arXiv subjects

Harvey M. Friedman

Publications and source records attributed to Harvey M. Friedman.

4 recordsLinked to original sources

arXiv2025

When Bi-interpretability implies Synonymy

Two salient notions of sameness of theories are synonymy, also known as definitional equivalence, and bi-interpretability. Of these two definitional equivalence is the strictest notion. In which cases can we infer synonymy from bi-interpretability? We study this question for the case of sequential theories. Our result is as follows. Suppose that two sequential theories are bi-interpretable and that the interpretations involved in the bi-interpretation are one-dimensional and identity preserving. Then, the theories are synonymous. The crucial ingredient of our proof is a version of the Schröder-Bernstein theorem under very weak conditions. We think this last result has some independent interest. We provide an example to show that this result is optimal. There are two finitely axiomatized sequential theories that are bi-interpretable but not synonymous, where precisely one of the interpretations involved in the bi-interpretation is not identity preserving.

math.LO↗

arXiv2015

Integration on the Surreals: a Conjecture of Conway, Kruskal and Norton

In his monograph On Numbers and Games, J. H. Conway introduced a real-closed field No of surreal numbers containing the reals and the ordinals, as well as a vast array of less familiar numbers. A longstanding aim has been to develop analysis on No as a powerful extension of ordinary analysis on the reals. This entails finding a natural way of extending important functions f from the reals to the reals to functions f* from the surreals to the surreals, and naturally defining integration on the f*. The usual square root, log, and exp were naturally extended to No by Bach, Conway, Kruskal, and Norton, retaining their usual properties. Later Norton also proposed a treatment of integration, but Kruskal discovered it has flaws. In his recent survey [2, p. 438], Siegel characterizes the question of the existence of a reasonable definition of surreal integration as "perhaps the most important open problem in the theory of surreal numbers." This paper addresses this and related unresolved issues with positive and negative results. In the positive direction, we show that semi-algebraic, semi-analytic, analytic, meromorphic, or more generally Écalle-Borel transseriable functions extend naturally to No, and an integral with good properties exists on them. In the negative direction, we show there is a fundamental set-theoretic obstruction to naturally extending many larger families of functions.

math.LO↗

arXiv2014

Foundational aspects of singular integrals

We investigate integration of classes of real-valued continuous functions on (0,1]. Of course difficulties arise if there is a non-$L^1$ element in the class, and the Hadamard finite part integral ({\em p.f.}) does not apply. Such singular integrals arise naturally in many contexts including PDEs and singular ODEs. The Lebesgue integral as well as $p.f.$, starting at zero, obey two fundamental conditions: (i) they act as antiderivatives and, (ii) if $f =g$ on $(0,a)$, then their integrals from $0$ to $x$ coincide for any $x\in (0,a)$. We find that integrals from zero with the essential properties of $p.f.$, plus positivity, exist by virtue of the Axiom of Choice (AC) on all functions on $(0,1]$ which are $L^1((ε,1])$ for all $ε>0$. However, this existence proof does not provide a satisfactory construction. Without some regularity at $0$, the existence of general antiderivatives which satisfy only (i) and (ii) above on classes with a non-$L^1$ element is independent of ZF (the usual ZFC axioms for mathematics without AC), and even of ZFDC (ZF with the Axiom of Dependent Choice). Moreover we show that there is no mathematical description that can be proved (within ZFC or even extensions of ZFC with large cardinal hypotheses) to uniquely define such an antiderivative operator. Such results are precisely formulated for a variety of sets of functions, and proved using methods from mathematical logic, descriptive set theory and analysis. We also analyze $p.f.$ on analytic functions in the punctured unit disk, and make the connection to singular initial value problems.

math.FA↗

arXiv1998

Finite functions and the necessary use of large cardinals

We present a coherent collection of finite mathematical theorems some of which can only be proved by going well beyond the usual axioms for mathematics. The proofs of these theorems illustrate in clear terms how one uses the well studied higher infinities of abstract set theory called large cardinals in an essential way in order to derive results in the context of the natural numbers. The findings raise the specific issue of what consitutes a valid mathematical proof and the general issue of objectivity in mathematics in a down to earth way. Large cardinal axioms, which go beyond the usual axioms for mathematics, have been commonly used in abstract set theory since the 1960's. We believe that the results reported on here are the early stages of an evolutionary process in which new axioms for mathematics will be commonly used in an essential way in the more concrete parts of mathematics.

math.LO↗