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4ee1feb6907a205a6746e81d2a23ec3d779c225a | subsection | 36 | 223 | The Rising Sun Lemma, and [PR:LPO]LPO and [PR:WLPO]WLPO | Since [PR:LPO]LPO implies that f is uniformly continuous (see REF ) and therefore S_x = \sup _{[x,1]} f exists, we can decide whether f(x) = S_x, or f(x) < S_x. In the second case, using Proposition REF .REF , we can find b_x = \inf \mbox{$\left\lbrace \,z \geqslant x \, | \, f(z) = S_x \,\right\rbrace $}. Using Propos... | {
"cite_spans": []
} | 1804.05495 | Constructive Reverse Mathematics | [
"Hannes Diener"
] | [
"math.LO"
] | 2,018 | en | Mathematics | [
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5cb2f531c4eeaad70cd360bd5a2945dd9f77a75d | subsection | 37 | 223 | The Rising Sun Lemma, and [PR:LPO]LPO and [PR:WLPO]WLPO | If we had followed Tao's formulation more closely we get an equivalence to [PR:WLPO]WLPO. Luckily, we can reuse most of the proof above.Proposition 3.9
[PR:WLPO]WLPO is equivalent to the following statement.Consider a continuous function f:[0,1] \rightarrow \mathbb {R}. Then one can find a, at most countableThis is on... | {
"cite_spans": []
} | 1804.05495 | Constructive Reverse Mathematics | [
"Hannes Diener"
] | [
"math.LO"
] | 2,018 | en | Mathematics | [
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f1ba8f76e5d1f2852840169e4efea9d4e7a85765 | subsection | 38 | 223 | [PR:LLPO]LLPO and [PR:WKL]WKL | The final limited omniscience principle is the lesser limited principle of omniscience.
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} | 1804.05495 | Constructive Reverse Mathematics | [
"Hannes Diener"
] | [
"math.LO"
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77b9ef4d2922289353d96636952b015046c318f7 | subsection | 39 | 223 | Completeness of Finite Sets | In Proposition REF we cited a paper by Mandelkern that among many other insights shows that [PR:LLPO]LLPO is equivalent to every two-element set of reals being closed.
That is constructively we cannot show that for any two reals a,b\overline{\left\lbrace a,b \right\rbrace } = \left\lbrace a,b \right\rbrace \ .Analysing... | {
"cite_spans": []
} | 1804.05495 | Constructive Reverse Mathematics | [
"Hannes Diener"
] | [
"math.LO"
] | 2,018 | en | Mathematics | [
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e4f0bd191fde6f74d3a9536394c2d07506e887f1 | subsection | 40 | 223 | Completeness of Finite Sets | Now if z_\infty \in \left\lbrace a,b, \inf \left\lbrace a,b \right\rbrace , \sup \left\lbrace a, b \right\rbrace \right\rbrace we can make the following decisions:z_\infty = 0:
there cannot be an even m such that a_m = 1, since in that case a \ne 0 and z_n \rightarrow a.
z_\infty = a:
there cannot be an odd m such that... | {
"cite_spans": []
} | 1804.05495 | Constructive Reverse Mathematics | [
"Hannes Diener"
] | [
"math.LO"
] | 2,018 | en | Mathematics | [
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30a0f6c2ab4b3db7e1bb518e4a2a2b9835771dff | subsection | 41 | 223 | Completeness of Finite Sets | Notice that there exists a realisability model (based on infinite time Turing machines) in which there is a surjection \mathbb {N}\rightarrow \mathbb {N}^{\mathbb {N}} , which means that the closure of \left\lbrace a,b \right\rbrace is countable, but that in that model also [PR:LPO]LPO and therefore [PR:LLPO]LLPO holds... | {
"cite_spans": []
} | 1804.05495 | Constructive Reverse Mathematics | [
"Hannes Diener"
] | [
"math.LO"
] | 2,018 | en | Mathematics | [
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f2533a3ecae565f7806256f91ab378b78012abb6 | subsection | 42 | 223 | [PR:WKL]WKL, Minima, and Fixed Points | The one principle in Constructive Reverse Mathematics that is possibly best known by non-constructivists is Weak Kőnig's Lemma.
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} | 1804.05495 | Constructive Reverse Mathematics | [
"Hannes Diener"
] | [
"math.LO"
] | 2,018 | en | Mathematics | [
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4124d4697c93a46a09e4ecf4c21ff3a73f4a53ef | subsection | 43 | 223 | [PR:WKL]WKL, Minima, and Fixed Points | Thus we can, using dependent choice, iteratively define a sequence \alpha \in 2^{\mathbb {N}} such that T_{\overline{\alpha }n} is a infinite, decidable tree for all n \in \mathbb {N}. In particular \overline{\alpha }n \in T.Remark 4.6Analysing the above proof we can see that [PR:WLPO]WLPO implies [PR:WKL]WKL, only ass... | {
"cite_spans": []
} | 1804.05495 | Constructive Reverse Mathematics | [
"Hannes Diener"
] | [
"math.LO"
] | 2,018 | en | Mathematics | [
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99b7db858211ea6a3458c0ebef13ad24d507c558 | subsection | 44 | 223 | [PR:WKL]WKL, Minima, and Fixed Points | Finally the equivalence of REF and [PR:WKL]WKL can be found in (and relies on the equivalence of [PR:WKL]WKL and REF ).Remark 4.8 Using Proposition REF we can replace uniform by point-wise continuity in the above proposition.This last proposition above also allows the following heuristic.Heuristic 4.9 Many fix point th... | {
"cite_spans": []
} | 1804.05495 | Constructive Reverse Mathematics | [
"Hannes Diener"
] | [
"math.LO"
] | 2,018 | en | Mathematics | [
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95f7f564f846dfbf8c929a590c6375f61237a20a | subsection | 45 | 223 | Space-filling curves | The content of this section might well be folklore among some constructivists. We do not claim to be the first to prove these results, but we are also not aware that they have been written down anywhere else.One of the stranger objects in (classical) analysis are space-filling curves, which are continuous functions [0,... | {
"cite_spans": []
} | 1804.05495 | Constructive Reverse Mathematics | [
"Hannes Diener"
] | [
"math.LO"
] | 2,018 | en | Mathematics | [
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3d1dfa401541ca99a893b65957876a3a35d4302a | subsection | 46 | 223 | Space-filling curves | The question of whether we can prove a stronger version of non-injectiveness for an arbitrary space-filling curve constructively appears to be a tricky one.
quQuestion 2Is every space-filling curve non-injective? More precise: if f is a space-filling curve, then does there exist x \ne y with f(x) = f(y)?It seems feasib... | {
"cite_spans": []
} | 1804.05495 | Constructive Reverse Mathematics | [
"Hannes Diener"
] | [
"math.LO"
] | 2,018 | en | Mathematics | [
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2d8e1675b0b5193e2476bb9bfae5f17d2070b97c | subsection | 47 | 223 | Space-filling curves | Without loss of generality we may assume that f([0,1]) \subset [-1,1] and that f(0)=-1 and f(1) = 1. Now define h,g:[0,1] \rightarrow [0,1]^2 piecewise linearly byh(x) = {\left\lbrace \begin{array}{ll}
-\frac{3}{2}x+1 & \text{ if } 0 \leqslant x \leqslant \frac{1}{3} \\
\frac{1}{2} & \text{ if } \frac{1}{3} \leqslant x... | {
"cite_spans": []
} | 1804.05495 | Constructive Reverse Mathematics | [
"Hannes Diener"
] | [
"math.LO"
] | 2,018 | en | Mathematics | [
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ff1b605e16406cec1af2f6c8ae0aebc80573cc20 | subsection | 48 | 223 | The Greedy Algorithm | A matroid (E, \mathcal {F}) consists of a finite set E together with a collection \mathcal {F} of subsets E, such that\emptyset \in \mathcal {F}
A \in \mathcal {F} \wedge B \subset A \Rightarrow B \in \mathcal {F}
A,B \in \mathcal {F} \wedge {B}={A}+1 \Rightarrow \exists {x \in B} : { A \cup \left\lbrace x \right\rbr... | {
"cite_spans": []
} | 1804.05495 | Constructive Reverse Mathematics | [
"Hannes Diener"
] | [
"math.LO"
] | 2,018 | en | Mathematics | [
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2923b1d4940f34254ff482dd68ebbdb909d8466b | subsection | 49 | 223 | The Greedy Algorithm | Z_{i} = \emptyset terminate and return F_{i} .
choose y \in Z_{i} such that w(y) \leqslant w(z_{i}) for all z_{i} \in Z_{i} let F_{i+1} = F_{i} \cup \left\lbrace y \right\rbraceNotice, that in step 6 we need (REF ).
