Abstract
Elementary arithmetic (also known as “elementary function arithmetic”) is a fragment of first-order arithmetic so weak that it cannot prove the totality of an iterated exponential function. Surprisingly, however, the theory turns out to be remarkably robust. I will discuss formal results that show that many theorems of number theory and combinatorics are derivable in elementary arithmetic, and try to place these results in a broader philosophical context.
Highlights
The relationship between mathematical logic and the philosophy of mathematics has long been a rocky one
The precision of a formal logical analysis represents the philosophical ideal, the paradigm of clarity and rigor; for others, it is just the point at which philosophy becomes uninteresting and sterile. Both formal and more broadly philosophical approaches can yield insight, and each is enriched by a continuing interaction: philosophical reflection can inspire mathematical questions and research programs, which, in turn, inform and illuminate philosophical discussion
I will start by describing an informal attitude that is commonly held by metamathematical proof theorists, and survey some of the formal results that support this point of view
Summary
The relationship between mathematical logic and the philosophy of mathematics has long been a rocky one. For example, the question as to whether the mean-value theorem of undergraduate calculus can be derived in a certain restricted theory may well depend on one’s definition of the real numbers, or of a continuous function When it comes to finitary objects like numbers, finite sets, and finite sequences, issues of representation seem less problematic, because it usually turns out that various natural definitions are shown to be equivalent, or at least have equivalent properties, on the basis of minimal systems of axioms. One should not view the present work as a survey of any particular research program but, rather, as an exploration of a tacit Weltanschauung that guides contemporary research in proof theory
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