Infinite

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The word infinity comes from the Latin infinitas or "unboundedness." It refers to several distinct concepts (usually linked to the idea of "without end") which arise in philosophy, mathematics, and theology.


In mathematics, "infinity" is often used in contexts where it is treated as if it were a number (i.e., it counts or measures things: "an infinite number of terms") but it is a different type of "number" than the real numbers. Infinity is related to limits, aleph numbers, classes in set theory, Dedekind-infinite sets, large cardinals,Large cardinals are quantitative infinities defining the number of things in a collection, which are so large that they cannot be proven to exist in the ordinary mathematics of Zermelo-Fraenkel plus Choice (ZFC).Russell's paradox, non-standard arithmetic, hyperreal numbers, projective geometry, extended real numbers and the absolute Infinite.



For lessons on the topic of Infinity, follow this link.

Logic

In logic an infinite regress argument is "a distinctively philosophical kind of argument purporting to show that a thesis is defective because it generates an infinite series when either (form A) no such series exists or (form B) were it to exist, the thesis would lack the role (e.g., of justification) that it is supposed to play."Cambridge Dictionary of Philosophy, Second Edition, p. 429

The word infinity comes from the Latin infinitas or "unboundedness." It refers to several distinct concepts (usually linked to the idea of "without end") which arise in philosophy, mathematics, and theology.

In mathematics, "infinity" is often used in contexts where it is treated as if it were a number (i.e., it counts or measures things: "an infinite number of terms") but it is a different type of "number" than the real numbers. Infinity is related to limits, aleph numbers, classes in set theory, Dedekind-infinite sets, large cardinals,<ref>Large cardinals are quantitative infinities defining the number of things in a collection, which are so large that they cannot be proven to exist in the ordinary mathematics of Zermelo-Fraenkel plus Choice (ZFC).</ref> Russell's paradox, non-standard arithmetic, hyperreal numbers, projective geometry, extended real numbers and the absolute Infinite.

Infinity symbol

The precise origin of the infinity symbol is unclear. One possibility is suggested by the name it is sometimes called — the lemniscate, from the Latin lemniscus, meaning "ribbon." One can imagine walking forever along a simple loop formed from a ribbon.

A popular explanation is that the infinity symbol is derived from the shape of a Möbius strip. Again, one can imagine walking along its surface forever. However, this explanation is improbable, since the symbol had been in use to represent infinity for over two hundred years before August Ferdinand Möbius and Johann Benedict Listing discovered the Möbius strip in 1858.

It is also possible that it is inspired by older religious/alchemical symbolism. For instance, it has been found in Tibetan rock carvings, and the ouroboros, or infinity snake, is often depicted in this shape. In the Rider-Waite tarot deck, the lemniscate represents the balance of forces and is often associated with the magician card.

John Wallis is usually credited with introducing ∞ as a symbol for infinity in 1655 in his De sectionibus conicis. One conjecture about why he chose this symbol is that he derived it from a Roman numeral for 1000 that was in turn derived from the Etruscan numeral for 1000, which looked somewhat like CIƆ and was sometimes used to mean "many." Another conjecture is that he derived it from the Greek letter ω (omega), the last letter in the Greek alphabet.<ref>The History of Mathematical Symbols, By Douglas Weaver, Mathematics Coordinator, Taperoo High School with the assistance of Anthony D. Smith, Computing Studies teacher, Taperoo High School.</ref>

Another possibility is that the symbol was chosen because it was easy to rotate an "8" character by 90° when typesetting was done by hand. The symbol is sometimes called a "lazy eight", evoking the image of an "8" lying on its side.

Another popular belief is that the infinity symbol is a clear depiction of the hour glass turned 90°. Obviously, this action would cause the hour glass to take infinite time to empty thus presenting a tangible example of infinity. The invention of the hourglass predates the existence of the infinite symbol allowing this theory to be plausible.

The infinity symbol is represented in Unicode by the character ∞ (U+221E).

History

Early Indian views of infinity

The Isha Upanishad of the Yajurveda (c. 4th to 3rd century BC) states that "if you remove a part from infinity or add a part to infinity, still what remains is infinity".

Pūrṇam adaḥ pūrṇam idam (That is full, this is full)
pūrṇāt pūrṇam udacyate (From the full, the full is subtracted)
pūrṇasya pūrṇam ādāya (When the full is taken from the full)
pūrṇam evāvasiṣyate (The full still will remain.) - Isha Upanishad

The Indian mathematical text Surya Prajnapti (c. 400 BC) classifies all numbers into three sets: enumerable, innumerable, and infinite. Each of these was further subdivided into three orders:

  • Enumerable: lowest, intermediate and highest
  • Innumerable: nearly innumerable, truly innumerable and innumerably innumerable
  • Infinite: nearly infinite, truly infinite, infinitely infinite

The Jains were the first to discard the idea that all infinites were the same or equal. They recognized different types of infinities: infinite in length (one dimension), infinite in area (two dimensions), infinite in volume (three dimensions), and infinite perpetually (infinite number of dimensions).

According to Singh (1987), Joseph (2000) and Agrawal (2000), the highest enumerable number N of the Jains corresponds to the modern concept of aleph-null (the cardinal number of the infinite set of integers 1, 2, ...), the smallest cardinal transfinite number. The Jains also defined a whole system of infinite cardinal numbers, of which the highest enumerable number N is the smallest.

In the Jaina work on the theory of sets, two basic types of infinite numbers are distinguished. On both physical and ontological grounds, a distinction was made between asaṃkhyāta ("countless, innumerable") and ananta ("endless, unlimited"), between rigidly bounded and loosely bounded infinities.