10,023 research outputs found
The Fan Theorem, its strong negation, and the determinacy of games
IIn the context of a weak formal theory called Basic Intuitionistic
Mathematics , we study Brouwer's Fan Theorem and a strong
negation of the Fan Theorem, Kleene's Alternative (to the Fan Theorem). We
prove that the Fan Theorem is equivalent to contrapositions of a number of
intuitionistically accepted axioms of countable choice and that Kleene's
Alternative is equivalent to strong negations of these statements. We also
discuss finite and infinite games and introduce a constructively useful notion
of determinacy. We prove that the Fan Theorem is equivalent to the
Intuitionistic Determinacy Theorem, saying that every subset of Cantor space
is, in our constructively meaningful sense, determinate, and show that Kleene's
Alternative is equivalent to a strong negation of a special case of this
theorem. We then consider a uniform intermediate value theorem and a
compactness theorem for classical propositional logic, and prove that the Fan
Theorem is equivalent to each of these theorems and that Kleene's Alternative
is equivalent to strong negations of them. We end with a note on a possibly
important statement, provable from principles accepted by Brouwer, that one
might call a Strong Fan Theorem.Comment: arXiv admin note: text overlap with arXiv:1106.273
Pincherle's theorem in Reverse Mathematics and computability theory
We study the logical and computational properties of basic theorems of
uncountable mathematics, in particular Pincherle's theorem, published in 1882.
This theorem states that a locally bounded function is bounded on certain
domains, i.e. one of the first 'local-to-global' principles. It is well-known
that such principles in analysis are intimately connected to (open-cover)
compactness, but we nonetheless exhibit fundamental differences between
compactness and Pincherle's theorem. For instance, the main question of Reverse
Mathematics, namely which set existence axioms are necessary to prove
Pincherle's theorem, does not have an unique or unambiguous answer, in contrast
to compactness. We establish similar differences for the computational
properties of compactness and Pincherle's theorem. We establish the same
differences for other local-to-global principles, even going back to
Weierstrass. We also greatly sharpen the known computational power of
compactness, for the most shared with Pincherle's theorem however. Finally,
countable choice plays an important role in the previous, we therefore study
this axiom together with the intimately related Lindel\"of lemma.Comment: 43 pages, one appendix, to appear in Annals of Pure and Applied Logi
Reverse Mathematics and parameter-free Transfer
Recently, conservative extensions of Peano and Heyting arithmetic in the
spirit of Nelson's axiomatic approach to Nonstandard Analysis, have been
proposed. In this paper, we study the Transfer axiom of Nonstandard Analysis
restricted to formulas without parameters. Based on this axiom, we formulate a
base theory for the Reverse Mathematics of Nonstandard Analysis and prove some
natural reversals, and show that most of these equivalences do not hold in the
absence of parameter-free Transfer.Comment: 22 pages; to appear in Annals of Pure and Applied Logi
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