On the strength of marriage theorems and uniformity

“…We shall study a 'sequential' version of HBC 0 involving sequences of (sub-)coverings. Such sequential theorems are well-studied in RM, starting with [62, IV.2.12], and also in [9,10,17,18,21,22,69]. Principle 3.2 (HBC seq 0 ).…”

Section: Countable Coveringsmentioning

confidence: 99%

On robust theorems due to Bolzano, Weierstrass, Cantor, and Jordan

Normann¹,

Sanders²

2021

Preprint

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Reverse Mathematics (RM hereafter) is a program in the foundations of mathematics where the aim is to identify the minimal axioms needed to prove a given theorem from ordinary, i.e. non-set theoretic, mathematics. This program has unveiled surprising regularities: the minimal axioms are very often equivalent to the theorem over the base theory, a weak system of 'computable mathematics', while most theorems are either provable in this base theory, or equivalent to one of only four logical systems. The latter plus the base theory are called the 'Big Five' and the associated equivalences are robust following Montalbán, i.e. stable under small variations of the theorems at hand. Working in Kohlenbach's higher-order RM, we obtain two long series of equivalences based on theorems due to Bolzano, Weierstrass, and Cantor; these equivalences are extremely robust and have no counterpart among the Big Five systems, as they are strictly between the base theory and the higherorder counterpart of weak König's lemma. In this light, higher-order RM is much richer than its second-order cousin, boasting as it does two extra 'Big' systems. Our study includes numerous variations of the Bolzano-Weierstrass theorem formulated as the existence of suprema for (third-order) countable sets in Cantor space. We similarly investigate many basic theorems about the real line, limit points, and countable unions, many going back to Cantor.

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“…Note that such 'sequential' theorems are well-studied in RM, starting with [100, IV.2.12], and can also be found in e.g. [21,22,29,30,42].…”

Section: Appendix B the Gödel Hierarchymentioning

confidence: 99%

Pincherle's theorem in Reverse Mathematics and computability theory

Normann

Sanders

2018

Preprint

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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/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 Lindelöf lemma.Theorem 1.1 (Pincherle, 1882). Let E be a closed and bounded subset of R n and let f : E → R be locally bounded. Then f is bounded on E.

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On the strength of marriage theorems and uniformity

Cited by 7 publications

References 15 publications

Pincherle's theorem in reverse mathematics and computability theory

Pincherle's theorem in reverse mathematics and computability theory

On robust theorems due to Bolzano, Weierstrass, Cantor, and Jordan

Pincherle's theorem in Reverse Mathematics and computability theory

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