# Foundations

Number systems |
The book and teaching material assume the validity of a certain number of principles applied to real numbers. This method is customary when introducing analysis. Assuming one has some familiarity with the formal construction of number systems in the classical setting, we show how this construction is performed using the extra axioms. The method is essentially the same with the addition of showing the existence of ultralarge numbers, ultrasmall numbers and observable neighbour. |

Consistency proof | When introducing a new method for performing analysis, by means of adding new axioms, it is important to prove that this extension cannot make contradictions appear. It is not necessary to study foundations in detail to be able to use analysis with ultrasmall numbers. One can very well use it based on trust. Nonetheless, a proof of the consistency of the system is given here. |

Closure | In the book, the principle of Closure is introduced. In the chapter about Foundtions, it is shown that it follows from transfer, even though in the book, it is mentioned that it follows directly from stability (p33). We prove this here. |

External sets | The axioms introduced for analysis with ultrasmall numbers determine internal sets only, sets that behave exactly like the usual sets of set theory. A study of external sets, with unusual but sometimes useful properties is also possible. |