![]() ![]() In 1960, Abraham Robinson provided an answer following the first approach. 2) Extend the axioms (or extend the language) so that the distinction between the infinitesimals and non-infinitesimals can be made in the real numbers themselves. There are in fact many ways to construct such a one-dimensional linearly ordered set of numbers, but fundamentally, there are two different approaches: 1) Extend the number system so that it contains more numbers than the real numbers. The question is: what is this model? What are its properties? Is there only one such model? Nonetheless, the theorem proves that there is a model (a number system) in which this will be true. This is not true in the real numbers ( R) given by ZFC. In the second expression, the statement says that there is an x (at least one), chosen first, which is between 0 and 1/ n for any n. Here, one chooses n first, then one finds the corresponding x. The first statement is true in the real numbers as given in ZFC set theory : for any positive integer n it is possible to find a real number between 1/ n and zero, but this real number will depend on n. The possibility to switch "for any" and "there exists" is crucial. A consequence of this theorem is that if there is a number system in which it is true that for any positive integer n there is a positive number x such that 0 < x < 1/ n, then there exists an extension of that number system in which it is true that there exists a positive number x such that for any positive integer n we have 0 < x < 1/ n. This theorem is fundamental for the existence of infinitesimals as it proves that it is possible to formalise them. In 1936 Maltsev proved the compactness theorem. We consider here systems where infinitesimals can be shown to exist. The method of constructing infinitesimals of the kind used in nonstandard analysis depends on the model and which collection of axioms are used.
0 Comments
Leave a Reply. |
AuthorWrite something about yourself. No need to be fancy, just an overview. ArchivesCategories |