For (\ref {greedy3}) \Rightarrow (\ref {greedy1}) let x,y \in \mathbb {R}. Then\mathcal {M} = \left(\le... | {
"cite_spans": []
} | 1804.05495 | Constructive Reverse Mathematics | [
"Hannes Diener"
] | [
"math.LO"
] | 2,018 | en | Mathematics | [
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49ef1901dce4e2761d20ea1111ea6fb8c02cba2d | subsection | 50 | 223 | Sharkovskii's Theorem | Sharkovskii'sThere are various alternative ways of spelling Sharkovskii. Among them Sharkovsky , Sarkovskii , Sarkovski . Theorem from 1964 is, in our opinion, one of the most entertaining results in mathematics: simple to state yet utterly surprising. It concerns itself with a very rudimentary type of dynamical system... | {
"cite_spans": []
} | 1804.05495 | Constructive Reverse Mathematics | [
"Hannes Diener"
] | [
"math.LO"
] | 2,018 | en | Mathematics | [
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61c76c0a79f9583353982f8872682a8c2b52fa9f | subsection | 51 | 223 | Sharkovskii's Theorem | \\
\end{array}\right.}
[Figure: We think of a as being very small. Depending on a f crosses the lines y = 0 and y=1 or misses both.]As f([0,\frac{1}{3}]) \supset [0, \frac{2}{3}] and f([\frac{2}{3},1]) \supset [\frac{1}{3},1] we see that f([0,1]) \supset [0,1]. So assume there exists an interval [c,d] \subset [0,1] su... | {
"cite_spans": []
} | 1804.05495 | Constructive Reverse Mathematics | [
"Hannes Diener"
] | [
"math.LO"
] | 2,018 | en | Mathematics | [
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ef22bed1886357b841cd670cfc2e446c6b2c3ef4 | subsection | 52 | 223 | Sharkovskii's Theorem | As we assume [PR:LLPO]LLPO either\sup \left\lbrace f *{a_0, \frac{1}{2} (a_n + b_n)} \right\rbrace \geqslant b \text{ or } \sup \left\lbrace f \left(*{a_0, \frac{1}{2} (a_n + b_n)} \right) \right\rbrace \leqslant b \ .In the first case set a_{n+1} = a_n and as we assumed [PR:LLPO]LLPO the minimum principle holds and so... | {
"cite_spans": []
} | 1804.05495 | Constructive Reverse Mathematics | [
"Hannes Diener"
] | [
"math.LO"
] | 2,018 | en | Mathematics | [
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789edc5b87c2f48a20adb0029ab1b406c6665106 | subsection | 53 | 223 | Sharkovskii's Theorem | \\
\end{array}\right.}
[Figure: We cannot decide whether the graph of f between \frac{2}{4} and \frac{3}{4} lies above or below the diagonal.]f(0) = \frac{1}{4}, f(\frac{1}{4}) = 1 and f(1)=0 so f has a 3–periodic point. If f has a fixed point x, then
either x < \frac{3}{4} or x > \frac{2}{4}. In the first case a \leq... | {
"cite_spans": []
} | 1804.05495 | Constructive Reverse Mathematics | [
"Hannes Diener"
] | [
"math.LO"
] | 2,018 | en | Mathematics | [
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fdf8fd02509e552bf3299a97b924fe595f27b29d | subsection | 54 | 223 | Graph colourings | We assume that the reader is familiar with basic graph theoretic definitions. A nice little theorem by Erdős and Bruijn *Theorem 8.1.3, reminiscent of the compactness theorem in logic. is
[leftmargin=2em,rightmargin=3em,skipabove=1em,skipbelow=1em, innerleftmargin=-1em,innerrightmargin=0em, nobreak=true, linecolor=blue... | {
"cite_spans": []
} | 1804.05495 | Constructive Reverse Mathematics | [
"Hannes Diener"
] | [
"math.LO"
] | 2,018 | en | Mathematics | [
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63a7cc4ddf05a89770f5f6f3b25fb7af5d16bc3b | subsection | 55 | 223 | Graph colourings | If all of G can be coloured, then if the vertices 1 and 2 get the same colour there cannot be an even n with a_n=1, which means \forall {n \in \mathbb {N}} : {a_{2n}=0}. Similarly if 1 and 2 are coloured with different colours, then \forall {n \in \mathbb {N}} : {a_{2n+1}=0}.
For an arbitrary k>2 we simply add one copy... | {
"cite_spans": []
} | 1804.05495 | Constructive Reverse Mathematics | [
"Hannes Diener"
] | [
"math.LO"
] | 2,018 | en | Mathematics | [
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6b8d82d4f373e3b4858c09f7f38995f2c1a810e3 | subsection | 56 | 223 | Graph colourings | In particular \overline{\alpha }n \in T for all n \in \mathbb {N}, which means we have found a path through T. | {
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} | 1804.05495 | Constructive Reverse Mathematics | [
"Hannes Diener"
] | [
"math.LO"
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ff7f6ecd41b846e5b7d6be8d02e899ce777cec5b | subsection | 57 | 223 | Variations of [PR:WKL]WKL | Reminiscent of the way [PR:WWKL]WWKL (Section ) is a weakening of [PR:FAN]FAN_{\Delta } is the following weakening of [PR:WKL]WKL.This version was communicated to us by M. Hendtlass by Email. We do not know who deserves credit for proposing it.
Let k<1. We now require that for each n our tree not only have at least one... | {
"cite_spans": []
} | 1804.05495 | Constructive Reverse Mathematics | [
"Hannes Diener"
] | [
"math.LO"
] | 2,018 | en | Mathematics | [
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0.... | |
9d984e4de011ee4f9e2c1cf98d75dd6dba309277 | subsection | 58 | 223 | Variations of [PR:WKL]WKL | It remains an open question whether the same holds for k \geqslant \frac{1}{2}.
quQuestion 3To what principle is [PR:WKLp]\textrm {WKL}^{\prime }(k) equivalent to for k > \frac{1}{2}?Notice that this question seems to be related to the question whether [PR:LLPOn]\textrm {LLPO}_{n} (which will be introduced in Section )... | {
"cite_spans": []
} | 1804.05495 | Constructive Reverse Mathematics | [
"Hannes Diener"
] | [
"math.LO"
] | 2,018 | en | Mathematics | [
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875431497e111fc1603e14bc84a3b55057e897a7 | subsection | 59 | 223 | Compactness of Propositional logic | The language of first order logic is, as usual, defined inductively viaEvery proposition symbol A_{0},A_{1},A_{2}, \dots is a formula.
If \alpha , \beta are formulas, then so is \lnot \alpha , \alpha \wedge \beta , \alpha \vee \beta , \alpha \rightarrow \beta .It is straightforward to show that an truth assignment \ma... | {
"cite_spans": []
} | 1804.05495 | Constructive Reverse Mathematics | [
"Hannes Diener"
] | [
"math.LO"
] | 2,018 | en | Mathematics | [
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4777dce68eaa18f7014b4d5577803d074613b902 | subsection | 60 | 223 | Compactness of Propositional logic | \end{array}\right.}For all j \leqslant i \leqslant m, sinceT is a tree \overline{u}i \in T, and therefore we have that \mathcal {B^{\prime }}\left(B_{\overline{u}i,j} \right) = \mathbf {t}.
Hence \mathcal {B^{\prime }}(\alpha _{i}) = \mathbf {t} for all i \leqslant m, which means that \mathcal {B^{\prime }} is a model ... | {
"cite_spans": []
} | 1804.05495 | Constructive Reverse Mathematics | [
"Hannes Diener"
] | [
"math.LO"
] | 2,018 | en | Mathematics | [
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153c3039c52bbcb0001e7c635f687ae3dd635095 | subsection | 61 | 223 | [PR:MP]MP and Below | We will start this chapter by introducing some notation. If a number x \in \mathbb {R} is such that \lnot \lnot (0 < x) we say that x is almost positive and write 0 \lessdot x. Furthermore, we write y \lessdot x instead of 0\lessdot x - y and x \gtrdot y interchangeably with y \lessdot x. Note that since x \leqslant y ... | {
"cite_spans": []
} | 1804.05495 | Constructive Reverse Mathematics | [
"Hannes Diener"
] | [
"math.LO"
] | 2,018 | en | Mathematics | [
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0.... | |
4e492e8376e49102077b7c0f262a0d11277ce758 | subsection | 62 | 223 | [PR:MP]MP | Markov's principle is a weak form of double negation elimination.
