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Theory

The Axiomatic Method has often been referred to as the most fundamental and important mathematical tool of our time. When Hilbert proposed his famous list of 23 Problems in Paris during the International Congress of Mathematicians of 1900, he paid special attention to axiomatic problems. Such are the cases of problems 1,2, and 6. There is a more metamathematical question that Hilbert decided not to include: a 24-th problem on simplicity criteria for proofs and being able to prove that the simplest proof is indeed the simplest proof.

Benacerraf's Identification Problem is related to Hilbert's 24-th Problem. It puts into question the validity of defining natural numbers as sets because there is more than one way of defining them. It has not been clear if there exists a set theory that supersedes all others. So, Benacerraf argues that natural numbers are not sets. We have defined sets to obey the rules of natural numbers we want them to. But that does not make a number a set. The number 2 is not the set {{0}} that Zermelo-Fraenkel say it is. The number 2 is also not the set {0,{0}} that Von Neumann says it is. Then the question is still open: What is a Natural Number? Is a number a set or not? And if so, what set is the number 2?

The general, long-standing consensus among mathematicians has been that the nature of a number is not important to describe, and that the only thing that is of interest is their behavior. In other words, the only focus is on the external properties of numbers, and the internal properties are ignored. The logical and algebraic relations of numbers and their structures are described in a novel and convenient way by means of a full description of mathematical objects given in terms of a proposed Canonical Set Theory.

The proposed set theory is transparent, clear, and simple. Apart from having numerous immediate applications it also exhibits advantages in describing other mathematical areas. The set theoretic description of mathematics here proposed uncovers connections between set theory, number theory, groups, linear spaces and analysis, among others. 

Other fundamental questions of mathematics will be addressed. Such as the question of whether or not there exists an intrinsic relation between the nature of numbers and the real world. The consensus on this question is that there is nothing about the nature of mathematical objects that will tell us anything about the laws of physics. Numbers are used to study physical phenomena, but they do not have any direct relation to these phenomena. The argument is the same as in the case of sets and numbers. Now, numbers are defined to study physical phenomena. But, that does not imply that particles or any other physical form are mathematical objects. Here, numerical operations are defined in such a way that they also model basic behaviors of waves and particles.

Course SetTheory and Finite Strutures

REAL NUMBERS AND THE CONTINUUM

No description of the natural numbers is complete if it cannot give a satisfactory description of the real number system and infinity.

OBJECT TYPES

Other types of objects are well defined and easy to work with. These object types include real functions, collections of functions, sequences of functions, functions of functions, matrices, etc.

SYSTEMS AND CATEGORIES

The world exhibits structural uniformity in its different scales, uniformities in the different levels of reality. Properties are maintained at all scales of nature. There are relations between different processes of different nature, that follow certain fixed patterns. Throughout space, these relations with numerical interpretation, resound and shape the world as we know it.

Objects and relations are viewed as complementary concepts in the understanding of General Theory of Systems, and in particular mathematics.

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