℶ α + 1 = 2 ℶ α , {\displaystyle \beth _{\alpha +1}=2^{\beth _{\alpha }},} ℶ λ = sup { ℶ α : α < λ } , {\displaystyle...

define ω α {\displaystyle \omega _{\alpha }} as the initial ordinal of the cardinal ℵ α {\displaystyle \aleph _{\alpha }} . The existence of ω 1 {\displaystyle...

+ , {\displaystyle \aleph _{\alpha ^{+}}=(\aleph _{\alpha })^{+}\,,} if λ is a successor ordinal then ℵ λ {\displaystyle \aleph _{\lambda }} is not a...

the set of all integers. More generally, the ℵ α {\displaystyle \aleph _{\alpha }} aleph number notation marks the ordered sequence of all distinct infinite...

{\displaystyle \omega _{\alpha }} with ℵ α {\displaystyle \aleph _{\alpha }} , except that the notation ℵ α {\displaystyle \aleph _{\alpha }} is used for writing...

{\displaystyle \aleph _{\alpha +1}=\aleph _{\alpha }^{+}} and for λ an infinite limit ordinal, ℵ λ = ⋃ β < λ ℵ β {\displaystyle \aleph _{\lambda }=\bigcup...

Aleph (or alef or alif, transliterated ʾ) is the first letter of the Semitic abjads, including Arabic ʾalif ا‎, Aramaic ʾālap 𐡀, Hebrew ʾālef א‎, North...

ℵ n − 1 {\displaystyle \aleph _{n-1}} ⁠. If α ≥ ω, the α-th number class has cardinality ⁠ ℵ α {\displaystyle \aleph _{\alpha }} ⁠. Therefore, the cardinalities...

{\displaystyle \aleph _{\alpha }} -saturated, and moreover two ηα real closed fields both of cardinality ℵ α {\displaystyle \aleph _{\alpha }} are order...

a Structured If (short for else if) First letter of the alphabet: A Aleph Alpha This disambiguation page lists articles associated with the title Elif...

{\displaystyle \aleph _{0}<\aleph _{1}<\aleph _{2}<\ldots .} For each ordinal α {\displaystyle \alpha } , ℵ α + 1 {\displaystyle \aleph _{\alpha +1}} is the...

iota) and [u] (Υ, upsilon); the glottal stop consonant /ʔ/ (aleph) was used for [a] (Α, alpha); the pharyngeal /ʕ/ (ʿayin) was turned into [o] (Ο, omicron);...

fraternities. The letters were chosen to spite the fraternity the Alephs were denied, Alpha Zeta Alpha, by naming their fraternity the Hebrew counterparts. However...

{\displaystyle \aleph _{G(\alpha )}} is greater than ℵ α {\displaystyle \aleph _{\alpha }} for each α in the domain of G, and ℵ α {\displaystyle \aleph _{\alpha }}...

demand for non-US cloud services. Schwarz also holds a minority stake in Aleph Alpha, a German AI startup. "Retail Index". Retrieved 2020-08-06. "10 biggest...

μ ℵ μ − 1 ℵ α . {\displaystyle \aleph _{\mu }^{\aleph _{\alpha }}=\aleph _{\mu }\;\aleph _{\mu -1}^{\aleph _{\alpha }}.} This formula was, together with...

examples of normal functions are given by the aleph numbers f ( α ) = ℵ α {\displaystyle f(\alpha )=\aleph _{\alpha }} , which connect ordinal and cardinal...

Aleph (also known as DOT or 2,5-dimethoxy-4-methylthioamphetamine) is a psychedelic hallucinogenic drug and a substituted amphetamine of the phenethylamine...

anarchy symbol ⟨𐤀⟩: Phoenician aleph, from which the following symbols originally derive: ⟨Α α⟩: Greek letter alpha, from which the following letters...

{\displaystyle \aleph _{\alpha }} , Cohen built a model in which the cardinality of the power set of ω is ℵ α {\displaystyle \aleph _{\alpha }} if the cofinality...

{\displaystyle \scriptstyle 2^{\aleph _{\alpha }}} friendship graphs of cardinal ℵ α {\displaystyle \scriptstyle \aleph _{\alpha }} ", Canadian Mathematical...

is proper. Crucially, all proper forcings preserve ℵ 1 {\displaystyle \aleph _{1}} . PFA directly implies its version for ccc forcings, Martin's axiom...

cofinality of ℵ 0 {\displaystyle \aleph _{0}} is ℵ 0 . {\displaystyle \aleph _{0}.} Thus ℵ 0 {\displaystyle \aleph _{0}} is a regular cardinal. The cofinality...

{\displaystyle \scriptstyle 2^{\aleph _{\alpha }}} friendship graphs of cardinal ℵ α {\displaystyle \scriptstyle \aleph _{\alpha }} ", Canadian Mathematical...

ℵ 0 {\displaystyle \aleph _{0}} , ℵ 1 {\displaystyle \aleph _{1}} , ℵ 2 {\displaystyle \aleph _{2}} , ℵ 3 {\displaystyle \aleph _{3}} , and so on. Its...

inaccessible cardinals to be uncountable (in which case ⁠ ℵ 0 {\displaystyle \aleph _{0}} ⁠ is strongly inaccessible). Weakly inaccessible cardinals were introduced...