[leftmargin=2em,rightmargin=3em,skipabove=1em,skipbelow=1em, innerleftmargin=-1em,innerrightmargin=0em, nobreak=true, linecolor=blueish, linewidth=5pt, bottomline=false, topline=false, rightline=false, backgroundcolor=blueish!10]([PR:MP]MP)
Every al... | {
"cite_spans": []
} | 1804.05495 | Constructive Reverse Mathematics | [
"Hannes Diener"
] | [
"math.LO"
] | 2,018 | en | Mathematics | [
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ef0dda117966899a226bf417cef8208b2a4e8475 | subsection | 63 | 223 | [PR:MP]MP | Given m \in \mathbb {N} we can consider the following binary sequence c_n defined byc_n = {\left\lbrace \begin{array}{ll} 0 & \text{if } m \notin \left\lbrace a_1, b_1, \dots , a_n,b_n \right\rbrace \\ 1 & \text{otherwise} \ . \end{array}\right.}Now we have \lnot \forall {n \in \mathbb {N}} : {c_n =0}, since the assump... | {
"cite_spans": []
} | 1804.05495 | Constructive Reverse Mathematics | [
"Hannes Diener"
] | [
"math.LO"
] | 2,018 | en | Mathematics | [
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aad4e4940bc29265503ef174ddf7607a60506246 | subsection | 64 | 223 | [PR:WMP]WMP | Weak Markov's Principle states that every pseudo-positive number is positive:
[leftmargin=2em,rightmargin=3em,skipabove=1em,skipbelow=1em, innerleftmargin=-1em,innerrightmargin=0em, nobreak=true, linecolor=blueish, linewidth=5pt, bottomline=false, topline=false, rightline=false, backgroundcolor=blueish!10]([PR:WMP]WMP)... | {
"cite_spans": []
} | 1804.05495 | Constructive Reverse Mathematics | [
"Hannes Diener"
] | [
"math.LO"
] | 2,018 | en | Mathematics | [
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3e070df464f617d7096a06795142f47b63b0a9bf | subsection | 65 | 223 | [PR:WMP]WMP | Consider a x^{\prime } = \min \left\lbrace x,y \right\rbrace \in (- \infty , x]. With the previous step we know that either x^{\prime } \gtrdot 0 or x^{\prime } \lessdot x. In the first case, since y \geqslant x^{\prime }, by part REF of Lemma REF , we have y \gtrdot 0.
In the second case, by part REF of Lemma REF , we... | {
"cite_spans": []
} | 1804.05495 | Constructive Reverse Mathematics | [
"Hannes Diener"
] | [
"math.LO"
] | 2,018 | en | Mathematics | [
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2bd13b3c37c21c37b6b4b71e46b98d48ba74d9ae | subsection | 66 | 223 | [PR:WMP]WMP | Define a sequence (y_n)_{n \geqslant 1}y_n = {\left\lbrace \begin{array}{ll}
x & \text{if } \forall {i \leqslant n} : {\alpha (i) = \beta (i) } \\
0 & \text{otherwise}
\end{array}\right.}
\ .It is easy to see that (y_n)_{n \geqslant 1} is a Cauchy sequence converging to a limit y. Since x is pseudo-positive by Lemma RE... | {
"cite_spans": []
} | 1804.05495 | Constructive Reverse Mathematics | [
"Hannes Diener"
] | [
"math.LO"
] | 2,018 | en | Mathematics | [
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41bb62aea77fd0bd4b48c61701995bc9cdf7e4bf | subsection | 67 | 223 | [PR:WMP]WMP | Define a sequence (x_n)_{n \geqslant 1} in \left\lbrace a,b \right\rbrace by the following algorithm. As long as \lambda _n=0 set x_n=a. If \lambda _n=1 for the first time we can decide whether x > p d(a,b) or x < q d(a,b) . In the first case we set x_n=a from then on, in the second case we set x_n = b from then on. It... | {
"cite_spans": []
} | 1804.05495 | Constructive Reverse Mathematics | [
"Hannes Diener"
] | [
"math.LO"
] | 2,018 | en | Mathematics | [
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... | |
021b171fa8ee778931820c1a8e6819b23103aa6b | subsection | 68 | 223 | [PR:WMP]WMP | If X= \overline{ \left\lbrace a,b \right\rbrace }, then any function f:X \rightarrow \mathbb {R} is strongly extensional.Obviously REF implies REF .Assume f:2^{\mathbb {N}}\rightarrow \mathbb {R} is a function with f(\alpha ) \ne f(\beta ) If \gamma \in \mathbb {N}^{\mathbb {N}}, then let \gamma ^{01} \in 2^{\mathbb {N... | {
"cite_spans": []
} | 1804.05495 | Constructive Reverse Mathematics | [
"Hannes Diener"
] | [
"math.LO"
] | 2,018 | en | Mathematics | [
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98fc1e79478c7632684a1ae26498d244e1581114 | subsection | 69 | 223 | [PR:WMP]WMP | Now we can consider the function g:\mathbb {N}_{\infty }\rightarrow \mathbb {R} defined by g(\alpha ) = f(x^\alpha _\infty ). Furthermore, it is easy to see that x^{(000\ldots )}_\infty = a and x^{\lambda }_\infty = b. Hence \lambda \ne 000\ldots and thus a \ne b.Next, we will show that REF implies REF . So let f:Y \ri... | {
"cite_spans": []
} | 1804.05495 | Constructive Reverse Mathematics | [
"Hannes Diener"
] | [
"math.LO"
] | 2,018 | en | Mathematics | [
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f4f0341b1b034a6b3c847bb8430edf62ba36679a | subsection | 70 | 223 | Body | The disjunctive version of Markov's principle [PR:MPv]{\textrm {MP}^{\vee }} can be seen as an instance of de'Morgan's laws (see Section ) as well as another weakening of [PR:MP]MP. It states that every almost positive number is pseudo-positive.[leftmargin=2em,rightmargin=3em,skipabove=1em,skipbelow=1em, innerleftmargi... | {
"cite_spans": []
} | 1804.05495 | Constructive Reverse Mathematics | [
"Hannes Diener"
] | [
"math.LO"
] | 2,018 | en | Mathematics | [
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7c7515447b2e0aa9cfb4b7a0feaf726117110ecb | subsection | 71 | 223 | Body | If \alpha is a binary sequence such that \lnot \forall {n \in \mathbb {N}} : {\alpha (n)=0}, then for any \beta we have
\forall {\beta \in 2^{\mathbb {N}}} : {\lnot (\alpha = \beta ) \vee \lnot (\beta = 0)}
For \alpha ,\beta \in 2^{\mathbb {N}}
\lnot (\forall {n \in \mathbb {N}} : {\alpha _n =0} \wedge \forall {n ... | {
"cite_spans": []
} | 1804.05495 | Constructive Reverse Mathematics | [
"Hannes Diener"
] | [
"math.LO"
] | 2,018 | en | Mathematics | [
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23609059ef607e95787dfb39a6514b41dac6758b | subsection | 72 | 223 | Body | To see that \ref {MPo:3} \Rightarrow \ref {MPo:4} let \alpha be a binary sequence such that \lnot \forall {n \in \mathbb {N}} : {\alpha _n=0}, and \beta \in 2^{\mathbb {N}} be arbitrary. Now define \gamma \in 2^{\mathbb {N}} by\gamma _{2n}=1 & \Rightarrow \alpha _n \ne \beta _n \ , \\
\gamma _{2n+1}=1 & \Rightarrow \be... | {
"cite_spans": []
} | 1804.05495 | Constructive Reverse Mathematics | [
"Hannes Diener"
] | [
"math.LO"
] | 2,018 | en | Mathematics | [
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a3b752d95211949b082e7fffe47dd510be8d5fbc | subsection | 73 | 223 | Body | So we have decided\lnot (\alpha = \beta ) \vee \lnot (\beta = 000\ldots ) \ .To see that \ref {MPo:4} \Rightarrow \ref {MPo:8} let \alpha , \beta be a binary sequences such that\lnot (\forall {n \in \mathbb {N}} : {\alpha _n =0} \wedge \forall {n \in \mathbb {N}} : {\beta _n =0}) \ .Now consider the sequences\gamma = \... | {
"cite_spans": []
} | 1804.05495 | Constructive Reverse Mathematics | [
"Hannes Diener"
] | [
"math.LO"
] | 2,018 | en | Mathematics | [
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0.... | |
8e3af1ed961b8fd6709dede6a55b5dc7de8ef192 | subsection | 74 | 223 | Body | Because of the similarity of [PR:LLPO]LLPO and [PR:MPv]{\textrm {MP}^{\vee }} it is not surprising that we can find a weakening of [PR:WKL]WKL that is equivalent to the latter.
