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*the '''hyperreals''': denoted ^*<math>\mathbb{R}</math>, these extend the real numbers with both infinite and infinitesimal quantities. They are used in non-standard analysis. Other extensions of the real numbers with infinite and infinitesimal quantities include the '''surreal numbers''', the '''superreal numbers''', and the '''Levi-Civita field'''.

*the '''hyperreals''': denoted ^*<math>\mathbb{R}</math>, these extend the real numbers with both infinite and infinitesimal quantities. They are used in non-standard analysis. Other extensions of the real numbers with infinite and infinitesimal quantities include the '''surreal numbers''', the '''superreal numbers''', and the '''Levi-Civita field'''.

The conclusion to be drawn from all of this, is that although in popular usage we speak about ''infinity'' as if it were a single thing, there are actually numerous diferent infinities, all belonging to different systems. If we want to talk about ''infinity'', we ought to be careful to specify which infinity we mean.

The conclusion to be drawn from all of this, is that although in popular usage we speak about ''infinity'' as if it were a single thing, there are actually numerous different infinities, all belonging to different systems. If we want to talk about ''infinity'', we ought to be careful to specify which infinity we mean.

==Archimedean property==

==Archimedean property==

An ordered field is said to be '''Archimedean''' if it has no infinitely large or infinitely small elements. One way of stating this: an ordered field is Archimedian, if for every element of the field, there is a greater natural number. This is true for the reals - for every real number, there is a greater natural number - but not for example for the affinely extended reals, since <math>+\infty</math> is greater than every natural number. So the affinely extended reals are said to be '''non-Archimedian'''. Likewise, the hyperreal numbers, the surreal numbers, the superreal numbers and the Levi-Civita field are all non-Archimedian. Any ordered field containing infinities must also contain infinitesimals as their inverse.

An ordered field is said to be '''Archimedean''' if it has no infinitely large or infinitely small elements. One way of stating this: an ordered field is Archimedean, if for every element of the field, there is a greater natural number. This is true for the reals - for every real number, there is a greater natural number - but not for example for the affinely extended reals, since <math>+\infty</math> is greater than every natural number. So the affinely extended reals are said to be '''non-Archimedean'''. Likewise, the hyperreal numbers, the surreal numbers, the superreal numbers and the Levi-Civita field are all non-Archimedean. Any ordered field containing infinities must also contain infinitesimals as their inverse.

   

   

To understand the notion of non-Archimedeanity, it is interesting to imagine what it would be like were space or time were non-Archimedean. If space was non-Archimedean, there could be areas of the universe infinitely distant from here. If time was non-Archimedean, there could be past times an infinite number of years before the present, and future times an infinite number of years into the future. It appears that no one has every seriously suggested that time or space actually are non-Archimedean, but it is nonetheless an interesting mental exercise to understand the notion.

To understand the notion of non-Archimedeanity, it is interesting to imagine what it would be like were space or time were non-Archimedean. If space was non-Archimedean, there could be areas of the universe infinitely distant from here. If time was non-Archimedean, there could be past times an infinite number of years before the present, and future times an infinite number of years into the future. It appears that no one has every seriously suggested that time or space actually are non-Archimedean, but it is nonetheless an interesting mental exercise to understand the notion.