Nonstandard arithmetic and recursive comprehension

Keisler, H. Jerome

doi:10.1016/j.apal.2010.01.001

Cited by 3 publications

(1 citation statement)

References 6 publications

Supporting

Mentioning

Contrasting

Order By: Relevance

“…In this section, we introduce the base theory NSA in which we shall work. The system NSA is inspired by the theories ns-WKL 0 , * RCA 0 , * RCA 0 , and * WKL 0 , introduced by Keita Yokoyama and Jerry Keisler [28,35,34,33,16,17] to apply techniques of Nonstandard Analysis in Reverse Mathematics. Like Yokoyama's system ns-WKL 0 , we consider more than two sorts, corresponding to the standard number variables, the nonstandard number variables, the standard set variables, and the nonstandard set variables.…”

Section: The Base System Nsamentioning

confidence: 99%

Algorithm and proof as Ω-invariance and transfer: A new model of computation in nonstandard analysis

Sanders¹

2014

Electron. Proc. Theor. Comput. Sci.

View full text Add to dashboard Cite

We propose a new model of computation based on Nonstandard Analysis. Intuitively, the role of 'algorithm' is played by a new notion of finite procedure, called Ω-invariance and inspired by physics, from Nonstandard Analysis. Moreover, the role of 'proof' is taken up by the Transfer Principle from Nonstandard Analysis. We obtain a number of results in Constructive Reverse Mathematics to illustrate the tight correspondence to Errett Bishop's Constructive Analysis and the associated Constructive Reverse Mathematics.

show abstract

Section: The Base System Nsamentioning

confidence: 99%

Algorithm and proof as Ω-invariance and transfer: A new model of computation in nonstandard analysis

Sanders¹

2014

Electron. Proc. Theor. Comput. Sci.

View full text Add to dashboard Cite

show abstract

Nonstandard second-order arithmetic and Riemannʼs mapping theorem

Horihata

Yokoyama

2014

Annals of Pure and Applied Logic

View full text Add to dashboard Cite

Computability Theory, Nonstandard Analysis, and Their Connections

Normann¹,

Sanders²

2019

J. symb. log.

View full text Add to dashboard Cite

We investigate the connections between computability theory and Nonstandard Analysis. In particular, we investigate the two following topics and show that they are intimately related.(T.1) A basic property of Cantor space$2^ $ is Heine–Borel compactness: for any open covering of $2^ $, there is a finite subcovering. A natural question is: How hard is it to compute such a finite subcovering? We make this precise by analysing the complexity of so-called fan functionals that given any $G:2^ \to $, output a finite sequence $\langle f_0 , \ldots ,f_n \rangle $ in $2^ $ such that the neighbourhoods defined from $\overline {f_i } G\left( {f_i } \right)$ for $i \le n$ form a covering of $2^ $.(T.2) A basic property of Cantor space in Nonstandard Analysis is Abraham Robinson’s nonstandard compactness, i.e., that every binary sequence is “infinitely close” to a standard binary sequence. We analyse the strength of this nonstandard compactness property of Cantor space, compared to the other axioms of Nonstandard Analysis and usual mathematics.Our study of (T.1) yields exotic objects in computability theory, while (T.2) leads to surprising results in Reverse Mathematics. We stress that (T.1) and (T.2) are highly intertwined, i.e., our study is holistic in nature in that results in computability theory yield results in Nonstandard Analysis and vice versa.

show abstract

Nonstandard arithmetic and recursive comprehension

Cited by 3 publications

References 6 publications

Algorithm and proof as Ω-invariance and transfer: A new model of computation in nonstandard analysis

Algorithm and proof as Ω-invariance and transfer: A new model of computation in nonstandard analysis

Nonstandard second-order arithmetic and Riemannʼs mapping theorem

Computability Theory, Nonstandard Analysis, and Their Connections

Contact Info

Product

Resources

About