A non-standard construction of Haar measure and weak König's lemma

2000 ◽  
Vol 65 (1) ◽  
pp. 173-186 ◽  
Author(s):  
Kazuyuki Tanaka ◽  
Takeshi Yamazaki
Keyword(s):  
Standard Model ◽  
Compact Group ◽  
Haar Measure ◽  
Standard Method ◽  
Initial Part ◽  
Standard Construction ◽  
Necessary And Sufficient ◽  
Weak König’S Lemma ◽  
König’S Lemma

AbstractIn this paper, we show within RCA0 that weak König's lemma is necessary and sufficient to prove that any (separable) compact group has a Haar measure. Within WKL0, a Haar measure is constructed by a non-standard method based on a fact that every countable non-standard model of WKL0 has a proper initial part isomorphic to itself [10].

scholarly journals The Brouwer invariance theorems in reverse mathematics

2020 ◽  
Vol 8 ◽  
Author(s):  
Takayuki Kihara
Keyword(s):  
Reverse Mathematics ◽  
Base System ◽  
Open Problems ◽  
Explicit Algorithm ◽  
Topological Embedding ◽  
Weak König’S Lemma ◽  
König’S Lemma

Abstract In [12], John Stillwell wrote, ‘finding the exact strength of the Brouwer invariance theorems seems to me one of the most interesting open problems in reverse mathematics.’ 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 $\mathbb {R}^3$ .

scholarly journals Compact groups with a set of positive Haar measure satisfying a nilpotent law

2021 ◽  
pp. 1-4
Author(s):  
ALIREZA ABDOLLAHI ◽  
MEISAM SOLEIMANI MALEKAN
Keyword(s):  
Compact Group ◽  
Haar Measure ◽  
Positive Answer ◽  
Nilpotent Subgroup ◽  
Compact Groups

Abstract The following question is proposed by Martino, Tointon, Valiunas and Ventura in [4, question 1·20]: Let G be a compact group, and suppose that \[\mathcal{N}_k(G) = \{(x_1,\dots,x_{k+1}) \in G^{k+1} \;|\; [x_1,\dots, x_{k+1}] = 1\}\] has positive Haar measure in $G^{k+1}$ . Does G have an open k-step nilpotent subgroup? We give a positive answer for $k = 2$ .

scholarly journals Reverse mathematics and a Ramsey-type König's Lemma

2012 ◽  
Vol 77 (4) ◽  
pp. 1272-1280 ◽  
Author(s):  
Stephen Flood
Keyword(s):  
Reverse Mathematics ◽  
Ramsey's Theorem ◽  
Weak Regularity ◽  
Ramsey’S Theorem ◽  
Ramsey Type ◽  
Weak König’S Lemma ◽  
König’S Lemma

AbstractIn this paper, we propose a weak regularity principle which is similar to both weak König's lemma and Ramsey's theorem. We begin by studying the computational strength of this principle in the context of reverse mathematics. We then analyze different ways of generalizing this principle.

scholarly journals The Computational Strength of Extensions of Weak König’s Lemma

1998 ◽  
Vol 5 (41) ◽  
Author(s):  
Ulrich Kohlenbach
Keyword(s):  
Natural Number ◽  
Arbitrary Function ◽  
The Other ◽  
Binary Trees ◽  
Subsequent Paper ◽  
Bounded Functions ◽  
The One ◽  
Weak König’S Lemma ◽  
König’S Lemma

