Transfinite numbers, also known as infinite numbers, are numbers that are not finite. These numbers were first considered by Indian Jaina mathematicians around the 4th century BC.[1] [2] Infinity is a term with very distinct, separate meanings which arise in theology, philosophy, mathematics and everyday life. ... JAIN is an activity within the Java Community Process, developing APIs for the creation of telephony (voice and data) services. ... (5th century BC - 4th century BC - 3rd century BC - other centuries) (2nd millennium BC - 1st millennium BC - 1st millennium AD) // Events Invasion of the Celts into Ireland Battle of the Allia and subsequent Gaulish sack of Rome 383 BCE Second Buddhist Councel at Vesali. ...

As with finite numbers, there are two ways of thinking of transfinite numbers, as ordinal and cardinal numbers. Unlike the finite ordinals and cardinals, the transfinite ordinals and cardinals define different classes of numbers.

• The lowest transfinite ordinal number is ω.

• The first transfinite cardinal number is aleph-null, , the cardinality of the infinite set of the integers. The next higher cardinal number is aleph-one, .

The continuum hypothesis states that there are no intermediate cardinal numbers between aleph-null and the cardinality of the continuum (the set of real numbers): that is to say, aleph-one is the cardinality of the set of real numbers. Commonly, ordinal numbers, or ordinals for short, are numbers used to denote the position in an ordered sequence: first, second, third, fourth, etc. ... In linguistics, cardinal numbers is the name given to number words that are used for quantity (one, two, three), as opposed to ordinal numbers, words that are used for order (first, second, third). ... In the branch of mathematics known as set theory, aleph usually refers to a series of numbers used to represent the cardinality (or size) of infinite sets. ... In mathematics, the cardinality of a set is a measure of the number of elements of the set. There are two approaches to cardinality – one which compares sets directly using bijections, injections, and surjections, and another which uses cardinal numbers. ... An infinite set in Cantors set theory is any set which is not finite. ... The integers consist of the positive natural numbers (1, 2, 3, …), their negatives (−1, −2, −3, ...) and the number zero. ... In linguistics, cardinal numbers is the name given to number words that are used for quantity (one, two, three), as opposed to ordinal numbers, words that are used for order (first, second, third). ... In the branch of mathematics known as set theory, aleph usually refers to a series of numbers used to represent the cardinality (or size) of infinite sets. ... In mathematics, the continuum hypothesis is a hypothesis about the possible sizes of infinite sets. ... In mathematics, the cardinality of the continuum is the cardinal number of the set of real numbers R (sometimes called the continuum). ... In mathematics, the real numbers are intuitively defined as numbers that are in one-to-one correspondence with the points on an infinite line—the number line. ...

In both the cardinal and ordinal number systems, the transfinite numbers can keep on going forever, with progressively more bizarre kinds of numbers.

Beyond all these, Georg Cantor's conception of the Absolute Infinite represents the absolute largest possible concept of "large number". Georg Cantor Georg Ferdinand Ludwig Philipp Cantor (March 3, 1845 – January 6, 1918) was a mathematician who was born in Russia and lived in Germany for most of his life. ... The Absolute Infinite is Georg Cantors concept of an infinity that transcended the transfinite numbers. ...

See also

• Beth number
• Large cardinal
• Mahlo cardinal
• Infinitesimal