In that case say that the entered order turns P into the ordered field. The ordered field P is not Archimedean in only case when when in it there are positive infinitesimal elements. The ordered field P is called as expansion of a field of real numbers of R if P contains all real numbers and, besides, operations and an order from P, their R considered on elements, coincide with usual arithmetic operations and a usual order on real numbers.
However, perhaps, the principal value of the non-standard analysis consists in other. Language of the non-standard analysis appeared a convenient construction tool of mathematical models of the physical phenomena. Ideas and methods of the non-standard analysis can become important part of future physical picture of the world. In any case already now many experts in mathematical physics actively use the non-standard analysis in the work.
Example Creation of an immeasurable set. Each real number satisfying to an inequality, we decompose in recurring binary decimal; for ensuring unambiguity we forbid decomposition with infinite number of the going in a row units. We fix any infinitely large natural number and we select those real numbers at which - the member of decomposition is equal to unit; a set of all real numbers which are selected thus immeasurably according to Lebesgue.
Thus, if the number is infinite a little, the number is infinitely great in the sense that it more than any of numbers: 1, 1+1, 1+1+1, 1+1+1+1, etc. From told it is possible to see that existence infinitesimal contradicts a so-called axiom of Archimedes which claims that for any two pieces And and In it is possible to postpone smaller of them (so many time that in the sum to receive the piece surpassing a bigger piece in length (.