The current foundations for mathematics rest on first order set theories using classical logic. This presents two avenues to question this foundation.

Classical logic has been known since antiquity to have issues with self-referential statements. The Liar’s paradox is a prototypical example. Foundational definitions are fundamentally self-referential, thus classical logic appears too rigid to be applied to foundational questions.

Fortunately, alternatives exist. Intuitionist logic is a viable competitor, having already established an alternate representation for the basis of our most common applications of natural and real numbers. Further, a minimal logic exists, containing only the axioms used to embody our intuitive understanding of the meanings of commonly used connectives. Both of these would be resistant to the paradoxes found in naive set theory using proofs by contradiction within classical logic.

First order logic is insufficient to uniquely define the natural numbers, so why should we expect it to be sufficient to uniquely define any other infinite object? More fundamentally, classical first order logic can not uniquely define its own underpinnings. The standard solution is to introduce successive higher order logics, each providing the tools needed to establish the validity of all prior logics.

Ideally, the supremum of such a hierarchy could be used as a basis for the foundational issues. In classical logic this leads to contradictions. It can be hoped that a weaker logic system can avoid these issues.

Thus what is desired is a set theory of unbounded logical order using a weaker logic than classical logic.

This rejection of classical logic then forces me to pragmatically retreat to formalism when dealing with foundational mathematical constructs. Mathematics is then reduced to a symbolic manipulation game with some type of rules. The question becomes which set of rules are most useful.

Mathematical structures, then, are just what we define them to be. Thus the fundamental units of study are the definitions that we chose to adopt. Adding axioms then either clarify the objects being defined or restrict our attention to those structures where the additional axiom holds.

Likewise, the rules are just what we define them to be. The rules defining the basic logical connectives then form a minimal logic, upon which we can restrict to intuitionistic and classical logic by adding additional axioms.

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