Formalizing non-standard arguments in second-order arithmetic

Yokoyama, Keita

doi:10.2178/jsl/1286198143

Cited by 19 publications

(18 citation statements)

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“…Moreover, several principles of measure theory and Fourier analysis are equivalent to WWKL (see e.g. [1,18,22]). Thus, it seems plausible to extend the 'Big Five' with WWKL 0 (which is RCA 0 plus WWKL) to the 'Big Six'.…”

Section: Reverse Mathematics Is a Program In Foundations Of Mathematimentioning

confidence: 99%

“…For the rest of this section, we adopt the usual definition for continuous functions on R in second-order arithmetic (see [18,Chapter II]). The notions of modulus of uniform continuity and derivative are also defined as usual in second-order arithmetic (see also [22]). We first define C 1 , C r and C ∞ -functions in R.…”

Section: The Dirac Delta In Classical Reverse Mathematicsmentioning

confidence: 99%

“…On the other hand, it can be shown, with substantial effort, that RCA 0 proves that the existence of the Riemann integral implies effective integrability (see [22]). …”

Section: The Dirac Delta In Classical Reverse Mathematicsmentioning

confidence: 99%

“…By [22], we know that Fourier analysis can be developed in RCA 0 or WWKL 0 , depending on the underlying class of functions. As the Dirac delta function is an important tool in Fourier analysis, we can expect the properties of this function to be provable in WWKL 0 .…”

Section: 2mentioning

confidence: 99%

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The Dirac delta function in two settings of Reverse Mathematics

Sanders

Yokoyama

2011

Arch. Math. Logic

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Abstract. The program of Reverse Mathematics ([18]) has provided us with the insight that most theorems of ordinary mathematics are either equivalent to one of a select few logical principles, or provable in a weak base theory. In this paper, we study the properties of the Dirac delta function ([3, 15]) in two settings of Reverse Mathematics. In particular, we consider the Dirac Delta Theorem, which formalizes the well-known property

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Section: Reverse Mathematics Is a Program In Foundations Of Mathematimentioning

confidence: 99%

Section: The Dirac Delta In Classical Reverse Mathematicsmentioning

confidence: 99%

“…On the other hand, it can be shown, with substantial effort, that RCA 0 proves that the existence of the Riemann integral implies effective integrability (see [22]). …”

Section: The Dirac Delta In Classical Reverse Mathematicsmentioning

confidence: 99%

Section: 2mentioning

confidence: 99%

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The Dirac delta function in two settings of Reverse Mathematics

Sanders

Yokoyama

2011

Arch. Math. Logic

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“…Yokoyama [6,7] introduced a theory that is stronger than * RCA 0 but, like * RCA 0 , is conservative with respect to RCA 0 and has a very weak form of the Downward STP.…”

Section: Definition 22mentioning

confidence: 99%

Nonstandard arithmetic and recursive comprehension

Keisler

2010

Annals of Pure and Applied Logic

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a b s t r a c tFirst order reasoning about hyperintegers can prove things about sets of integers. In the author's paper Nonstandard Arithmetic and Reverse Mathematics, Bulletin of Symbolic Logic 12 (2006) 100-125, it was shown that each of the ''big five'' theories in reverse mathematics, including the base theory RCA 0 , has a natural nonstandard counterpart. But the counterpart * RCA 0 of RCA 0 has a defect: it does not imply the Standard Part Principle that a set exists if and only if it is coded by a hyperinteger. In this paper we find another nonstandard counterpart, * RCA 0 , that does imply the Standard Part Principle.

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Some Nonstandard Equivalences in Reverse Mathematics

Sanders

2018

Sailing Routes in the World of Computation

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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.From the above it is clear that the [Big Five] five basic systems RCA 0 , WKL 0 , ACA 0 , ATR 0 , Π 1 1 -CA 0 arise naturally from investigations of the Main Question. The proof that these systems are mathematically natural is provided by Reverse Mathematics. ⋆

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Formalizing non-standard arguments in second-order arithmetic

Cited by 19 publications

References 10 publications

The Dirac delta function in two settings of Reverse Mathematics

The Dirac delta function in two settings of Reverse Mathematics

Nonstandard arithmetic and recursive comprehension

Some Nonstandard Equivalences in Reverse Mathematics

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