Topic
Elementary function arithmetic
About: Elementary function arithmetic is a research topic. Over the lifetime, 4 publications have been published within this topic receiving 67 citations.
Papers
Posted Content•
TL;DR: It is shown that a number of axioms from NonStandard Analysis are equivalent to theorems of ordinary mathematics not involving Nonstandard Analysis, proved in a weak base theory recently introduced by van den Berg and the author.
Abstract: Reverse Mathematics (RM) is a program in the foundations of mathematics founded by Friedman and developed extensively by Simpson. The aim of RM is finding the minimal axioms needed to prove a theorem of ordinary (i.e. non-set theoretical) mathematics. In the majority of cases, one also obtains an equivalence between the theorem and its minimal axioms. This equivalence is established in a weak logical system called the base theory; four prominent axioms which boast lots of such equivalences are dubbed mathematically natural by Simpson. In this paper, we show that a number of axioms from Nonstandard Analysis are equivalent to theorems of ordinary mathematics not involving Nonstandard Analysis. These equivalences are proved in a weak base theory recently introduced by van den Berg and the author. In particular, our base theories have the first-order strength of elementary function arithmetic, in contrast to the original version of this paper [22]. Our results combined with Simpson's criterion for naturalness suggest the controversial point that Nonstandard Analysis is actually mathematically natural.
8 citations
Patent•
9 Sep 1987TL;DR: In this article, a function arithmetic unit which performs elementary operations at high speeds and which enables required memory capacity to be reduced is presented. But it does not specify the number of elementary functions to be found.
Abstract: A function arithmetic unit which performs elementary operations at high speeds and which enables required memory capacity to be reduced. The function arithmetic unit includes a constant memory which stores a constant that corresponds to the number of successive iterative operations; a controller which causes addition or subtraction in the successive iterative operations; and arithmetic units which receive, as initial values, an argument of an elementary function value to be found and two constants determined depending upon the elementary functions to be found, and which subject these initial values to n iterative operations to produce elementary function values. A further arithmetic unit subjects the elementary function values to multiplication, addition, subtraction, or division, or a combination thereof, thereby to produce the thus obtained value.
8 citations
1 Jan 2017
1 citations
TL;DR: The authors showed that many theorems of number theory and combinatorics are derivable in elementary function arithmetic, and tried to place these results in a broader philosophical context, and showed that the theory turns out to be remarkably robust.
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.
Related Topics (5)
Performance Metrics
| Year | Papers |
|---|---|
| 2018 | 1 |
| 2017 | 1 |
| 2003 | 1 |
| 1987 | 1 |