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Weak König's lemma implies the uniform continuity theorem: a direct proof [article]

Matthew Hendtlass
2016 arXiv   pre-print
We show in Bishop's constructive mathematics---in particular, using countable choice---that weak König's lemma implies the uniform continuity theorem.  ...  Theorem 1. Weak König's lemma implies the uniform continuity theorem.  ...  the witnesses x, y that δ is not a modulus of uniform continuity for ε.  ... 
arXiv:1611.02527v1 fatcat:f6rmbcf6uzgidh3jy4egq5icx4

A Note on the Sequential Version of ${\rm \Pi^1_2}$ Statements [chapter]

Makoto Fujiwara, Keita Yokoyama
2013 Lecture Notes in Computer Science  
Moreover our results suggest the optimality of Dorais's uniformization theorems.  ...  In connection with uniform computability and intuitionistic provability, the strength of the sequential version of Π 1 2 theorems has been investigated in reverse mathematics.  ...  For instance, the intermediate value theorem is provable in RCA 0 , but the sequential version of it is equivalent to WKL (weak König's lemma), and so, not provable in RCA 0 .  ... 
doi:10.1007/978-3-642-39053-1_20 fatcat:e2lt4xoqgvfudkgfachr3mrvyy

On the Computational Content of the Brouwer Fixed Point Theorem [chapter]

Vasco Brattka, Stéphane Le Roux, Arno Pauly
2012 Lecture Notes in Computer Science  
It easily follows from a meta theorem presented in [3] that the Brouwer Fixed Point Theorem BFT n is reducible to Weak Kőnig's Lemma WKL for any dimension n, i.e. BFT n ≤ W WKL.  ...  Constructions similar to those used for the above counterexamples have been utilized in order to prove that the Brouwer Fixed Point Theorem is equivalent to Weak Kőnig's Lemma in reverse mathematics [  ... 
doi:10.1007/978-3-642-30870-3_7 fatcat:sw75b4h5lbe5ddhto3skhgixgm

The Brouwer Fixed Point Theorem Revisited [chapter]

Vasco Brattka, Stéphane Le Roux, Joseph S. Miller, Arno Pauly
2016 Lecture Notes in Computer Science  
Constructions similar to those used for the above counterexamples have been utilized in order to prove that the Brouwer Fixed Point Theorem is equivalent to Weak Kőnig's Lemma in reverse mathematics [  ...  A superficial reading of the results of Orevkov [16] and Baigger [1] can lead to the wrong conclusion that they actually provide a reduction of Weak Kőnig's Lemma to the Brouwer Fixed Point Theorem  ... 
doi:10.1007/978-3-319-40189-8_6 fatcat:yny2hizfwzfivfoyhpuao3budi

Non-standard Nonstandard Analysis and the computational content of standard mathematics [article]

Sam Sanders
2015 arXiv   pre-print
In particular, we show that from classical and ineffective existence proofs (not involving Nonstandard Analysis but using weak Koenig's lemma), one can 'automatically' extract approximations to the objects  ...  This system validates so-called non-standard uniform boundedness principles which are central to Kohlenbach's approach to proof mining ([14]).  ...  This research was supported by the following funding bodies: FWO Flanders, the John Templeton Foundation, the Alexander von Humboldt Foundation, and the Japan Society for the Promotion of Science.  ... 
arXiv:1509.00282v2 fatcat:giwpuvq4oneepgooxisopfl74a

The Brouwer invariance theorems in reverse mathematics

Takayuki Kihara
2020 Forum of Mathematics, Sigma  
In this article, we solve Stillwell's problem by showing that (some forms of) the Brouwer invariance theorems are equivalent to the weak König's lemma over the base system ${\sf RCA}_0$ .  ...  In particular, there exists an explicit algorithm which, whenever the weak König's lemma is false, constructs a topological embedding of $\mathbb {R}^4$ into $\mathbb {R}^3$ .  ...  Advanced Research Networks) and the Young Scholars Overseas Visit Program of Nagoya University. The author would like to thank Keita Yokoyama for valuable discussions.  ... 
doi:10.1017/fms.2020.52 fatcat:taj6tulstjawto7f44kjvqckwa

OPEN QUESTIONS ABOUT RAMSEY-TYPE STATEMENTS IN REVERSE MATHEMATICS

LUDOVIC PATEY
2016 Bulletin of Symbolic Logic  
The strength of consequences of Ramsey's theorem has been extensively studied in reverse mathematics and under various reducibilities, namely, computable reducibility and uniform reducibility.  ...  The inability to answer those questions reveals some gaps in our understanding of the combinatorics of Ramsey's theorem.  ...  The opinions expressed in this publication are those of the author(s) and do not necessarily reflect the views of the John Templeton Foundation.  ... 
doi:10.1017/bsl.2015.40 fatcat:oxgrrazljfdzdccisbvnw3ivyi

Open questions about Ramsey-type statements in reverse mathematics [article]

Ludovic Patey
2015 arXiv   pre-print
The strength of consequences of Ramsey's theorem has been extensively studied in reverse mathematics and under various reducibilities, namely, computable reducibility and uniform reducibility.  ...  The inability to answer those questions reveals some gaps in our understanding of the combinatorics of Ramsey's theorem.  ...  The opinions expressed in this publication are those of the author(s) and do not necessarily reflect the views of the John Templeton Foundation.  ... 
arXiv:1506.04780v3 fatcat:g3zaguharjbx7kwzv74q5p4fvi