The weak König's lemma WKL is of crucial significance in the study of fragments of mathematics which on the one hand are mathematically strong but on the other hand have a low proof-theoretic and computational strength. In addition to the restriction to binary trees (or equivalently bounded trees), WKL<br />is also `weak' in that the tree predicate is quantifier-free. Whereas in general the computational and proof-theoretic strength increases when logically more complex trees are allowed, we show that this is not the case for trees which are<br />given by formulas in a class Phi where we allow an arbitrary function quantifier prefix over bounded functions in front of a Pi^0_1-formula. This results in a schema Phi-WKL.<br />Another way of looking at WKL is via its equivalence to the principle<br /> For all x there exists y<=1 for all z A0(x; y; z) -> there exists f <= lambda x.1 for all x, z A0(x, fx, z);<br />where A0 is a quantifier-free formula (x, y, z are natural number variables). <br /> We generalize this to Phi-formulas as well and allow function quantifiers `there exists g <= s'<br />instead of `there exists y <= 1', where g <= s is defined pointwise. The resulting schema is called Phi-b-AC^0,1.<br />In the absence of functional parameters (so in particular in a second order context), the corresponding versions of Phi-WKL and Phi-b-AC^0,1 turn out to<br />be equivalent to WKL. This changes completely in the presence of functional<br />variables of type 2 where we get proper hierarchies of principles Phi_n-WKL and<br />Phi_n-b-AC^0,1. Variables of type 2 however are necessary for a direct representation<br />of analytical objects and - sometimes - for a faithful representation of<br />such objects at all as we will show in a subsequent paper. By a reduction of<br />Phi-WKL and Phi-b-AC^0,1 to a non-standard axiom F (introduced in a previous paper) and a new elimination result for F relative to various fragment of arithmetic in all finite types, we prove that Phi-WKL and Phi-b-AC^0,1 do<br />neither contribute to the provably recursive functionals of these fragments nor to their proof-theoretic strength. In a subsequent paper we will illustrate the greater mathematical strength of these principles (compared to WKL).

scholarly journals Comparing DNR and WWKL

2004 ◽  
Vol 69 (4) ◽  
pp. 1089-1104 ◽  
Author(s):  
Klaus Ambos-Spies ◽  
Bjørn Kjos-Hanssen ◽  
Steffen Lempp ◽  
Theodore A. Slaman
Keyword(s):  
Reverse Mathematics ◽  
Axiom System ◽  
Recursive Functions ◽  
Weak König’S Lemma ◽  
König’S Lemma

Abstract.In Reverse Mathematics, the axiom system DNR. asserting the existence of diagonally non-recursive functions, is strictly weaker than WWKL0 (weak weak König's Lemma).

Translation-Invariant Operators On Lp(G), 0 < p < 1 (II)

1977 ◽  
Vol 29 (3) ◽  
pp. 626-630 ◽  
Author(s):  
Daniel M. Oberlin
Keyword(s):  
Compact Group ◽  
Haar Measure ◽  
Lebesgue Space ◽  
Locally Compact Group ◽  
Locally Compact ◽  
Translation Invariant ◽  
Translation Operators ◽  
Left And Right

For a locally compact group G, let LP(G) be the usual Lebesgue space with respect to left Haar measure m on G. For x ϵ G define the left and right translation operators Lx and Rx by Lx f(y) = f(xy), Rx f(y) = f(yx)(f ϵ Lp(G),y ϵ G). The purpose of this paper is to prove the following theorem.

scholarly journals Connected choice and the Brouwer fixed point theorem

2019 ◽  
Vol 19 (01) ◽  
pp. 1950004 ◽  
Author(s):  
Vasco Brattka ◽  
Stéphane Le Roux ◽  
Joseph S. Miller ◽  
Arno Pauly
Keyword(s):  
Fixed Point ◽  
Fixed Point Theorem ◽  
Point Theorem ◽  
Unit Cube ◽  
Closed Set ◽  
Brouwer Fixed Point Theorem ◽  
Fixed Dimension ◽  
Lipschitz Constants ◽  
Weak König’S Lemma ◽  
König’S Lemma

We study the computational content of the Brouwer Fixed Point Theorem in the Weihrauch lattice. Connected choice is the operation that finds a point in a non-empty connected closed set given by negative information. One of our main results is that for any fixed dimension the Brouwer Fixed Point Theorem of that dimension is computably equivalent to connected choice of the Euclidean unit cube of the same dimension. 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. While we can present two independent proofs for dimension three and upward that are either based on a simple geometric construction or a combinatorial argument, the proof for dimension two is based on a more involved inverse limit construction. The connected choice operation in dimension one is known to be equivalent to the Intermediate Value Theorem; we prove that this problem is not idempotent in contrast to the case of dimension two and upward. We also prove that Lipschitz continuity with Lipschitz constants strictly larger than one does not simplify finding fixed points. Finally, we prove that finding a connectedness component of a closed subset of the Euclidean unit cube of any dimension greater than or equal to one is equivalent to Weak Kőnig’s Lemma. In order to describe these results, we introduce a representation of closed subsets of the unit cube by trees of rational complexes.

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