[leftmargin=2em,rightmargin=3em,skipabove=1em,skipbelow=1em, innerleftmargin=-1em,innerrightmargin=0em, nobreak=true, linecolor=blueish, linewi... | {
"cite_spans": []
} | 1804.05495 | Constructive Reverse Mathematics | [
"Hannes Diener"
] | [
"math.LO"
] | 2,018 | en | Mathematics | [
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c2a67539ae7e9adec2a04069ea0cff5b5e4a9df0 | subsection | 75 | 223 | Body | In particular \overline{\alpha }n \in T.Conversely consider a binary sequence (a_n)_{n \geqslant 1} with at most one 1 and such that \lnot \forall {n \in \mathbb {N}} : {a_n = 0}. Now consider the decidable tree T defined byu \in T \iff \exists {n \in \mathbb {N}} : {u = 0^n \wedge \forall {i \leqslant n} : {a_{2i} = 0... | {
"cite_spans": []
} | 1804.05495 | Constructive Reverse Mathematics | [
"Hannes Diener"
] | [
"math.LO"
] | 2,018 | en | Mathematics | [
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0.... | |
6047ae48ec85e4b5d97c0c7dce009d96a7da23be | subsection | 76 | 223 | Body | Finally, it is also easy to see that T has at most one path. However if T actually admits a path \alpha we can check whether \alpha (1)=0, in which case \lnot \forall {n \in \mathbb {N}} : {a_n=0}, or \alpha (1)=1, in which case \forall {n \in \mathbb {N}} : {a_n=0}. Thus [PR:WLPO]WLPO holds, which contradicts the exis... | {
"cite_spans": []
} | 1804.05495 | Constructive Reverse Mathematics | [
"Hannes Diener"
] | [
"math.LO"
] | 2,018 | en | Mathematics | [
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48ffaf2969c02426bcd514fadfb182e89c59ea37 | subsection | 77 | 223 | Body | \forall { x,y \in \mathbb {R}} : { (x > 0 \iff y = 0) \Rightarrow \lnot (x = 0) \vee y=0}.There are no other, more interesting, equivalences known, at this stage.tocsection1½ [PR:BD]BD
At the same time as investigating [PR:BDN]BD-N H. Ishihara also introduced the stronger principle [PR:BD]BD which does not require a co... | {
"cite_spans": []
} | 1804.05495 | Constructive Reverse Mathematics | [
"Hannes Diener"
] | [
"math.LO"
] | 2,018 | en | Mathematics | [
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0... | |
06b456cd1c38bbe0ec182f5b6ee2b380726d199d | subsection | 78 | 223 | Body | Notice that \Pi _{1}^0 statements are stable, that is we can eliminate preceding double negations.The equivalencies for the restricted versions of [Pr:DGP]DGP are non-trivial, and the proofs are entertaining: First assume that [PR:LLPO]LLPO holds. And consider two binary sequences a_n and b_n. Now let \alpha be the seq... | {
"cite_spans": []
} | 1804.05495 | Constructive Reverse Mathematics | [
"Hannes Diener"
] | [
"math.LO"
] | 2,018 | en | Mathematics | [
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843f08c90411670f244e9d8cd1dfccbd256270a3 | subsection | 79 | 223 | Body | Similarly in the second case we can show that \forall {n \in \mathbb {N}} : {a_{2n+1} =0}. Thus we have shown that \Sigma _1^0-[Pr:DGP]DGP implies [PR:LLPO]LLPO.Next, let a_n and b_n be binary sequences and construct \alpha and \alpha ^\prime as above. Using [PR:LLPO]LLPO we can, again, make the decision in Equation (R... | {
"cite_spans": []
} | 1804.05495 | Constructive Reverse Mathematics | [
"Hannes Diener"
] | [
"math.LO"
] | 2,018 | en | Mathematics | [
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265602cbe1b88a80e9bc3f877fbed44f6f28f00c | subsection | 80 | 223 | Body | In the first case, also \exists {n \in \mathbb {N}} : {b_n =1} \Rightarrow \forall {n \in \mathbb {N}} : {a_n =0}. In the second case \forall {n \in \mathbb {N}} : {a_n =0} \Rightarrow \exists {n \in \mathbb {N}} : {b_n =1}, since the antecedent contradicts our assumption.Conversely, let a_n be an arbitrary binary sequ... | {
"cite_spans": []
} | 1804.05495 | Constructive Reverse Mathematics | [
"Hannes Diener"
] | [
"math.LO"
] | 2,018 | en | Mathematics | [
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340b7a49c850717c3bfddeac927402de26a2012b | subsection | 81 | 223 | Body | Every searchable subset of \mathbb {N} is bounded.Assume \lnot \textrm {LPO} and let S \subset \mathbb {N} be searchable. Assume furthermore that S is unbounded, that is, with a bit of work, there exists an bijection s: \mathbb {N}\rightarrow S. Then, if a_n is an arbitrary binary sequence consider p: S \rightarrow 2 d... | {
"cite_spans": []
} | 1804.05495 | Constructive Reverse Mathematics | [
"Hannes Diener"
] | [
"math.LO"
] | 2,018 | en | Mathematics | [
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e4dd5d744e83de7f28b14fafcd656faa3dc3b761 | subsection | 82 | 223 | The Fan Theorems | As a way to re-capture the unit interval's compactness—that is cover compactness—which was lost when rejecting the law of excluded middle, L.E.J. Brouwer made generous use of the fan theorem. Since he also made free use of the principle of continuous choice the complexity of the sets involved did not make a difference ... | {
"cite_spans": []
} | 1804.05495 | Constructive Reverse Mathematics | [
"Hannes Diener"
] | [
"math.LO"
] | 2,018 | en | Mathematics | [
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3f57c55c0ca48f8c1fb4bbfa7f9325b41065eb14 | subsection | 83 | 223 | The Fan Theorems | We can now formally state the four versions of the fan theorem, that are going to be of interest to us.
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"cite_spans": []
} | 1804.05495 | Constructive Reverse Mathematics | [
"Hannes Diener"
] | [
"math.LO"
] | 2,018 | en | Mathematics | [
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5a4caa91b8c9657b8953c364926a132d3ea39b7f | subsection | 84 | 223 | The Fan Theorems | Since \overline{\alpha }M \in B, in particular, \overline{\alpha }M \in B_{M+{w}}. Since {w} \leqslant (M + {w}) - M, we have \overline{\alpha }M \ast w \in C^\prime , and since w was arbitrary that means that \overline{\alpha }M \in C. Thus C is a uniform bar.[PR:FAN]FAN_{\Delta }, and therefore all of the fan theorem... | {
"cite_spans": []
} | 1804.05495 | Constructive Reverse Mathematics | [
"Hannes Diener"
] | [
"math.LO"
] | 2,018 | en | Mathematics | [
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c814d612ea237831c669e3780478042fced61be1 | subsection | 85 | 223 | The Fan Theorems | Notice that some early papers on the topic have used a more refined notation: what is labeled [PR:AS]\textrm {AS}_{}^{\, } there is what we label [PR:AS]\textrm {AS}_{[0,1]}^{\, \mathbb {R}}, and which is equivalent to [PR:AS]\textrm {AS}_{[0,1]}^{\, 1}. | {
"cite_spans": []
} | 1804.05495 | Constructive Reverse Mathematics | [
"Hannes Diener"
] | [
"math.LO"
] | 2,018 | en | Mathematics | [
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aae1cdd6536c1a68916a1cd75e6953ca80991731 | subsection | 86 | 223 | Linking | In the following we want to establish strong links between 2^{\mathbb {N}} and [0,1].
We will first adapt Cantor's middle third set construction for our purposes.
First consider a fixed p \in (0,1).
LetI^{p}_{u} = [a^{p}_{u},b^{p}_{u}] = *{ (1-p) \sum _{n \leqslant {u}} p^{n-1}u(n), (1-p) \sum _{n \leqslant {u}} p^{n-1... | {
"cite_spans": []
} | 1804.05495 | Constructive Reverse Mathematics | [
"Hannes Diener"
] | [
"math.LO"
] | 2,018 | en | Mathematics | [
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fa86825a9bec2afb0ae5bb8bd17397564b0ac14a | subsection | 87 | 223 | Linking | Then
*{F^{p}(\alpha ) - F^{p}(\beta )} & \leqslant (1-p) \sum _{i \geqslant 1} p^{i-1}*{\alpha (i) - \beta (i)} \\ & \leqslant (1-p) \sum _{i > n} p^{i} \\ & = (1-p) p^n \sum _{i \geqslant 0} p^{i} \\
& = \frac{1-p}{1-p} p^n = p^n \ .
Consider 0< p <\frac{1}{2}. We will define subsets J^{p}_{u} and positive numbers ... | {
"cite_spans": []
} | 1804.05495 | Constructive Reverse Mathematics | [
"Hannes Diener"
] | [
"math.LO"
] | 2,018 | en | Mathematics | [
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5ea5a529777b12f022d8a3cc877f434c6e6f6b51 | subsection | 88 | 223 | Linking | Since F(\beta ) \in I^{p}_{\overline{\beta }n} and x \in I^{p}_{\overline{\alpha }n}, by REF we have F(\beta )< x. The case \alpha <\beta can be treated analogously.
