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The Brouwer invariance theorems in reverse mathematics
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...
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Published in: | Forum of mathematics. Sigma 2020, Vol.8, Article e51 |
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creator | Kihara, Takayuki |
description | 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
$\mathbb {R}^3$
. |
doi_str_mv | 10.1017/fms.2020.52 |
format | article |
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${\sf RCA}_0$
. 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
$\mathbb {R}^3$
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${\sf RCA}_0$
. 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
$\mathbb {R}^3$
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${\sf RCA}_0$
. 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
$\mathbb {R}^3$
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subjects | 03B30 54F45 55M10 Algorithms Foundations Invariance Mathematical analysis Theorems |
title | The Brouwer invariance theorems in reverse mathematics |
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