Numbers between real numbers

Question

I wonder if there can be numbers (in some extended theory) for which all reals are either smaller or larger than this number, but no real number is equal to that number?!

Is there some extension of number which allows that? Under what conditions (axiom etc.) there is no such number.

Answer 1

Under the axioms of the real numbers this cannot occur. You must add new elements to the real numbers, note that if $\varepsilon$ is smaller than all $\frac1n$ but still positive then $\frac1\varepsilon$ is larger than any real number.

Such $\varepsilon$ is called infinitesimal and their existence is incompatible with the real numbers per se. There is a branch, however, called non-standard analysis in which these numbers play an important role.

One example to such field is called Hyperreal numbers.

Answer 2

You want a "non-archimedean" ordered field. Examples are hyperreals and surreals. My favorite one: the transseries (G. A. Edgar, "Transseries for Beginners," http://www.math.ohio-state.edu/~edgar/preprints/trans_begin/). Also try the Levi-Civita numbers: http://en.wikipedia.org/wiki/Levi-Civita_field . See here http://en.wikipedia.org/wiki/Infinitesimal for many examples.

Answer 3

Consider the field $\mathbb{R}(x)$ of all (formal) rational functions in one variable with real coefficients. While this is not an ordered field, it is an orderable field -- it is possible to define an ordering $<$ on rational functions that is consistent with the usual laws of arithmetic.

For any ordering $<$ of $\mathbb{R}(x)$, we can define sets $L = \{ a \in \mathbb{R} \mid a < x\}$ and $R = \{ a \in \mathbb{R} \mid a > x\}$, and we have $\mathbb{R} = L \cup R$ -- under this ordering, every real number is either less than or greater than the polynomial $x$.

It turns out the ordering $<$ is completely determined by $L$ and $R$, and conversely each way to choose $L$ and $R$ corresponds to an ordering of $\mathbb{R}(x)$.

The complete list of orderings are:

• The ordering "$+\infty$" - $x$ is larger than every real number
• The ordering "$-\infty$" - $x$ is smaller than every real number
• The ordering "$a^+$" - $x$ is infinitesimally larger than $a$
• The ordering "$a^-$" - $x$ is infinitesimally smaller than $a$

The labels I've chosen for the orderings refer to "where" $x$ is placed in relation to the real line.

Some good buzzwords that relate to this sort of topic are:

• Real closed field
• Formally real field
• Real algebraic geometry
• Semi-algebraic geometry

There is an easy way to write down a first-order theory whose models are examples of the sort of number system you ask for. For example,

• Start with the language of ordered fields
• Add a new constant symbol $\varepsilon$
• Add in all of the ordered field axioms
• Add in one axiom $0 < \varepsilon$
• For every positive integer $n$, add in one axiom $\varepsilon < n$

Every model of this theory will have a number $\varepsilon$ with the property that it is larger than every non-positive real number, and smaller than every positive real number.