To see that F^{p} is injective let \gamma \ne \beta . Now let \alpha be as constructed above for x=F^{p}(\beta ). Then the assumption that \alpha \ne \bet... | {
"cite_spans": []
} | 1804.05495 | Constructive Reverse Mathematics | [
"Hannes Diener"
] | [
"math.LO"
] | 2,018 | en | Mathematics | [
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5920b8ced069a696730883e4cebcc373cde70f48 | subsection | 89 | 223 | Linking | It is easy to see that therefore F^{p}(\alpha ) = x, which means we have shown surjectivity.Remark 1.2 Notice that for p=\frac{1}{2} the function F^{p} cannot be shown to be surjective, constructively, since that would be a restatement of the fact that every real number x\in [0,1] has a binary expansion and therefore e... | {
"cite_spans": []
} | 1804.05495 | Constructive Reverse Mathematics | [
"Hannes Diener"
] | [
"math.LO"
] | 2,018 | en | Mathematics | [
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286c662415eacb87652ae0667d14923f3a3a72d3 | subsection | 90 | 223 | Linking | Below, we need a slightly stronger version of this result.Definition 1.4 A map g:X \rightarrow Y between to metric spaces X and Y is called uniformly surjective if\forall {\varepsilon >0} : {\exists { \delta >0} : { \forall {x,y \in Y} : {d(x,y)<\delta \Rightarrow \exists {\alpha ,\beta \in Y } : { d(\alpha ,\beta )<\v... | {
"cite_spans": []
} | 1804.05495 | Constructive Reverse Mathematics | [
"Hannes Diener"
] | [
"math.LO"
] | 2,018 | en | Mathematics | [
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08edfb073079462beee932f55c71cd4c36af9389 | subsection | 91 | 223 | Linking | But the situation is actually much worse since even a function as well behaved and canonical as F^p with p > {1}{2} doesn't have the property of mapping complete sets to complete sets.Lemma 1.7 If, for p > {1}{2}, the function F^p:2^{\mathbb {N}}\rightarrow [0,1] maps complete sets to complete sets (i.e. is a closed ma... | {
"cite_spans": []
} | 1804.05495 | Constructive Reverse Mathematics | [
"Hannes Diener"
] | [
"math.LO"
] | 2,018 | en | Mathematics | [
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e45147e1f88368803dc35fe069ed57e922b6f779 | subsection | 92 | 223 | Linking | The above example suggests that connectedness may play a role.Definition 1.8 We call a sequence (x_n)_{n \geqslant 1} tail-located if the distances d(x,\left\lbrace x_n, x_{n+1}, \dots \right\rbrace ) exist for every n \geqslant 1 and x \in \mathbb {R}.Lemma 1.9
If (x_n)_{n \geqslant 1} is a sequence of real numbers t... | {
"cite_spans": []
} | 1804.05495 | Constructive Reverse Mathematics | [
"Hannes Diener"
] | [
"math.LO"
] | 2,018 | en | Mathematics | [
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640123fc34e33a18af5434ef53b76850ac2c8657 | subsection | 93 | 223 | Linking | Hencew_n \in 2^{\ast }\Rightarrow *{F^{{1}{2}}(v_n) - x_n} < 2^{-n+1} \for all n \in \mathbb {N}. Since *{v_n} = n for all n, the set D=\left\lbrace v_{1}, v_{2}, \dots \right\rbrace \cap 2^{\ast } is decidable. Therefore the setB = \mbox{$\left\lbrace \,u \in 2^{\ast } \, | \,\lnot \exists {w \in 2^{\ast }} : {u \ast ... | {
"cite_spans": []
} | 1804.05495 | Constructive Reverse Mathematics | [
"Hannes Diener"
] | [
"math.LO"
] | 2,018 | en | Mathematics | [
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a75b6083329088090e1dcae2c5c7eb9fddc837f0 | subsection | 94 | 223 | Linking | Assume that (x_n)_{n \geqslant 1} is tail-located, and let u \in 2^{\ast } be arbitrary. Let a,b be the endpoints of F^{{1}{2}}(B_u) that is a = F^{{1}{2}}(u) and b =F^{{1}{2}}(u\ast 1 \ast 1 \ast \dots ).That is with the notation of Section a=a^{{1}{2}}_u and b=b^{{1}{2}}_u.
Choose N and \delta >0 such that x_n is bou... | {
"cite_spans": []
} | 1804.05495 | Constructive Reverse Mathematics | [
"Hannes Diener"
] | [
"math.LO"
] | 2,018 | en | Mathematics | [
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ced109ef6b355b03e08947b1341f067e05d2d16e | subsection | 95 | 223 | Linking | Let \eta : \mathbb {N}\rightarrow 2^{\ast } be a bijection. In particular, that means that we have\forall {n \in \mathbb {N}} : {\exists {m \in \mathbb {N}} : {i \geqslant m \Rightarrow *{\eta (i)} \geqslant n}} \ .We may also assume that i \leqslant j \Rightarrow *{\eta (i)} \leqslant *{\eta (j)}.The function mapping ... | {
"cite_spans": []
} | 1804.05495 | Constructive Reverse Mathematics | [
"Hannes Diener"
] | [
"math.LO"
] | 2,018 | en | Mathematics | [
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84cf6f6240ef411517527d70f1f407df6a5d62e6 | subsection | 96 | 223 | Linking | In both cases *{x_{i} - x} > \min \left\lbrace \delta , 3^{-(n+2)} \right\rbrace for all i \geqslant m, so (x_n)_{n \geqslant 1} is eventually bounded away from every x \in [0,1].So let us tackle the three numbered assertions.Assume that (x_n)_{n \geqslant 1} is eventually bounded away from the entire set [0,1]; say x_... | {
"cite_spans": []
} | 1804.05495 | Constructive Reverse Mathematics | [
"Hannes Diener"
] | [
"math.LO"
] | 2,018 | en | Mathematics | [
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7183db219ecc33a0d38267442d013abe4626b9e9 | subsection | 97 | 223 | Linking | The sets F_{n,m} are, for m \geqslant M, a finite 2^{-m}-approximation of A_n: for let i \geqslant n with \eta (i) \notin B and let j be such that \eta (j) = \overline{\eta (i)} m. By REF and REF we have n \leqslant j \leqslant k_{m}. Since B is closed under extensions \eta (j) \notin B. Hence \eta (j) \in F_{n,M} and ... | {
"cite_spans": []
} | 1804.05495 | Constructive Reverse Mathematics | [
"Hannes Diener"
] | [
"math.LO"
] | 2,018 | en | Mathematics | [
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8e7fddf22cc793bf4859eac32fef53d451e4ef0a | subsection | 98 | 223 | [PR:WWKL]WWKL | We start with a principle that is a weakening of [PR:FAN]FAN_{\Delta }.
