True arithmetic

Abstract: A system ALPO of set theory is presented and proved to form a conservative extension of Peano arithmetic. The system has sufficient strength to allow a very substantial amount of analysis to be directly formalized, including Lebesgue integration theory, without resorting to wholesale padding of objects with extra information.

Abstract: We introduce a natural set of arithmetic expressions and define the complexity class AE to consist of all those arithmetic functions (over the fieldsF 2n) that are described by these expressions. We show that AE coincides with the class of functions that are computable with constant depth and polynomial-size unbounded fan-in arithmetic circuits satisfying a natural uniformity constraint (DLOGTIME-uniformity). A 1-input and 1-output arithmetic function over the fieldsF2n may be identified with ann-input andn-output Boolean function when field elements are represented as bit strings. We prove that if some such representation is X-uniform (where X is P or DLOGTIME), then the arithmetic complexity of a function (measured with X-uniform unbounded fan-in arithmetic circuits) is identical to the Boolean complexity of this function (measured with X-uniform threshold circuits). We show the existence of a P-uniform representation and we give partial results concerning the existence of representations with more restrictive uniformity properties.

Abstract: In this paper two different formulations of Robinson's arithmetic based on relevant logic are examined. The formulation based on the natural numbers (including zero) is shown to collapse into classical Robinson's arithmetic, whereas the one based on the positive integers (excluding zero) is shown not to similarly collapse. Relations of these two formulations to R. K. Meyer's system R# of relevant Peano arithmetic are examined, and some remarks are made about the role of constant functions (e.g., multiplication by zero) in relevant arithmetic.

Abstract: There is an analogy between concepts such as end-extension types and minimal types in the model theory of Peano arithmetic and concepts such as P-points and selective ultrafilters in the theory of ultrafilters on N. Using the notion of conservative extensions of models, we prove some theorems clarifying the relation between these pairs of analogous concepts. We also use the analogy to obtain some modeltheoretic results with techniques originally used in ultrafilter theory. These results assert that every countable nonstandard model of arithmetic has a bounded minimal extension and that some types in arithmetic are not 2-isolated.. Introduction. In this paper we shall analyze (in ?2) and exploit (in ??3, 4) a certain analogy between some concepts in the theory of models of Peano arithmetic and some concepts in the theory of ultrafilters on the set N of natural numbers. As might be expected, we shall make considerable use of the fact that an ultrafilter on N may be viewed as the type realized by an element in a suitable nonstandard model of arithmetic, provided that our "arithmetic" contains symbols for all relations on N. (But we shall not use the related fact that a type in Peano arithmetic may be viewed as an ultrafilter in a suitable Lindenbaum algebra.) The difference between the uncountable language with symbols for all relations and the countable language of Peano arithmetic affects the spirit of proofs in the two theories. Constructions of ultrafilters with interesting properties are usually transfinite inductions of length the power of the continuum. Difficulties often occur at limit stages of the induction, and many theorems depend on the continuum hypothesis (or some related assumptions, like Martin's axiom) to make the limit stages tractable. On the other hand, constructions of models or types in Peano arithmetic are usually ordinary inductions (of length co), so there are no limit stages to worry about, but the successor stages require more work than in ultrafilter theory because one has to know that sets with certain properties not only exist but are definable in the language under consideration (and provably have the properties one wants). These features of typical proofs in the two theories can be clearly seen in Rudin's proof [12] that the continuum hypothesis implies the existence of Ppoints and MacDowell and Specker's proof [9] that every model of Peano arithmetic has an elementary end-extension. If one ignores the differences indicated above between these two proofs, one finds that they both depend on the same simple fact: If a function f has an infinite domain, then f is either constant or one-to-one on some infinite subset of the domain. The connection between the two existence proofs may be taken as evidence of some connection between P-points and end-extensions (or, better, end-extension Received July 2, 1973. (D 1974, Association for Symbolic Logic