However, in the 1960s Abraham Robinson showed how infinitely large and infinitesimal numbers can be rigourously defined and developed what is called nonstandard analysis. Because his theory in its fullfledged form makes unrestricted use of classical logic and set theory and, in particular, of the axiom of choice, it is suspected to be nonconstructive[?] from the outset. The construction given below is a simplified version of Robinson's more general construction and is due to Lindstrom.
The hyperreals *R form an ordered field containing the reals R as a subfield. Unlike the reals, the hyperreals do not form a metric space, but by virtue of their order they carry an order topology.
The hyperreals are defined in such a way that every firstorder logic statement that uses basic arithmetic (the natural numbers, plus, times, comparison) and quantifies only over the real numbers is also true if we presume that they quantify over hyperreal numbers. For example, we can state that for every real number there is another number greater than it:
The same will then also hold for hyperreals:
Another example is the statement that if you add 1 to a number you get a bigger number:
which will also hold for hyperreals:
This however doesn't mean that R and *R behave the same. For instance, in *R there exists an element w such that
We are going to construct the hyperreals via sequences of reals. This is nice, because we can immediately identify the real number r with the sequence (r, r, r, ...) and we can also add and multiply sequences: (a_{0}, a_{1}, a_{2}, ...) + (b_{0}, b_{1}, b_{2}, ...) = (a_{0} + b_{0}, a_{1} + b_{1}, a_{2} + b_{2}, ...) and analogously for multiplication.
We also need to be able to compare sequences, and there we run into trouble: some entries of the first sequence may be bigger than the corresponding entries of the second sequence, and some others may be smaller. We have to specify "which positions matter". Since there are infinitely many indices, we don't want finite sets of indices to matter. A consistent choice of "index sets that matter" is given by any free ultrafilter U on the natural numbers which does not contain any finite sets. Such an U exists by the axiom of choice. (In fact, there are many such U, but it turns out that it doesn't matter which one we take.) We think of U as singling out those sets of indices that "matter": We write (a_{0}, a_{1}, a_{2}, ...) ≤ (b_{0}, b_{1}, b_{2}, ...) if and only if the set of natural numbers { n : a_{n} ≤ b_{n} } is in U. This is a total preorder and it turns into a total order if we agree not to distinguish between two sequences a and b if a≤b and b≤a. With this identification, the ordered field *R of hyperreals is constructed.
A nonstandard real number e is called infinitesimal if it is smaller than every positive real number and bigger than every negative real number. Zero is an infinitesimal, but nonzero infinitesimals also exist: take for instance the class of the sequence (1, 1/2, 1/3, 1/4, 1/5, 1/6, ...) (this works because U contains all index sets whose complement is finite).
A nonstandard real number x is called finite if there exists a natural number n such that – n < x < +n; otherwise, x is called infinite. Infinite numbers exist; take for instance the class of the sequence (1, 2, 3, 4, 5, ...). A nonzero number x is infinite if and only if 1/x is infinitesimal.
Now it turns out that every finite nonstandard real number is "very close" to a unique real number, in the following sense: if x is a finite nonstandard real, then there exists one and only one real number st(x) such that x – st(x) is infinitesimal. This number st(x) is called the standard part of x. This operation has nice properties:
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