The so called weak weak König's lemma ([PR:WWKL]WWKL) plays a role in Simpson style reverse mathematics . The name is somewhat misleading since it is not resembling weak König's lemma (see Section ) but rather its contrapositive i.e. the fan theore... | {
"cite_spans": []
} | 1804.05495 | Constructive Reverse Mathematics | [
"Hannes Diener"
] | [
"math.LO"
] | 2,018 | en | Mathematics | [
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6d3ff4e0f5223a9bb95bfb7e8f0fd98550b9627f | subsection | 99 | 223 | [PR:WWKL]WWKL | It seems reasonable to conjecture that WWKL has more equivalents in measure theory, which at the moment has not received much attention in constructive analysis.Proposition 2.2 [PR:WWKL]WWKL is equivalent to the following “weaker” version for every k>0
[leftmargin=2em,rightmargin=3em,skipabove=1em,skipbelow=1em, innerl... | {
"cite_spans": []
} | 1804.05495 | Constructive Reverse Mathematics | [
"Hannes Diener"
] | [
"math.LO"
] | 2,018 | en | Mathematics | [
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0.0062144217081367... | |
66e7ad627b6346e9de0f97bd4d47b8a025f17310 | subsection | 100 | 223 | [PR:WWKL]WWKL | Hence*{\mbox{$\left\lbrace \,u \in 2^{N+M} \, | \, u \notin B \,\right\rbrace $}}
& = *{ \bigcup _{i=1}^{m} \mbox{$\left\lbrace \, u_{i}w \in 2^{N+M} \, | \, w \notin B^{(i)} \,\right\rbrace $}} \\
& = \sum _{i=1}^{m}*{\mbox{$\left\lbrace \, w \in 2^{M} \, | \, w \notin B^{(i)} \,\right\rbrace $}} \\
& < m k2^{M} < k2^... | {
"cite_spans": []
} | 1804.05495 | Constructive Reverse Mathematics | [
"Hannes Diener"
] | [
"math.LO"
] | 2,018 | en | Mathematics | [
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c250b7512204d04ca6d13ed187f97ba28e94a3f1 | subsection | 101 | 223 | [PR:FAN]FAN | As mentioned in the introduction [PR:FAN]FAN_{\Delta } deserves the prominence of being involved in the first “proper” equivalence of constructive reverse mathematics. It is also a fairly robust statement, as the next lemma shows.Lemma 3.1
If B is decidable bar, then there exists a decidable bar B^{\prime } that is cl... | {
"cite_spans": []
} | 1804.05495 | Constructive Reverse Mathematics | [
"Hannes Diener"
] | [
"math.LO"
] | 2,018 | en | Mathematics | [
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791897f224053831c386588689c30e457e6dd164 | subsection | 102 | 223 | [PR:FAN]FAN | To see that REF implies REF let A,B be subsets such that d(a,b)>0 for all a \in A and b \in B. By *Chapter 7 Corollary 4.4 there exists surjective, uniformly continuous functions g_1:2^{\mathbb {N}}\rightarrow A and g_2:2^{\mathbb {N}}\rightarrow B. Then the function h: 2^{\mathbb {N}}\rightarrow \mathbb {R} defined by... | {
"cite_spans": []
} | 1804.05495 | Constructive Reverse Mathematics | [
"Hannes Diener"
] | [
"math.LO"
] | 2,018 | en | Mathematics | [
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b65e7c969edef8dace3a8fe8ca557c681ae0a65d | subsection | 103 | 223 | [PR:FAN]FAN | Notice that either u_n = 0 \ast \dots \ast 0 or we can find its immediate “left” neighbour u_{L(n)} that is there exists w such that u_n=w\ast 1 \ast 0^k and u_{L(n)} = w \ast 0 \ast 1^\ell for some k,\ell \geqslant 0. We may assume that u_1 = 1 \ast \dots \ast 1. Now define a function f(0,1) \rightarrow \mathbb {R} by... | {
"cite_spans": []
} | 1804.05495 | Constructive Reverse Mathematics | [
"Hannes Diener"
] | [
"math.LO"
] | 2,018 | en | Mathematics | [
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9201db591ca30a4c17582831668a3ef5940460c0 | subsection | 104 | 223 | [PR:FAN]FAN | First we need a lemma, whose proof idea is based on the proof of the main result in the paper mentioned.Lemma 3.6
For every f:2^{\mathbb {N}}\rightarrow \mathbb {R} that has a continuous (functional) modulus of continuity and for every e \in \mathbb {N}, there exists a decidable bar B that is uniform only if
there exi... | {
"cite_spans": []
} | 1804.05495 | Constructive Reverse Mathematics | [
"Hannes Diener"
] | [
"math.LO"
] | 2,018 | en | Mathematics | [
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0.0093... | |
55fa5e37565db3a1bd90189264a240c95475cf54 | subsection | 105 | 223 | [PR:FAN]FAN | So there exists N \in \mathbb {N} such that\forall {\alpha \in 2^{\mathbb {N}}} : { \exists {n \leqslant N} : {\overline{\alpha }n \in B} } \ .So for any \alpha \in 2^{\mathbb {N}} there is n \leqslant N such that \mu (\overline{\alpha }n\ast 000\ldots ,e ) \leqslant n \leqslant N, and hence for all \beta \in 2^{\mathb... | {
"cite_spans": []
} | 1804.05495 | Constructive Reverse Mathematics | [
"Hannes Diener"
] | [
"math.LO"
] | 2,018 | en | Mathematics | [
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bfa5318e937d170fa39864d0ff8340d3affaeac4 | subsection | 106 | 223 | [PR:FAN]FAN | Using countable choice we can therefore construct \tau : \mathbb {N}\rightarrow \mathbb {N} such that\forall {\alpha , \beta \in 2^{\mathbb {N}}} : {\forall {n \in \mathbb {N}} : {\overline{\alpha }\tau (n) = \overline{\beta }\tau (n) \Rightarrow d(g(\alpha ),g(\beta )) < 2^{-n}}} \ .Furthermore, for \alpha \in 2^{\mat... | {
"cite_spans": []
} | 1804.05495 | Constructive Reverse Mathematics | [
"Hannes Diener"
] | [
"math.LO"
] | 2,018 | en | Mathematics | [
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f726489baea792caf8335274c3285c0f47a29852 | subsection | 107 | 223 | [PR:FAN]FAN | Furthermore \overline{\gamma }n = \overline{\tau }n, and hence\varepsilon & > *{h(\gamma ) - h(\tau )} \\
& = *{ d(f(h(\gamma ^{e})),f(h(\gamma ^{o}))) - d(f(h(\tau ^{e})),f(h(\tau ^{o})))} \\
& = *{ d(f(h(\alpha )),f(h(\beta ))) - d(f(h(\alpha )),f(h(\alpha )))} \\
& = d(f(h(\alpha )),f(h(\beta ))) = d(f(x),f(y)) \ .T... | {
"cite_spans": []
} | 1804.05495 | Constructive Reverse Mathematics | [
"Hannes Diener"
] | [
"math.LO"
] | 2,018 | en | Mathematics | [
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0.0... | |
757a542172565b0b2e24b14942a276a1f13c3003 | subsection | 108 | 223 | [PR:FAN]FAN | Straightforward.Lemma 3.10
For every decidable bar B there exists a point-wise continuous, fully located function f:[0,1] \rightarrow \mathbb {R} such that f is bounded if and only if B is uniform; and vice versa.First, start with a decidable bar B that is without loss of generality closed under extension. We are goin... | {
"cite_spans": []
} | 1804.05495 | Constructive Reverse Mathematics | [
"Hannes Diener"
] | [
"math.LO"
] | 2,018 | en | Mathematics | [
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bc93ddd4e7cc889b02f668a9a752d54a94521e69 | subsection | 109 | 223 | [PR:FAN]FAN | Since F^{{2}{3}} is surjective on I_{\overline{\alpha }M} (Lemma REF ) there exists \beta \in 2^{\mathbb {N}} such that F^{{2}{3}}(\beta ) = y and \overline{\beta }M = \overline{\alpha }M. But that means thatf \circ F^{{2}{3}}(\alpha )+ *{f \circ F^{{2}{3}}(\alpha ) - f \circ F^{{2}{3}}(\beta )} > f \circ F^{{2}{3}}(\b... | {
"cite_spans": []
} | 1804.05495 | Constructive Reverse Mathematics | [
"Hannes Diener"
] | [
"math.LO"
] | 2,018 | en | Mathematics | [
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d0d7c04a75dbd6c511ae12ff0efbd5e0cbcd4e91 | subsection | 110 | 223 | [PR:FAN]FAN | Every point-wise continuous mapping of [0, 1] into a metric space is uniformly sequentially continuous.Even though the equivalence {[PR:FAN]{\textrm {FAN}_{c}} \iff [PR:AS]{\textrm {AS}_{[0,1]}^{\, \mathbb {R}}}} has been shown in it also follows from Lemmas REF and REF . For the equivalence between [PR:AS]\textrm {AS}... | {
"cite_spans": []
} | 1804.05495 | Constructive Reverse Mathematics | [
"Hannes Diener"
] | [
"math.LO"
] | 2,018 | en | Mathematics | [
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0e4689008e04da2479ce4abaf9cee317a2297830 | subsection | 111 | 223 | [PR:FAN]FAN | But by the choice of M, also for any \beta we have g(\overline{\alpha }(2M) \ast \beta ) = g(\alpha ), and hence g is point-wise continuous. By our assumption it is therefore uniformly continuous, which means we can find an M as above which is independent of \alpha , which immediately gives us uniform continuity of f.F... | {
"cite_spans": []
} | 1804.05495 | Constructive Reverse Mathematics | [
"Hannes Diener"
] | [
"math.LO"
] | 2,018 | en | Mathematics | [
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0... | |
96d734afbc2c18abe90ae299350b897d86fe16f0 | subsection | 112 | 223 | [PR:FAN]FAN | Then the convergence is uniform.One direction follows from the fact that [PR:FAN]FAN_{c} implies [PR:FAN]FAN_{\Delta }, that [PR:FAN]FAN_{c} implies that a point-wise continuous function is uniformly continuous (see REF of REF ), so we can use Proposition REF .REF .For the other direction, by the previous proposition, ... | {
"cite_spans": []
} | 1804.05495 | Constructive Reverse Mathematics | [
"Hannes Diener"
] | [
"math.LO"
] | 2,018 | en | Mathematics | [
