Infinity

   Infinity (from Latin in and finis, without end) is a quantity so
   unimaginably large that it has no end. It plays a prominent role
   especially in [1]mathematics and [2]philosophy. As a "largest imaginable
   quantity" it is sometimes seen to be the opposite of the number [3]zero,
   the "smallest possible quantity", though other "opposites" can be though
   of too, such as minus infinity or an infinitely small non-zero number
   ([4]infinitesimal). The symbol for infinity is lemniscate, the symbol 8
   turned 90 degrees ([5]unicode U+221E, looking a bit like oo). Keep in mind
   that mere lack of boundaries doesn't imply infinity -- a [6]circle has no
   end but is not infinite; an infinity implies there is always more, no
   matter how much we get.

   The concept of infinity came to firstly be explored by philosophers -- as
   an abstract concept (similar to those of e.g. [7]zero or negative numbers)
   it took a while for it to evolve, be explored and accepted. We can't say
   who first "discovered" infinity, civilizations often had concepts similar
   to it that were connected for example to their gods. Zeno of Elea (5th
   century BC) was one of the earliest to tackle the issue of infinity
   mathematically by proposing [8]paradoxes such as that of Achilles and the
   tortoise.

   The term infinity has two slightly distinct meanings:

     * potential infinity: The unboundedness, lack of upper limit. For
       example the sequence of odd numbers 1, 3, 5, ... is potentially
       infinite. This is the less problematic kind of infinity as we know
       what's going on: we simply lack any limit and can keep going on
       forever.
     * actual infinity: Infinity as an actual "object" (for example a number)
       that's somehow "endlessly large", larger beyond any limits, largest
       possible etc. This type of infinity poses more issues as we don't know
       anything like this from [9]real life, we lack experience and intuition
       about it, we don't know how such an object should behave and we
       encounter [10]paradoxes. Stuff can get pretty weird and things we take
       for granted stop working, such as being able to just randomly pick
       elements from sets (see [11]axiom of choice). For example if we have
       the largest object possible, what happens if we put two of such
       objects together, will we get yet a larger object or not? How about
       two infinities minus one infinity -- is that an infinity or zero? What
       if we shrink infinity to half, what size will it have?

   It could be argued that potential infinity is really the reason for the
   existence of true, high level mathematics as we know it, as that is
   concerned with constructing mathematical [12]proofs -- such proofs are
   needed anywhere where there exist infinitely many possibilities, as if
   there was only a finite number of possibilities, we could simply enumerate
   and check them all without much thinking (e.g. with the help of a
   [13]computer). For example to confirm [14]Fermat's Last Theorem ("for
   whole numbers and n > 2 the equation a^n + b^n = c^n doesn't have a
   solution") we need a logical proof because there are infinitely many
   numbers; if there were only finitely many numbers, we could simply check
   them all and see if the theorem holds. So infinity, in a sense, is really
   what forces mathematicians to think.

   Is infinity a [15]number? Usually no, but it depends on the context.
   Infinity is not a [16]real number (which we usually understand by the term
   "number"), nor does it belong to any traditionally used set of numbers
   like integers or rational numbers, because including infinity would break
   the mathematical structure of these sets (e.g. real numbers would seize to
   be a [17]field), so the safe implicit answer to the question is no,
   infinity is not a traditional number, it is rather a concept closely
   related to numbers. However infinity may sometimes behave like a number
   and we may want to treat it so, so there also exist "special" number sets
   that include it -- see for example [18]transfinite numbers that are used
   to work with infinite sets and the numbers can be thought of as "sort of
   infinity numbers", but again, they are separated from the realm of the
   "traditional" numbers. This comes to play for example when computing
   [19]limits with which we want to be able to get infinity as a result. The
   first infinite ordinal number [20]omega is often seen as "the infinity
   number", but this always comes with asterisks, with infinities we have to
   start distinguishing between cardinal and ordinal numbers, we have to
   define all the basic operations again, check if they actually work, we
   also may have to give up some convenient assumptions we could use before
   as a tradeoff and so on. So ultimately everything depends on our
   definition of what number is and we can declare infinity to be a number in
   some systems, see also extended real number line and so on.

   An important term related to the term infinite is [21]infinitesimal, or
   infinitely small, a concept very important e.g. for [22]calculus. While
   the "traditional" concept of infinity looks beyond the greatest numbers
   imaginable, the concept of infinitely small is about being able to divide
   (or "zoom in", see also [23]fractals) without end, i.e. it appears while
   we start dividing by infinity -- this is important for [24]limits with
   which we explore values of functions that get "infinitely close" to some
   value without actually reaching it.

   When treated as [25]cardinality (i.e. size of a [26]set), we conclude that
   there are many infinities, some larger than others, for example there are
   infinitely many [27]rational numbers and infinitely many [28]real numbers,
   but in a sense there are more real numbers than rational ones -- this is
   very counter intuitive, but nevertheless was proven by [29]Georg Cantor in
   1874. He showed that it is possible to create a 1 to 1 pairing of natural
   numbers and rational numbers and so that these sets are of the same size
   -- he called this kind of infinity [30]countable -- then he showed it is
   not possible to make such pairing with real numbers and so that there are
   more real numbers than rational ones -- he called this kind of infinity
   [31]uncountable. Furthermore this hierarchy of "larger and larger
   infinities" goes on forever, as for any set we can always create a set
   with larger cardinality e.g. by taking its [32]power set (a set of all
   subsets).

   In regards to [33]programming: programmers are often just engineers and so
   simplify the subject of infinity in a way which to a mathematician would
   seem unacceptable. For example it is often a [34]good enough approximation
   of infinity to just use an extremely large number value, e.g. the largest
   one storable in given data type, which of course has its limitations, but
   in practice [35]just werks (just watch out for [36]overflows). Programmers
   also often resort to breaking the mathematical rules, e.g. they may accept
   that x / 0 = infinity, infinity + infinity = infinity etc. Systems based
   on [37]symbolic computation may be able to handle infinity with exact
   mathematical precision. Advanced data types, such as [38]floating point,
   often have a special value for infinity -- IEEE 754 floating point, for
   example, is capable of representing positive and negative infinity.

   WATCH OUT: infinite universe doesn't imply existence of everything -- this
   is a common fallacy to think it does. For example people tend to think
   that since the decimal expansion of the digits of [39]pi is infinite and
   basically "random", there should always exist any finite string of digits
   somewhere in it; this doesn't follow from the mere fact that the series is
   infinite (though the conclusion MAY or may not be true, we don't actually
   know this about pi yet). Imagine for example the infinite series of even
   numbers -- there are infinitely many numbers in it, but you will never
   find any odd number there.

See Also

     * [40]zero
     * [41]thrembo

Links:
1. math.md
2. philosophy.md
3. zero.md
4. infinitesimal.md
5. unicode.md
6. circle.md
7. zero.md
8. paradox.md
9. irl.md
10. paradox.md
11. axiom_of_choice.md
12. proof.md
13. computer.md
14. fermats_last_theorem
15. number.md
16. real_number.md
17. field.md
18. transfinite_number.md
19. limit.md
20. omega.md
21. infinitesimal.md
22. calculus.md
23. fractal.md
24. limit.md
25. cardinality.md
26. set.md
27. rational_number.md
28. real_number.md
29. cantor.md
30. countable.md
31. uncountable.md
32. power_set.md
33. programming.md
34. good_enough.md
35. just_werkd.md
36. overflow.md
37. symbolic_computation.md
38. float.md
39. pi.md
40. zero.md
41. thrembo.md