Editing Infinitesimal (section)

== First-order properties ==
In extending the real numbers to include infinite and infinitesimal quantities, one typically wishes to be as conservative as possible by not changing any of their elementary properties. This guarantees that as many familiar results as possible are still available. Typically, ''elementary'' means that there is no [[quantification (logic)|quantification]] over [[set (mathematics)|sets]], but only over elements. This limitation allows statements of the form "for any number x..." For example, the axiom that states "for any number&nbsp;''x'', ''x''&nbsp;+&nbsp;0&nbsp;=&nbsp;''x''" would still apply. The same is true for quantification over several numbers, e.g., "for any numbers&nbsp;''x'' and ''y'', ''xy''&nbsp;=&nbsp;''yx''." However, statements of the form "for any ''set''&nbsp;''S''&nbsp;of numbers&nbsp;..." may not carry over. Logic with this limitation on quantification is referred to as [[first-order logic]].

The resulting extended number system cannot agree with the reals on all properties that can be expressed by quantification over sets, because the goal is to construct a non-Archimedean system, and the Archimedean principle can be expressed by quantification over sets. One can conservatively extend any theory including reals, including set theory, to include infinitesimals, just by adding a countably infinite list of axioms that assert that a number is smaller than 1/2, 1/3, 1/4, and so on. Similarly, the [[Complete metric space|completeness]] property cannot be expected to carry over, because the reals are the unique complete ordered field up to isomorphism.

We can distinguish three levels at which a non-Archimedean number system could have first-order properties compatible with those of the reals:

# An [[ordered field]] obeys all the usual axioms of the real number system that can be stated in first-order logic. For example, the [[commutativity]] axiom ''x''&nbsp;+&nbsp;''y''&nbsp;=&nbsp;''y''&nbsp;+&nbsp;''x'' holds.
# A [[real closed field]] has all the first-order properties of the real number system, regardless of whether they are usually taken as axiomatic, for statements involving the basic ordered-field relations +,&nbsp;×, and&nbsp;≤. This is a stronger condition than obeying the ordered-field axioms. More specifically, one includes additional first-order properties, such as the existence of a root for every odd-degree polynomial. For example, every number must have a [[cube root]].
# The system could have all the first-order properties of the real number system for statements involving ''any'' relations (regardless of whether those relations can be expressed using&nbsp;+,&nbsp;×, and&nbsp;≤). For example, there would have to be a [[sine]] function that is well defined for infinite inputs; the same is true for every real function.

Systems in category 1, at the weak end of the spectrum, are relatively easy to construct but do not allow a full treatment of classical analysis using infinitesimals in the spirit of Newton and Leibniz. For example, the [[transcendental functions]] are defined in terms of infinite limiting processes, and therefore there is typically no way to define them in first-order logic. Increasing the analytic strength of the system by passing to categories&nbsp;2 and&nbsp;3, we find that the flavor of the treatment tends to become less constructive, and it becomes more difficult to say anything concrete about the hierarchical structure of infinities and infinitesimals.