THE HITCHHIKER`S GUIDE TO THE INCOMPLETENESS
... foundation logic? Now let’s first have a look at the first (informal) version of the incompleteness theorem: Incompleteness Theorem, First Version: There is a true sentence that mathematics cannot prove. All right, this is even worse: is it a theorem? It feels more like an assertion in philosophy ra ...
... foundation logic? Now let’s first have a look at the first (informal) version of the incompleteness theorem: Incompleteness Theorem, First Version: There is a true sentence that mathematics cannot prove. All right, this is even worse: is it a theorem? It feels more like an assertion in philosophy ra ...
Section 1.1 Introduction to Sets Set: A set is a collection of objects
... A set is finite if it is possible, given enough time, to write down every element in the set. A set is infinite if it is not finite. Example: Consider the set A = {1, 2, 3, …, 20} Is A finite or infinite? (Note: The three periods in this definition are called ellipses and mean that you should contin ...
... A set is finite if it is possible, given enough time, to write down every element in the set. A set is infinite if it is not finite. Example: Consider the set A = {1, 2, 3, …, 20} Is A finite or infinite? (Note: The three periods in this definition are called ellipses and mean that you should contin ...
Normal numbers and the Borel hierarchy
... tuple hx+1, 1i or an appending tuple hx0 , yi with x0 < x+1 and x0 +y > x+1. In the first case, the appending tuple hx+1, 1i is processed after hx, 1i, which appends x to the output. In the second case, the processing of an appending tuple hx0 , yi with x0 < x + 1 and x0 + y > x + 1 appends x right ...
... tuple hx+1, 1i or an appending tuple hx0 , yi with x0 < x+1 and x0 +y > x+1. In the first case, the appending tuple hx+1, 1i is processed after hx, 1i, which appends x to the output. In the second case, the processing of an appending tuple hx0 , yi with x0 < x + 1 and x0 + y > x + 1 appends x right ...
![arXiv:math/0407326v1 [math.CO] 19 Jul 2004](http://s1.studyres.com/store/data/016678102_1-7a26a9b4445f7030b68419a5be89f8d8-300x300.png)
Non-standard analysis
The history of calculus is fraught with philosophical debates about the meaning and logical validity of fluxions or infinitesimal numbers. The standard way to resolve these debates is to define the operations of calculus using epsilon–delta procedures rather than infinitesimals. Non-standard analysis instead reformulates the calculus using a logically rigorous notion of infinitesimal numbers.Non-standard analysis was originated in the early 1960s by the mathematician Abraham Robinson. He wrote:[...] the idea of infinitely small or infinitesimal quantities seems to appeal naturally to our intuition. At any rate, the use of infinitesimals was widespread during the formative stages of the Differential and Integral Calculus. As for the objection [...] that the distance between two distinct real numbers cannot be infinitely small, Gottfried Wilhelm Leibniz argued that the theory of infinitesimals implies the introduction of ideal numbers which might be infinitely small or infinitely large compared with the real numbers but which were to possess the same properties as the latterRobinson argued that this law of continuity of Leibniz's is a precursor of the transfer principle. Robinson continued:However, neither he nor his disciples and successors were able to give a rational development leading up to a system of this sort. As a result, the theory of infinitesimals gradually fell into disrepute and was replaced eventually by the classical theory of limits.Robinson continues:It is shown in this book that Leibniz's ideas can be fully vindicated and that they lead to a novel and fruitful approach to classical Analysis and to many other branches of mathematics. The key to our method is provided by the detailed analysis of the relation between mathematical languages and mathematical structures which lies at the bottom of contemporary model theory.In 1973, intuitionist Arend Heyting praised non-standard analysis as ""a standard model of important mathematical research"".