Update

infinity.md

@@ -9,7 +9,7 @@ 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). 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 [axiom of choice](axiom_of_choice.md)). 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 [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.
+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.md) ("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 [number](number.md)?** Usually no, but it depends on the context. Infinity is not a [real number](real_number.md) (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 [field](field.md)), 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 [transfinite numbers](transfinite_number.md) 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 [limits](limit.md) with which we want to be able to get infinity as a result. The first infinite ordinal number **[omega](omega.md)** 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.