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

2016
*
arXiv
*
pre-print

We show in Bishop's constructive mathematics---in particular, using countable choice---that

arXiv:1611.02527v1
fatcat:f6rmbcf6uzgidh3jy4egq5icx4
*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. ...##
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A Note on the Sequential Version of ${\rm \Pi^1_2}$ Statements
[chapter]

2013
*
Lecture Notes in Computer Science
*

Moreover our results suggest

doi:10.1007/978-3-642-39053-1_20
fatcat:e2lt4xoqgvfudkgfachr3mrvyy
*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 . ...##
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On the Computational Content of the Brouwer Fixed Point Theorem
[chapter]

2012
*
Lecture Notes in Computer Science
*

It easily follows from a meta

doi:10.1007/978-3-642-30870-3_7
fatcat:sw75b4h5lbe5ddhto3skhgixgm
*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 [ ...##
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The Brouwer Fixed Point Theorem Revisited
[chapter]

2016
*
Lecture Notes in Computer Science
*

Constructions similar to those used for

doi:10.1007/978-3-319-40189-8_6
fatcat:yny2hizfwzfivfoyhpuao3budi
*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*...##
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Non-standard Nonstandard Analysis and the computational content of standard mathematics
[article]

2015
*
arXiv
*
pre-print

In particular, we show that from classical and ineffective existence proofs (not involving Nonstandard Analysis but using

arXiv:1509.00282v2
fatcat:giwpuvq4oneepgooxisopfl74a
*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. ...##
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The Brouwer invariance theorems in reverse mathematics

2020
*
Forum of Mathematics, Sigma
*

In this article, we solve Stillwell's problem by showing that (some forms of)

doi:10.1017/fms.2020.52
fatcat:taj6tulstjawto7f44kjvqckwa
*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. ...##
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OPEN QUESTIONS ABOUT RAMSEY-TYPE STATEMENTS IN REVERSE MATHEMATICS

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. ...

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

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. ...

##
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Uniform and nonstandard existence in Reverse Mathematics
[article]

2015
*
arXiv
*
pre-print

Reverse Mathematics is a program in

arXiv:1502.03618v2
fatcat:u7vpd5wgpnef3mxpdr6xua4tzq
*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. ...##
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On the logical structure of choice and bar induction principles
[article]

2021
*
arXiv
*
pre-print

Boolean Prime Filter

arXiv:2105.08951v2
fatcat:r7f62hxb2jd77aa6rmwlgdm74q
*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 ...##
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Reverse Mathematics in Bishop's Constructive Mathematics

2006
*
Philosophia Scientiæ
*

and

doi:10.4000/philosophiascientiae.406
fatcat:bp5dqdaysbepbfl5vjxusodtaa
*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. ...##
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Connected Choice and the Brouwer Fixed Point Theorem
[article]

2018
*
arXiv
*
pre-print

Another main result is that connected choice is complete for dimension greater than or equal to two in

arXiv:1206.4809v3
fatcat:ybktamzncbeoxk2agabqiylziq
*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 ...##
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The Vitali Covering Theorem in the Weihrauch Lattice
[chapter]

2016
*
Lecture Notes in Computer Science
*

These versions are either computable or closely related to

doi:10.1007/978-3-319-50062-1_14
fatcat:jy6zkwcayrbj3n5ku6ssq4cynu
*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. ...##
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The reverse mathematics of the Tietze extension theorem
[article]

2016
*
arXiv
*
pre-print

We prove that several versions of

arXiv:1602.05398v1
fatcat:m5atkfoq2faypaqv27dnde5ify
*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 ...##
###
The reverse mathematics of the Tietze extension theorem

2016
*
Proceedings of the American Mathematical Society
*

We prove that several versions of

doi:10.1090/proc/13217
fatcat:37bspxvwnra67b7xhbsips7zcq
*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 ...
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