The Foundations of Mathematics: Axiomatic Systems and Incredible Infinities

Often, people who study mathematics learn theorems to prove results in and about the vast array of branches of mathematics (Algebra, Analysis, Topology, Geometry, Combinatorics, etc.). This helps them move forward in their understanding; but few ever question the basis for these theorems or whether those foundations are sucient or even secure. Theorems come from our foundations of mathematics, Axioms, Logic and Set Theory. In the early20th century, mathematicians set out to formalize the methods, operations and techniques people were assuming. In other words, they were formulating axioms. The most common axiomatic system is known as the Zermelo-Fraenkel axioms with the addition of the Axiom of Choice (AC), although AC is still a controversial axiom. These axioms helped build our notion of infinity, which turns out to be much more complicated than what had been suspected. Earlier mathematicians from the Greeks to Gauss even refused to view infinity as an actuality, and only referred to “potential infinities”. Now our axioms allow multiple sizes of infinity. With the help of ordinals and cardinals, we can start to see a framework for these previously obscure notions. This leads to the even more difficult question: what exactly are the real numbers? The Continuum Hypothesis provides one approach, yet the full story remains “unsolved” by mathematicians to this day

Derived rules for predicative set theory: an application of sheaves

We show how one may establish proof-theoretic results for constructive Zermelo-Fraenkel set theory, such as the compactness rule for Cantor space and the Bar Induction rule for Baire space, by constructing sheaf models and using their preservation properties

Mathematical Infinity, Its Inventors, Discoverers, Detractors, Defenders, Masters, Victims, Users, and Spectators

"The definitive clarification of the nature of the infinite has become necessary, not merely for the special interests of the individual sciences, but rather for the honour of the human understanding itself. The infinite has always stirred the emotions of mankind more deeply than any other question; the infinite has stimulated and fertilized reason as few other ideas have ; but also the infinite, more than other notion, is in need of clarification." (David Hilbert 1925

Study of logical paradoxes

By a paradox we understand a seemingly true statement or set of statements which lead by valid deduction to contradictory statements. Logical paradoxes - paradoxes which involve logical concepts - are in fact as old as the history of logic. The Liar paradox, for instance, goes back to Epimenides (6th century B.C.?). In the late 19th century a new impetus v/as given to the investigation of logical paradoxes by the discovery of new logico-mathematical paradoxes such as those of Russell and Burali- Porti. This came about in the course of attempts to give mathematics a rigorous axiomatic foundation. Sometimes a distinction is maintained between a paradox and an antinomy. In a paradox, it is said, semantical notions are involved and a certain "oddity", "strangeness", or what may be called "paradoxical situation", resides in its construction. The resolution of a paradox is therefore not simply a matter of removing contradiction, but also requires clarifying and removing the "oddity". On the other hand, an antinomy is said to consist in the derivation of a contradiction in an axiomatic system and its resolution lies in revising the system so as to avoid the contradiction. In discussing paradoxes and antinomies, we shall not be strictly bound by this usage of these terms: we use "paradox" and "antinomy" interchangeably. Indeed, from our point of view, even antinomies in an axiomatic system ultimately need semantic clarification and thus removal of paradoxical situations

Preliminary investigations on induction over real numbers

The induction principle for natural numbers expresses that when a property holds for some natural number a and is hereditary, then it holds for all numbers greater than or equal to a. We present a similar principle for real numbers

Why the classes P and NP are not well-defined finitarily

We distinguish finitarily between algorithmic verifiability, and algorithmic computability, to show that Goedel's 'formally' unprovable, but 'numeral-wise' provable, arithmetical proposition [(Ax)R(x)] can be finitarily evidenced as: algorithmically verifiable as 'always' true, but not algorithmically computable as 'always' true. Hence, though [R(x)] is algorithmically verifiable as a tautology, it is not algorithmically computable as a tautology by any Turing machine, whether deterministic or non-deterministic. By interpreting the PvNP problem arithmetically, rather than set-theoretically, we conclude that the clkasses P and NP are not well-defined finitarily since it immediately follows that SAT is neither in P nor in NP.Comment: 28 page