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da24d8cf927ca3b66e91b60521e060e6244f1e5d | subsection | 113 | 223 | [PR:FAN]FAN | The unproven (and probably false) assumption that [PR:FANst]FAN_{\textrm {stable}} and [PR:FAN]FAN_{\smash{\Pi ^{0}_{1}}} are equivalent can be explained by Lemma REF taken together with the following observation.Proposition 6.1 If a stable bar B is closed under extensions, then B is a \Pi _{1}^{0}-bar.This proof is b... | {
"cite_spans": []
} | 1804.05495 | Constructive Reverse Mathematics | [
"Hannes Diener"
] | [
"math.LO"
] | 2,018 | en | Mathematics | [
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0... | |
dc0699bfb6db75fe0a39a04e052f6fab7888a9bc | subsection | 114 | 223 | [PR:FAN]FAN | Notice that it would be enough to show that the closure (under extensions) of a stable bar is stable.Our first [PR:FAN]FAN_{\smash{\Pi ^{0}_{1}}}-equivalence is a sequential version of REF .REF .Proposition 6.2
[PR:FAN]FAN_{\smash{\Pi ^{0}_{1}}} is equivalent to the following statement:Every equi-continuous, and equi-... | {
"cite_spans": []
} | 1804.05495 | Constructive Reverse Mathematics | [
"Hannes Diener"
] | [
"math.LO"
] | 2,018 | en | Mathematics | [
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a71293c63a8176650dbf5ad6d6f50323c4c36763 | subsection | 115 | 223 | [PR:FAN]FAN | For any k \in \mathbb {N} and w \in 2^{\ast } we get thatf_{k}(\overline{\alpha }K \ast w )
& > f_{k}(\alpha ) - *{f_{k}(\overline{\alpha }K \ast w) - f_{k}(\alpha )} \ , \\
& > 2^{-(N-1)} - 2^{- N} \\
& > 2^{-N} > 2^{-K} \ .Hence \overline{\alpha }K \in B_{n,k}, since case REF is ruled out. Therefore B is a bar. By co... | {
"cite_spans": []
} | 1804.05495 | Constructive Reverse Mathematics | [
"Hannes Diener"
] | [
"math.LO"
] | 2,018 | en | Mathematics | [
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37ce189eb56dd34db78f42493873e971d526da78 | subsection | 116 | 223 | [PR:FAN]FAN | Since the sets B_{k} are closed under extensions also \overline{\alpha }N \in B_{k} for all n \in \mathbb {N} and \alpha \in 2^{\mathbb {N}}. Therefore \overline{\alpha }N \in B for all \alpha \in 2^{\mathbb {N}}, which means B is uniform.Proposition 6.3 The following are equivalent to [PR:FAN]FAN_{\smash{\Pi ^{0}_{1}}... | {
"cite_spans": []
} | 1804.05495 | Constructive Reverse Mathematics | [
"Hannes Diener"
] | [
"math.LO"
] | 2,018 | en | Mathematics | [
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74961807b4f3fce181b89371cfe3524b41ec5ee7 | subsection | 117 | 223 | [PR:FAN]FAN | A proof can be found in .Proposition 6.5 [PR:FAN]FAN_{\smash{\Pi ^{0}_{1}}} is equivalent to the statement that every equi-continous sequence of functions [0,1] \rightarrow \mathbb {R} is uniformly equi-continuous.It seems feasible to also prove this theorem for functions functions of type 2^{\mathbb {N}}\rightarrow \m... | {
"cite_spans": []
} | 1804.05495 | Constructive Reverse Mathematics | [
"Hannes Diener"
] | [
"math.LO"
] | 2,018 | en | Mathematics | [
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0.0... | |
9e560db5ac5de487b875fb7affe878a49557f291 | subsection | 118 | 223 | [PR:FAN]FAN | Since g^{-1}(U_i) is open there exists n such that \overline{\alpha }n \ast \beta \in g^{-1}(U_i) for all \beta \in 2^{\mathbb {N}}. So B is a bar. Applying [PR:FAN]FAN_{\textrm {full}} yields n such that u \in B for all u \in 2^n. That means we can find finitely many i_u such that (g^{-1}(U_{i_u}))_{u \in 2^n} covers ... | {
"cite_spans": []
} | 1804.05495 | Constructive Reverse Mathematics | [
"Hannes Diener"
] | [
"math.LO"
] | 2,018 | en | Mathematics | [
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0... | |
97ca13aa38c5657d1dbbc0970ba7689fc1c16028 | subsection | 119 | 223 | [PR:UCT]UCT | The uniform continuity theorem is the standard first step in numerous theorems of classical analysis. It states:
[leftmargin=2em,rightmargin=3em,skipabove=1em,skipbelow=1em, innerleftmargin=-1em,innerrightmargin=0em, nobreak=true, linecolor=blueish, linewidth=5pt, bottomline=false, topline=false, rightline=false, backg... | {
"cite_spans": []
} | 1804.05495 | Constructive Reverse Mathematics | [
"Hannes Diener"
] | [
"math.LO"
] | 2,018 | en | Mathematics | [
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0.0... | |
9d7892144e111a03a8449b08018d55b4a2a507f4 | subsection | 120 | 223 | [PR:UCT]UCT | Somehow, just the fact that there are countably many of them suffices.Proposition 5.4
[PR:UCT]UCT is equivalent to C([0,1]) being separable.More precise: [PR:UCT]UCT is equivalent to the statement that there exists a sequence of point-wise continuous functions f_n:[0,1] \rightarrow \mathbb {R} such that for all \varep... | {
"cite_spans": []
} | 1804.05495 | Constructive Reverse Mathematics | [
"Hannes Diener"
] | [
"math.LO"
] | 2,018 | en | Mathematics | [
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0.010213470086455345,... | |
f545174b796d6c08cf69f0f2b2798b66dd2848b7 | subsection | 121 | 223 | [PR:UCT]UCT | It is not too surprising that we can also prove the following version.Proposition (5.5½ version of Dini's theorem) [PR:UCT]UCT is equivalent to the statement that
if (f_n)_{n \geqslant 1}:[0,1] \rightarrow \mathbb {R} is a decreasing sequence of point-wise continuous functions converging point-wise to a point-wise cont... | {
"cite_spans": []
} | 1804.05495 | Constructive Reverse Mathematics | [
"Hannes Diener"
] | [
"math.LO"
] | 2,018 | en | Mathematics | [
-0.037960510700941086,
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0.008246646262705326,
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9f3d54521ce04ddbad38d99eec09a3c63a15ef38 | subsection | 122 | 223 | Comparing the Fan Theorems | The differences between the fan theorems are overall very subtle and often confusingly minute. Since many of the results in this section are variations of each other, we hope that the following table might highlight some of the differences.1mylightgray
[Table: NO_CAPTION]The following abbreviations are used:
2pwc: poin... | {
"cite_spans": []
} | 1804.05495 | Constructive Reverse Mathematics | [
"Hannes Diener"
] | [
"math.LO"
] | 2,018 | en | Mathematics | [
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0.03444407507777214,
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-0.02817458286881447,
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... | |
d417fd4e59c22af566983b221cd5e81595d34954 | subsection | 123 | 223 | [PR:BDN]BD-N and Below | Future work: include
Together with Weak Markov's principle (Section ), [PR:BDN]BD-N occupies a special and rare place in constructive mathematics: it is accepted in CLASS, in INT, as well as in RUSS. However, it is not accepted in BISH.
One might think that after reading Proposition REF , which says that [PR:BDN]BD-N ... | {
"cite_spans": []
} | 1804.05495 | Constructive Reverse Mathematics | [
"Hannes Diener"
] | [
"math.LO"
] | 2,018 | en | Mathematics | [
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0.040276285260915756,
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0.018520988523960114... | |
9270fbd162b343597f1b4bcb7bcf755ba9b9f69c | subsection | 124 | 223 | [PR:BDN]BD-N | A subset S of \mathbb {N} is pseudobounded if\lim _{n \rightarrow \infty } \frac{s_n}{n} = 0for each sequence (s_n)_{n \geqslant 1} in S. This is equivalent to assuming that for every sequence (s_n)_{n \geqslant 1} in S we haves_n < neventually.Trivially, every bounded set is pseudobounded, but the converse is more sub... | {
"cite_spans": []
} | 1804.05495 | Constructive Reverse Mathematics | [
"Hannes Diener"
] | [
"math.LO"
] | 2,018 | en | Mathematics | [
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0.058038532733917236,
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0.013861200772225857,
-0.009627317078411579,
... | |
5afdabf212ac4e21286c715c1114bcdcccc19974 | subsection | 125 | 223 | [PR:BDN]BD-N | Every sequentially continuous map f:\mathbb {N}^{\mathbb {N}}\rightarrow \mathbb {N} is locally bounded.[PR:BDN]{\textrm {BD-N}} \iff \ref {BDNequiv2} is shown in . Moreover, it is clear that REF \Rightarrow REF , and that REF \Rightarrow REF , so it remains to show that REF \Rightarrow [PR:BDN]BD-N. Let A =\left\lbrac... | {
"cite_spans": []
} | 1804.05495 | Constructive Reverse Mathematics | [
"Hannes Diener"
] | [
"math.LO"
] | 2,018 | en | Mathematics | [
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0.020813580602407455,
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0.019012993201613426,
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0.0... | |
ea5ef6340221bf37da1215f4d60ca03adfb77f37 | subsection | 126 | 223 | [PR:BDN]BD-N | That means that \max \left\lbrace a_1, \dots , a_M, M+1 \right\rbrace is an upper bound of A and therefore [PR:BDN]BD-N holds.Proposition 1.2 The following are equivalent to [PR:BDN]BD-N.Every uniformly sequentially continuous mapping of a separable metric space into a metric space is uniformly continuous.
