Hilary Putnam on Logic and Mathematics pp 179-234 | Cite as

# Concrete Mathematical Incompleteness: Basic Emulation Theory

## Abstract

By the modern form of Gödel’s First Incompleteness Theorem, we know that there are sentences (in the language of ZFC) that are neither provable nor refutable from the usual ZFC axioms for mathematics (assuming, as is generally believed, that ZFC is free of contradiction). Yet it is clear that the usual examples are radically different from normal mathematical statements in several glaring ways such as the mathematically remote subject matter and the essential involvement of uncharacteristically intangible objects. Starting in 1967, we embarked on the Concrete Mathematical Incompleteness program with the principal aim of developing readily accessible thematic mathematical research areas with familiar mathematical subject matter replete with examples of such incompleteness involving only characteristically tangible objects. The many examples developed over the years represent Concrete Mathematical Incompleteness ranging from weak fragments of finite set theory through ZFC and beyond. The program has reached a mature stage with the development of Emulation Theory. Emulation Theory, in its present basic developed form, involves finite length tuples of rational numbers. Only the usual ordering of rationals is used, and there is no use of even addition or multiplication. The basics are fully accessible to early undergraduate mathematics majors and gifted high school mathematics students, who will be able to engage with some simple nontrivial examples in two and three dimensions, with illustrations. In this paper, we develop the positive side of the theory, using various levels of set theory for systematic development. Some of these levels lie beyond ZFC and include familiar large cardinal hypotheses. The necessity of the various levels of set theory will be established in a forthcoming book (Concrete Mathematical Incompleteness in preparation).

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