In mathematics, transfinite numbers or infinite numbers are numbers that are "infinite" in the sense that they are larger than all finite numbers. These...

defined the transfinite ordinal numbers and transformed the infinite indices into ordinals by replacing ∞ with ω, the first transfinite ordinal number. Cantor...

numbers are a certain sequence of infinite cardinal numbers (also known as transfinite numbers), conventionally written ℶ 0 , ℶ 1 , ℶ 2 , ℶ 3 , … {\displaystyle...

Transfinite may refer to: Transfinite number, a number larger than all finite numbers, yet not absolutely infinite Transfinite induction, an extension...

Transfinite induction is an extension of mathematical induction to well-ordered sets, for example to sets of ordinal numbers or cardinal numbers. Its...

general, but all its elements do.) Beth number Gimel function Regular cardinal Infinity Transfinite number Ordinal number Given the axiom of choice, every infinite...

realized as subfields of the surreals. The surreals also contain all transfinite ordinal numbers; the arithmetic on them is given by the natural operations...

function F at a finite number of distinct points, these methods match F at a non-denumerable (transfinite) number of points. Transfinite interpolation is similar...

published a book about his new set theory, introducing, among other things, transfinite numbers and formulating the continuum hypothesis. In the 1960s, Abraham...

"Zur Einführung der transfiniten Zahlen" [On the Introduction of the Transfinite Numbers]. Acta Litterarum AC Scientiarum Ragiae Universitatis Hungaricae...

number line. Aleph null Suslin's problem Transfinite number Weisstein, Eric W. "Continuum". mathworld.wolfram.com. Retrieved 2020-08-12. "Transfinite...

number Algebraic number Transcendental number Hypercomplex number Transfinite number Indefinite and fictitious numbers Mean Weighted mean Median Mode Range...

interest, a fact he was well aware of. Originally, Cantor's theory of transfinite numbers was regarded as counter-intuitive – even shocking. This caused...

In mathematics, the epsilon numbers are a collection of transfinite numbers whose defining property is that they are fixed points of an exponential map...

cardinal number. This cardinal number is called ℵ 0 {\displaystyle \aleph _{0}} , aleph-null. He called the cardinal numbers of infinite sets transfinite cardinal...

Transcomputational problem – Class of computational problems Transfinite number – Number that is larger than all finite numbers Zeno machine – Hypothetical...

Philip Jourdain (ed.). Contributions to the Founding of the Theory of Transfinite Numbers. New York: Dover. ISBN 978-0-486-60045-1. {{cite book}}: ISBN...

879–90. doi:10.1137/0513062. Erdos, P.; Gillis, J. (2009). "Note on the Transfinite Diameter". Journal of the London Mathematical Society. s1-12 (3): 185...

transfinite 1.  An infinite ordinal or cardinal number (see Transfinite number) 2.  Transfinite induction is induction over ordinals 3.  Transfinite recursion...

{c}}=\beth _{2}} (beth two) Cardinal characteristic of the continuum "Transfinite number | mathematics". Encyclopedia Britannica. Retrieved 2020-08-12. Weisstein...

hypothesis. Besides transfinite cardinals, there also exists transfinite ordinals, like described in Fast Growing Hierarchy. Transfinite ordinals operate...

constructing models of set theory by repeating Cohen's forcing method a transfinite number of times. Iterated forcing was introduced by Solovay and Tennenbaum (1971)...

less than α. Thus there is one set Vα for each ordinal number α. Vα may be defined by transfinite recursion as follows: Let V0 be the empty set: V 0 :=...

representing an ordinal number is well-founded, the set of natural numbers is one of them. Applied to a well-founded set, transfinite induction can be formulated...

{\displaystyle \aleph _{0}} (the cardinal number of the infinite set of integers 1, 2, ...), the smallest cardinal transfinite number. The Jains also defined a whole...

[精密科学]; Prime number [素数]; Theory of quanta [素量説]; Algebraic number [代数的数]; Field of force [力の場]; Transcendental number [超越的数]; Transfinite aggregate [超限集合];...

proof theory). He adopted and defended Georg Cantor's set theory and transfinite numbers. In 1900, he presented a collection of problems that set a course...

can continue it to infinite ordinal numbers via transfinite recursion, thereby obtaining the transfinite derived series, which eventually terminates at...

which is called transfinite induction. Given a property (predicate) ⁠ P ( n ) {\displaystyle P(n)} ⁠ depending on a natural number, mathematical induction...

infinite number of functions. So instead let us merge the unary functions into a binary function. By transfinite recursion on α, we can use transfinite recursion...