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Nobuyuki Sakamoto, Takeshi Yamazaki (2004)
Uniform versions of some axioms of second order arithmeticMathematical Logic Quarterly, 50
Xiaokang Yu, S. Simpson (1990)
Measure theory and weak König's lemmaArchive for Mathematical Logic, 30
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The Principles of Quantum Mechanics, 1st edn
Emil Jeřábek (2010)
SATURATION AND Σ2-TRANSFER FOR ERNA
Sam Sanders (2011)
ERNA and Friedman's Reverse MathematicsThe Journal of Symbolic Logic, 76
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Some systems of second order arithmetic and their use
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P. Suppes (1993)
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P.A.M. Dirac (1927)
The Principles of Quantum Mechanics
C. Impens, Sam Sanders (2008)
Transfer and a supremum principle for ERNAJournal of Symbolic Logic, 73
L. Schwartz (1966)
Théorie des distributions
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THEORIE RELATIVE DES ENSEMBLES INTERNESOsaka Journal of Mathematics, 29
Sam Sanders (2010)
More infinity for a better finitismAnn. Pure Appl. Log., 161
K. Yokoyama (2009)
Standard and Non-standard analysis in second order arithmeticTohoku Mathematical Publications, 34
H. Friedman, S. Simpson, Rick Smith (1983)
Countable algebra and set existence axiomsAnn. Pure Appl. Log., 25
R. Chuaqui, P. Suppes (1995)
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Douglas Brown, Mariagnese Giusto, S. Simpson (2002)
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S. Simpson (1984)
Which set existence axioms are needed to prove the Cauchy/Peano theorem for ordinary differential equations?Journal of Symbolic Logic, 49
Richard, SommerStanford (1996)
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ERNA at Work
The program of Reverse Mathematics (Simpson 2009) 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 (Dirac 1927; Schwartz 1951) in two settings of Reverse Mathematics. In particular, we consider the Dirac Delta Theorem, which formalizes the well-known property $${\int_\mathbb{R}f(x)\delta(x)\,dx=f(0)}$$ of the Dirac delta function. We show that the Dirac Delta Theorem is equivalent to weak König’s Lemma (see Yu and Simpson in Arch Math Log 30(3):171–180, 1990) in classical Reverse Mathematics. This further validates the status of WWKL0 as one of the ‘Big’ systems of Reverse Mathematics. In the context of ERNA’s Reverse Mathematics (Sanders in J Symb Log 76(2):637–664, 2011), we show that the Dirac Delta Theorem is equivalent to the Universal Transfer Principle. Since the Universal Transfer Principle corresponds to WKL, it seems that, in ERNA’s Reverse Mathematics, the principles corresponding to WKL and WWKL coincide. Hence, ERNA’s Reverse Mathematics is actually coarser than classical Reverse Mathematics, although the base theory has lower first-order strength.
Archive for Mathematical Logic – Springer Journals
Published: Nov 23, 2011
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