Every unif... | {
"cite_spans": []
} | 1804.05495 | Constructive Reverse Mathematics | [
"Hannes Diener"
] | [
"math.LO"
] | 2,018 | en | Mathematics | [
-0.04597153887152672,
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0.026526737958192825,
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0.03736332431435585,
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0.01... | |
9421e8d5ff6f8fe1c4f001380b26328e3bccde89 | subsection | 127 | 223 | [PR:BDN]BD-N | But that means that for all n \geqslant N we cannot be in the first case, which means that we must have f(\alpha _n) = f(\beta _n). So f is uniformly sequentially continuous.That means that if can apply REF to get that f is locally bounded, and continue as in the proof of Proposition REF .We can also show that \ref {BD... | {
"cite_spans": []
} | 1804.05495 | Constructive Reverse Mathematics | [
"Hannes Diener"
] | [
"math.LO"
] | 2,018 | en | Mathematics | [
-0.06603806465864182,
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0.03485003113746643,
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0.027312414720654488,
-0.03988528251647949,
0... | |
849c830d08d5764e313014f00c81bd398db18597 | subsection | 128 | 223 | [PR:BDN]BD-N | Given \gamma \in \mathbb {N}^{\mathbb {N}}, we can slice up \gamma into countably many sequences and turn it into a double sequence (\gamma ) : \mathbb {N}\times \mathbb {N}\rightarrow \mathbb {N}
defined by(\gamma ) (m,n) = \gamma (\varphi (m,n)) \ .We can also zip a double indexed sequence \sigma :\mathbb {N}\times \... | {
"cite_spans": []
} | 1804.05495 | Constructive Reverse Mathematics | [
"Hannes Diener"
] | [
"math.LO"
] | 2,018 | en | Mathematics | [
-0.04671117663383484,
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0.03209676966071129,
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0.005777876358479261,
-0.03804626688361168,
0.... | |
78909f9a2969c76473dbeac05ce18813fca644c3 | subsection | 129 | 223 | [PR:BDN]BD-N | Let \mu , \eta such that \overline{\mu }(L) = \overline{\eta }(L) , but f(\mu ) \ne f(\eta ). Now consider the following sequence of sequences:\underbrace{000\ldots ,000\ldots , \dots , 000\ldots }_{2M}, \underbrace{\mu , \eta , \mu , \eta , \dots , \mu , \eta }_{2K}, 000\ldots , 000\ldots , \dots \ .If we combine thes... | {
"cite_spans": []
} | 1804.05495 | Constructive Reverse Mathematics | [
"Hannes Diener"
] | [
"math.LO"
] | 2,018 | en | Mathematics | [
-0.029387488961219788,
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0.049681030213832855,
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0.011077526956796646,
0.005554021801799536,
0.... | |
f639b600eacef8440c5321ff60d6feaa3ed633b5 | subsection | 130 | 223 | Below [PR:BDN]BD-N | Between, approximately, 2007 and 2010 a couple of statements were considered by researchers working in CRM for which a proof in BISH could not be found, but that were all implied by [PR:BDN]BD-N. Naturally a considerable amount of time was spent trying to prove that they were in fact equivalent to [PR:BDN]BD-N. As it t... | {
"cite_spans": []
} | 1804.05495 | Constructive Reverse Mathematics | [
"Hannes Diener"
] | [
"math.LO"
] | 2,018 | en | Mathematics | [
-0.024483833461999893,
-0.0029098386876285076,
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0.01617000810801983,
0.055313631892204285,
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0.005789168179035187,
-0.009328263811767101... | |
e580d5f55e619dd90fd36dda6509ff1bd71f4596 | subsection | 131 | 223 | Below [PR:BDN]BD-N | Now define the setA= \mbox{$\left\lbrace \,n \in \mathbb {N} \, | \,\exists {i,j \geqslant n} : {d(z_i,x_j) < 2^{-n}} \,\right\rbrace $} \cup \left\lbrace 0 \right\rbrace \ .The set A is easily seen to be countable; and, unsurprisingly, we are going to show that it is also pseudobounded. To this end let a_n be a sequen... | {
"cite_spans": []
} | 1804.05495 | Constructive Reverse Mathematics | [
"Hannes Diener"
] | [
"math.LO"
] | 2,018 | en | Mathematics | [
0.0035897991620004177,
0.04989859089255333,
-0.024872997775673866,
0.0016051094280555844,
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0.03946108743548393,
-0.028016455471515656,
-0.014466012828052044,
-0.0115743363276124,
... | |
380c907aafb817c124891a45eca88392a548d007 | subsection | 132 | 223 | Below [PR:BDN]BD-N | Now there cannot be an x \in X and i \geqslant M with d(x,z_i) < 2^{-(M+2)}, since otherwise by the density we can find j \geqslant M with d(x_j,x) < 2^{-(M+2)}, which would imply that d(x_i,x_j) < 2^{-(M+1)} and therefore M+1 \in A, which is a contradiction. So for all x \in X and i \geqslant M we have d(x,z_i) \geqsl... | {
"cite_spans": []
} | 1804.05495 | Constructive Reverse Mathematics | [
"Hannes Diener"
] | [
"math.LO"
] | 2,018 | en | Mathematics | [
-0.020172802731394768,
0.017593979835510254,
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-0.030472833663225174,
-0.019562430679798126,... | |
87700b8f210fadef466ecc5f6aba21898ed6ae33 | subsection | 133 | 223 | Below [PR:BDN]BD-N | For the other direction notice that if f:X \rightarrow \mathbb {R} is point-wise continuous, then f(X) is also separable and that [PR:AS]{\textrm {AS}_{f(X)}^{\, 1}} follows from [PR:AS]{\textrm {AS}_{X}^{\, 1}}, whence f(X) is totally bounded and therefore, in particular, bounded (also see *Theorem 11).To see that REF... | {
"cite_spans": []
} | 1804.05495 | Constructive Reverse Mathematics | [
"Hannes Diener"
] | [
"math.LO"
] | 2,018 | en | Mathematics | [
-0.01201077364385128,
0.02597501501441002,
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0.03357522562146187,
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-0.0018666681135073304,
-0.015200420282781124... | |
c565bb8a87d179a8e5c211d5b585ff24116b3be0 | subsection | 134 | 223 | Below [PR:BDN]BD-N | In it is shown that it cannot be proved in BISH. That REF , REF , and REF are strength-wise also between [PR:BDN]BD-N and unadorned BISH was shown in . It is unknown, whether this also holds for REF and REF , but it seems likely.Another natural principle which falls into the same category is
[leftmargin=2em,rightmargin... | {
"cite_spans": []
} | 1804.05495 | Constructive Reverse Mathematics | [
"Hannes Diener"
] | [
"math.LO"
] | 2,018 | en | Mathematics | [
-0.07618129253387451,
0.02330305241048336,
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0.0067871855571866035,
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0... | |
1144425cc93cf43ed767eae8bc003ceabc852bb0 | subsection | 135 | 223 | Below [PR:BDN]BD-N | Moreover, it even seems impossible to get an equivalence analogous to the one of Proposition REF .Proposition 2.4
[PR:wBDN]wBD-N implies that every uniformly sequentially continuous mapping f:2^{\mathbb {N}}\rightarrow \mathbb {N} is uniformly continuous.This follows easily from the next lemma, or alternatively, from ... | {
"cite_spans": []
} | 1804.05495 | Constructive Reverse Mathematics | [
"Hannes Diener"
] | [
"math.LO"
] | 2,018 | en | Mathematics | [
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0.004826073534786701,
... |
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