Transtheoretic Foundations of Mathematics Arithmetics Book

From the Author:

While the main target of volumes IA, IB & IC is the Goldbach Conjecture, the volumes also involve earlier three volumes, entitled Foundations of Semiological Number Theory. The target of the latter three volumes was to fill serious semiological gaps in the famous heavily topological Elements of N. Bourbaki in order to extend this development of mathematics to include computability theory and nonalgebraic natural number theory. There is no doubt that Bourbaki's work is euclidean in scope in the sense of being a foundational depiction of the mathematics of the first half if not the whole 20th century just as Euclid's Elements was the foundational depiction of the mathematics of his century in Ancient Greece.

The above-mentioned Bourbaki semiological extension is assumed as the first of three crucial parts of the transtheoretic foundations of mathematics developed in volumes IA, IB & IC. However, it should be noted, unlike the traditional conceptually starved set-theoretic foundations of mathematics, the transtheoretic development of mathematics, or transtheory, introduced in IA, IB & IC depicts a distinctly different conceptual foundation involving a whole range of foundational concepts which is generally not explicated in the traditional foundations of mathematics, but nonetheless implicitly used. The theory involving such foundational concepts forms the second part of transtheory, while transtheory forms the last part. The involved foundational concepts are aligned with well-defined objects in the extended Bourbaki development of mathematics, depicting transtheory also as a kind of theory of meaning. The famous Church's Thesis of theoretical computer science is an example of such an alignment, where the vague intuitive concept of computability is aligned with the well-defined notion of general recursion in the extended Bourbaki development within transtheory of mathematics. Transtheory is not to be confused with Hilbert's metamathematics as many are tempted to do.

The preliminaries for transtheory are introduced in volume IA. In turn, volume IB introduces a number of abstract arithmetics that lead to certain diophantine models, in particular, general recursive equational arithmetics, where all its well-formed formulas are equations of the form F=G, where F and G are general recursive functions. Volume IC, on the one hand, shows how certain sets of well-formed formulas of equational arithmetic give rise to an assertion of the consistency problem about arithmetic expressed within the equational arithmetic itself. On the other hand, volume IC shows that certain sets of sums of two number-theoretic partitions of odd prime numbers give rise to an assertion of the Goldbach problem within equational arithmetic. When all this is planted inside transtheory, it is given in IC that both the preceding problems are models of each other, and in turn it is shown that the Goldbach Conjecture is unprovable in equational arithmetic, which of course implies its truth.