Changes

*the '''affinely extended reals''': these are elements of the set <math>\mathbb{R} \cup \{ +\infty, -\infty \}</math>. In other words, the real numbers extended by the addition of positive and negative infinity. <math>+\infty</math> is greater than every real number, and <math>-\infty</math> is less than every real number.

*the '''affinely extended reals''': these are elements of the set <math>\mathbb{R} \cup \{ +\infty, -\infty \}</math>. In other words, the real numbers extended by the addition of positive and negative infinity. <math>+\infty</math> is greater than every real number, and <math>-\infty</math> is less than every real number.

*the '''projectively extended reals''': these are elements of the set <math>\mathbb{R} \cup \{ \infty \}</math>. In other words, the real numbers extended by the addition of a single signless infinity. The ordering relation is not defined for the projectively extended reals - but informally, <math>\infty</math> can be said to be both greater than ''and'' less than every real number.

*the '''projectively extended reals''': these are elements of the set <math>\mathbb{R} \cup \{ \infty \}</math>. In other words, the real numbers extended by the addition of a single signless infinity. The ordering relation is not defined for the projectively extended reals - but informally, <math>\infty</math> can be said to be both greater than ''and'' less than every real number.

*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. In this light, claims such as ''God is infinite'' are particularly incomprehnsible, because those who propose them never define which infinity they are using to describe God, or in what way that infinity describes him.

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. In this light, claims such as ''God is infinite'' are particularly incomprehnsible, because those who propose them never define which infinity they are using to describe God, or in what way that infinity describes him.

 

==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 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.