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infinity.md

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 # Infinity
 
-Infinity (from Latin *in* and *finis*, *without end*) is a quantity so unimaginably large that it has no end. It plays a large role especially in [mathematics](math.md) and [philosophy](philosophy.md). As a "largest imaginable quantity" it is sometimes seen as the opposite to the number [zero](zero.md), the "smallest possible quantity", though other "opposites" can be though of too, such as minus infinity or an infinitely small non-zero number ([infinitesimal](infinitesimal.md)). The symbol for infinity is *lemniscate*, the symbol 8 turned 90 degrees ([unicode](unicode.md) U+221E). Keep in mind that mere lack of boundaries doesn't imply infinity -- a [circle](circle.md) has no end but is not infinite; an infinity implies there is always more, no matter how much we get.
+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 [mathematics](math.md) and [philosophy](philosophy.md). As a "largest imaginable quantity" it is sometimes seen to be the opposite of the number [zero](zero.md), the "smallest possible quantity", though other "opposites" can be though of too, such as minus infinity or an infinitely small non-zero number ([infinitesimal](infinitesimal.md)). The symbol for infinity is *lemniscate*, the symbol 8 turned 90 degrees ([unicode](unicode.md) U+221E). Keep in mind that mere lack of boundaries doesn't imply infinity -- a [circle](circle.md) 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. [zero](zero.md) 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 [paradoxes](paradox.md) 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 [real life](irl.md), we lack experience and intuition about it, we don't know how such an object should behave and we encounter [paradoxes](paradox.md). For example if we have the largest object possible, what happens if we put two of such objects together, will we get yet larger object or not? How about two infinities minus one infinity -- is that an infinity or zero?
+- **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 [real life](irl.md), we lack experience and intuition about it, we don't know how such an object should behave and we encounter [paradoxes](paradox.md). For example if we have the largest object possible, what happens if we put two of such objects together, will we get yet 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 [proofs](proof.md) -- 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 [computer](computer.md)). For example to confirm [Fermat's Last Theorem](fermats_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.