Let's consider other method of creation of a field of hyper real numbers. But before we have to discuss concept of logical language and concept of interpretation of this language. Let's consider the general concept of one-sortable language of the first order.
Definition of a limit. The standard number is called as a sequence limit if all infinitely far members of this sequence are infinitely close to, i.e. for any non-standard hyper natural number the difference is infinitely small.
Let's give two more examples of "non-standard definitions" of standard concepts. Let - sequence of real numbers, or, in other words, function from N in R. Its non-standard analog represents function from * N in * R; it is natural to designate value of this function on hyper natural number m.
So, we entered addition, multiplication and an order on a set of hyper real numbers. It is easy to check that we received the ordered field, i.e. that in a set of hyper real numbers all usual properties of addition, multiplication and an order are carried out. Archimedes's axiom, however, in this field is not carried out.
Let the character set which elements we will call predicate symbols, and a set which elements we will call functional symbols be fixed. Let some natural number called by number of arguments, or valency, the corresponding symbol be compared to each predicate and functional symbol. In that case say that some language is set.
There are two ways of the proof. One of them uses nontrivial ultrafilters, and other method consists in application of one of the central theorems of logic - Gödel-Malzew's theorems of completeness, we will consider it in more detail. The concept of deductibility of this judgment from this set of judgments of T. Vyvodimost from T is defined means that there is a sequence of formulas, each of which belongs to either T, or in advance fixed set, or it turns out from the pregoing members of sequence by certain rules, and the last formula of this sequence is the formula. The sequence of formulas possessing the described properties is called as a formula conclusion from a set of formulas T.
Definition of a limit point. The standard number is called as a limit point of sequence if some infinitely far members of sequence are infinitely close to, i.e. there is such non-standard hyper natural number that the difference is infinitely small.
( if t and s terms, (t=s) - a formula; ( if - terms, and P - a predicate symbol with m arguments, P) - a formula; if P - a predicate symbol with zero arguments, P - a formula; ( if formula P and Q-, - formulas; ( if P - a formula, and - a variable, and - formulas.
Let's define concept of a formula of this language now. Let's choose and will record infinite sequence of the symbols called by variables. Let it will be for example symbols Will define at the beginning concept of a term. (T any variable and any functional symbol with zero arguments an essence terms;
Now we are able to tell precisely that we call hyper real numbers., any interpretation of P considered RL languages in which the same judgments, as in standard interpretation but for which Archimedes's axiom is not executed are true is called as system of hyper real numbers. Elements of the carrier of this interpretation are also called as hyper real numbers. Thus, perhaps there are a lot of systems of hyper real numbers.
Let's enter one more term relating to any language L and any set of T of judgments of the L. language a set of T joint if there is its model i.e. if there is an interpretation of language L in which all formulas from T. Teper are true everything it is ready to formulate the theorem of compactness of Maltsev.