Hyperreal number

Element of a nonstandard model of the reals, which can be infinite or infinitesimal
Infinitesimals (ε) and infinities (ω) on the hyperreal number line (1/ε = ω/1)

Similarly, the casual use of 1/0 = ∞ is invalid, since the transfer principle applies to the statement that division by zero is undefined. The rigorous counterpart of such a calculation would be that if ε is a non-zero infinitesimal, then 1/ε is infinite.

The hyperreals can be developed either axiomatically or by more constructively oriented methods. The essence of the axiomatic approach is to assert (1) the existence of at least one infinitesimal number, and (2) the validity of the transfer principle. In the following subsection we give a detailed outline of a more constructive approach. This method allows one to construct the hyperreals if given a set-theoretic object called an ultrafilter, but the ultrafilter itself cannot be explicitly constructed.

Comparing sequences is thus a delicate matter. We could, for example, try to define a relation between sequences in a componentwise fashion:

For more information about this method of construction, see ultraproduct.

Now if we take a nontrivial ultrafilter (which is an extension of the Fréchet filter) and do our construction, we get the hyperreal numbers as a result.