169 research outputs found
The Dirac delta function in two settings of Reverse Mathematics
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 integral(R) f(x)delta(x)dx = f (0) of the Dirac delta function. We show that the Dirac Delta Theorem is equivalent to weak Konig'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
The proof-theoretic strength of Ramsey's theorem for pairs and two colors
Ramsey's theorem for -tuples and -colors () asserts
that every k-coloring of admits an infinite monochromatic
subset. We study the proof-theoretic strength of Ramsey's theorem for pairs and
two colors, namely, the set of its consequences, and show that
is conservative over . This
strengthens the proof of Chong, Slaman and Yang that does not
imply , and shows that is
finitistically reducible, in the sense of Simpson's partial realization of
Hilbert's Program. Moreover, we develop general tools to simplify the proofs of
-conservation theorems.Comment: 32 page
Categorical characterizations of the natural numbers require primitive recursion
Simpson and the second author asked whether there exists a characterization
of the natural numbers by a second-order sentence which is provably categorical
in the theory RCA. We answer in the negative, showing that for any
characterization of the natural numbers which is provably true in WKL,
the categoricity theorem implies induction. On the other hand, we
show that RCA does make it possible to characterize the natural numbers
categorically by means of a set of second-order sentences. We also show that a
certain -conservative extension of RCA admits a provably
categorical single-sentence characterization of the naturals, but each such
characterization has to be inconsistent with WKL+superexp.Comment: 17 page
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