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The reverse mathematics of elementary recursive nonstandard analysis: a robust contribution to the foundations of mathematics

Sam Sanders (UGent)
(2010)
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(UGent) and (UGent)
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Abstract
Reverse Mathematics (RM) is a program in the Foundations of Mathematics founded by Harvey Friedman in the Seventies ([17, 18]). The aim of RM is to determine the minimal axioms required to prove a certain theorem of ‘ordinary’ mathematics. In many cases one observes that these minimal axioms are also equivalent to this theorem. This phenomenon is called the ‘Main Theme’ of RM and theorem 1.2 is a good example thereof. In practice, most theorems of everyday mathematics are equivalent to one of the four systems WKL0, ACA0, ATR0 and Π1-CA0 or provable in the base theory RCA0. An excellent introduction to RM is Stephen Simpson’s monograph [46]. Nonstandard Analysis has always played an important role in RM. ([32,52,53]). One of the open problems in the literature is the RM of theories of first-order strength I∆0 + exp ([46, p. 406]). In Chapter I, we formulate a solution to this problem in theorem 1.3. This theorem shows that many of the equivalences from theorem 1.2 remain correct when we replace equality by infinitesimal proximity ‘≈’ from Nonstandard Analysis. The base theory now is ERNA, a nonstandard extension of I∆0 + exp. The principle that corresponds to ‘Weak K ̈onig’s lemma’ is the Universal Transfer Principle (see axiom schema 1.57). In particular, one can say that the RM of ERNA+Π1-TRANS is a ‘copy up to infinitesimals’ of the RM of WKL0. This implies that RM is ‘robust’ in the sense this term is used in Statistics and Computer Science ([25,35]). Furthermore, we obtain applications of our results in Theoretical Physics in the form of the ‘Isomorphism Theorem’ (see theorem 1.106). This philosophical excursion is the first application of RM outside of Mathematics and implies that ‘whether reality is continuous or discrete is undecidable because of the way mathematics is used in Physics’ (see paragraph 3.2.4, p. 53). We briefly explore a connection with the program ‘Constructive Reverse Mathematics’ ([30,31]) and in the rest of Chapter I, we consider the RM of ACA0 and related systems. In particular, we prove theorem 1.161, which is a first step towards a ‘copy up to infinitesimals’ of the RM of ACA0. However, one major aesthetic problem with these results is the introduction of extra quantifiers in many of the theorems listed in theorem 1.3 (see e.g. theorem 1.94). To overcome this hurdle, we explore Relative Nonstandard Analysis in Chapters II and III. This new framework involves many degrees of infinity instead of the classical ‘binary’ picture where only two degrees ‘finite’ and ‘infinite’ are available. We extend ERNA to a theory of Relative Nonstandard Analysis called ERNAA and show how this theory and its extensions allow for a completely quantifier- free development of analysis. We also study the metamathematics of ERNAA, motivated by RM. Several ERNA-theorems would not have been discovered without considering ERNAA first.
Keywords
Reverse Mathematics, Nonstandard Analysis.

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MLA
Sanders, Sam. “The Reverse Mathematics of Elementary Recursive Nonstandard Analysis: a Robust Contribution to the Foundations of Mathematics.” 2010 : n. pag. Print.
APA
Sanders, S. (2010). The reverse mathematics of elementary recursive nonstandard analysis: a robust contribution to the foundations of mathematics. Ghent University. Faculty of Sciences, Ghent, Belgium.
Chicago author-date
Sanders, Sam. 2010. “The Reverse Mathematics of Elementary Recursive Nonstandard Analysis: a Robust Contribution to the Foundations of Mathematics”. Ghent, Belgium: Ghent University. Faculty of Sciences.
Chicago author-date (all authors)
Sanders, Sam. 2010. “The Reverse Mathematics of Elementary Recursive Nonstandard Analysis: a Robust Contribution to the Foundations of Mathematics”. Ghent, Belgium: Ghent University. Faculty of Sciences.
Vancouver
1.
Sanders S. The reverse mathematics of elementary recursive nonstandard analysis: a robust contribution to the foundations of mathematics. [Ghent, Belgium]: Ghent University. Faculty of Sciences; 2010.
IEEE
[1]
S. Sanders, “The reverse mathematics of elementary recursive nonstandard analysis: a robust contribution to the foundations of mathematics,” Ghent University. Faculty of Sciences, Ghent, Belgium, 2010.
@phdthesis{898469,
  abstract     = {Reverse Mathematics (RM) is a program in the Foundations of Mathematics founded by Harvey Friedman in the Seventies ([17, 18]). The aim of RM is to determine the minimal axioms required to prove a certain theorem of ‘ordinary’ mathematics. In many cases one observes that these minimal axioms are also equivalent to this theorem. This phenomenon is called the ‘Main Theme’ of RM and theorem 1.2 is a good example thereof. In practice, most theorems of everyday mathematics are equivalent to one of the four systems WKL0, ACA0, ATR0 and Π1-CA0 or provable in the base theory RCA0. An excellent introduction to RM is Stephen Simpson’s monograph [46]. Nonstandard Analysis has always played an important role in RM. ([32,52,53]).
One of the open problems in the literature is the RM of theories of first-order strength I∆0 + exp ([46, p. 406]). In Chapter I, we formulate a solution to this problem in theorem 1.3. This theorem shows that many of the equivalences from theorem 1.2 remain correct when we replace equality by infinitesimal proximity ‘≈’ from Nonstandard Analysis. The base theory now is ERNA, a nonstandard extension of I∆0 + exp. The principle that corresponds to ‘Weak K ̈onig’s lemma’ is the Universal Transfer Principle (see axiom schema 1.57). In particular, one can say that the RM of ERNA+Π1-TRANS is a ‘copy up to infinitesimals’ of the RM of WKL0. This implies that RM is ‘robust’ in the sense this term is used in Statistics and Computer Science ([25,35]).
Furthermore, we obtain applications of our results in Theoretical Physics in the form of the ‘Isomorphism Theorem’ (see theorem 1.106). This philosophical excursion is the first application of RM outside of Mathematics and implies that ‘whether reality is continuous or discrete is undecidable because of the way mathematics is used in Physics’ (see paragraph 3.2.4, p. 53). We briefly explore a connection with the program ‘Constructive Reverse Mathematics’ ([30,31]) and in the rest of Chapter I, we consider the RM of ACA0 and related systems. In particular, we prove theorem 1.161, which is a first step towards a ‘copy up to infinitesimals’ of the RM of ACA0. However, one major aesthetic problem with these results is the introduction of extra quantifiers in many of the theorems listed in theorem 1.3 (see e.g. theorem 1.94). To overcome this hurdle, we explore Relative Nonstandard Analysis in Chapters II and III. This new framework involves many degrees of infinity instead of the classical ‘binary’ picture where only two degrees ‘finite’ and ‘infinite’ are available. We extend ERNA to a theory of Relative Nonstandard Analysis called ERNAA and show how this theory and its extensions allow for a completely quantifier- free development of analysis. We also study the metamathematics of ERNAA, motivated by RM. Several ERNA-theorems would not have been discovered without considering ERNAA first.},
  author       = {Sanders, Sam},
  keywords     = {Reverse Mathematics,Nonstandard Analysis.},
  language     = {eng},
  pages        = {123},
  publisher    = {Ghent University. Faculty of Sciences},
  school       = {Ghent University},
  title        = {The reverse mathematics of elementary recursive nonstandard analysis: a robust contribution to the foundations of mathematics},
  year         = {2010},
}