Update

infinity.md

@@ -11,9 +11,9 @@ The term *infinity* has two slightly distinct meanings:
 
 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.
 
-**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") because that would break the nice [field](field.md) structure of real numbers, 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 -- 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", though they mostly live in a separate realm from the traditional numbers. Also for example the result of computing a [limit](limit.md) may be a real number but also infinity; so ultimately everything depends on our definition of what number is and we can declare infinity to be a number in some systems, for example there exists so called *extended real number line* which consists of real numbers and plus/minus infinity, which ARE treated as numbers.
+**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.
 
-An important term related to the term *infinite* is **[infinitesimal](infinitesimal.md)**, or *infinitely small*, a  concept very important e.g. for [calculus](calculus.md). 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 [fractals](fractal.md)) without end; for example in the realm of [real numbers](real_number.md) we may start at number 1 and keep moving closer and closer towards zero without ever reaching the "smallest nonzero number", as no matter how close to zero we are, we may always divide our distance by two. A term also related to this is [limit](limit.md), which helps us explore values "infinitely close", "infinitely far" etc.
+An important term related to the term *infinite* is **[infinitesimal](infinitesimal.md)**, or *infinitely small*, a  concept very important e.g. for [calculus](calculus.md). 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 [fractals](fractal.md)) without end, i.e. it appears while we start dividing by infinity -- this is important for [limits](limit.md) with which we explore values of functions that get "infinitely close" to some value without actually reaching it.
  
 When treated as [cardinality](cardinality.md) (i.e. size of a [set](set.md)), we conclude that **there are many infinities, some larger than others**, for example there are infinitely many [rational numbers](rational_number.md) and infinitely many [real numbers](real_number.md), but in a sense there are more real numbers than rational ones -- this is very counter intuitive, but nevertheless was proven by [Georg Cantor](cantor.md) 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 **[countable](countable.md)** -- 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 **[uncountable](uncountable.md)**. 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 [power set](power_set.md) (a set of all subsets).