Uniform and nonstandard existence in Reverse Mathematics [article]

Sam Sanders
2015 arXiv   pre-print
Reverse Mathematics is a program in the foundations of mathematics which provides an elegant classification of theorems of ordinary mathematics based on computability.  ...  Our aim is to provide an alternative classification of theorems based on the central tenet of Feferman's Explicit Mathematics, namely that a proof of existence of an object yields a procedure to compute  ...  This research was supported by the following funding bodies: FWO Flanders, the John Templeton Foundation, the Alexander von Humboldt Foundation, and the Japan Society for the Promotion of Science.  ... 
arXiv:1502.03618v2 fatcat:u7vpd5wgpnef3mxpdr6xua4tzq

On the logical structure of choice and bar induction principles [article]

Nuria Brede, Hugo Herbelin
2021 arXiv   pre-print
Boolean Prime Filter Theorem / Ultrafilter Theorem if B is the two-element set 𝔹 (for a constructive definition of prime filter), ∙ the axiom of dependent choice if A = ℕ, ∙ Weak König's Lemma if A =  ...  ℕ and B = 𝔹 (up to weak classical reasoning) GBI_A,B,T intuitionistically captures the strength of ∙ Gödel's completeness theorem in the form validity implies provability for entailment relations if  ...  ACKNOWLEDGMENTS We thank the communities of researchers who contributed to develop the material we built on, and in particular Camille Noûs, from the Cogitamus Lab, who embodies the collective and collaborative  ... 
arXiv:2105.08951v2 fatcat:r7f62hxb2jd77aa6rmwlgdm74q

Reverse Mathematics in Bishop's Constructive Mathematics

Hajime Ishihara
2006 Philosophia Scientiæ  
and continuous properties.  ...  We will overview the results in an informal approach to constructive reverse mathematics, that is reverse mathematics in Bishop's constructive mathematics, especially focusing on compactness properties  ...  Weak König's lemma (WKL) [Ishihara 1990 ]: every infinite tree has an infinite path. 6.  ... 
doi:10.4000/philosophiascientiae.406 fatcat:bp5dqdaysbepbfl5vjxusodtaa

Connected Choice and the Brouwer Fixed Point Theorem [article]

Vasco Brattka, Stéphane Le Roux, Joseph S. Miller, Arno Pauly
2018 arXiv   pre-print
Another main result is that connected choice is complete for dimension greater than or equal to two in the sense that it is computably equivalent to Weak Kőnig's Lemma.  ...  Finally, we prove that finding a connectedness component of a closed subset of the Euclidean unit cube of any dimension greater or equal to one is equivalent to Weak Kőnig's Lemma.  ...  Theorems that have been compared and classified in this sense include Weak Kőnig's Lemma WKL, the Hahn-Banach Theorem [26] , the Baire Category Theorem [14] , Banach's Inverse Mapping Theorem, the Open  ... 
arXiv:1206.4809v3 fatcat:ybktamzncbeoxk2agabqiylziq

The Vitali Covering Theorem in the Weihrauch Lattice [chapter]

Vasco Brattka, Guido Gherardi, Rupert Hölzl, Arno Pauly
2016 Lecture Notes in Computer Science  
These versions are either computable or closely related to uniform variants of Weak Weak Kőnig's Lemma.  ...  We study the uniform computational content of the Vitali Covering Theorem for intervals using the tool of Weihrauch reducibility.  ...  We have summarized the results in the diagram in Figure 2 . The diagram also indicates some equivalence classes in the neighborhood that are related to Weak Kőnig's Lemma WKL.  ... 
doi:10.1007/978-3-319-50062-1_14 fatcat:jy6zkwcayrbj3n5ku6ssq4cynu

The reverse mathematics of the Tietze extension theorem [article]

Paul Shafer
2016 arXiv   pre-print
We prove that several versions of the Tietze extension theorem for functions with moduli of uniform continuity are equivalent to WKL_0 over RCA_0.  ...  Thus, in RCA 0 we can formulate (but not prove) weak König's lemma, which is the statement "every infinite subtree of 2 <N has an infinite path." WKL 0 is then the system RCA 0 + weak König's lemma.  ...  Reversing the strong Tietze extension theorem to weak König's lemma In their analysis of sTET [0,1] , Giusto and Simpson first show that RCA 0 sTET [0,1] by showing that sTET [0,1] fails in REC, the model  ... 
arXiv:1602.05398v1 fatcat:m5atkfoq2faypaqv27dnde5ify

The reverse mathematics of the Tietze extension theorem

Paul Shafer
2016 Proceedings of the American Mathematical Society  
We prove that several versions of the Tietze extension theorem for functions with moduli of uniform continuity are equivalent to WKL0 over RCA0.  ...  Thus, in RCA 0 we can formulate (but not prove) weak König's lemma, which is the statement "every infinite subtree of 2 <N has an infinite path." WKL 0 is then the system RCA 0 + weak König's lemma.  ...  Reversing the strong Tietze extension theorem to weak König's lemma In their analysis of sTET [0,1] , Giusto and Simpson first show that RCA 0 sTET [0,1] by showing that sTET [0,1] fails in REC, the model  ... 
doi:10.1090/proc/13217 fatcat:37bspxvwnra67b7xhbsips7zcq
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