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9cb144c4a2d798ad8e34f1b81ead25f5d2de03fc | subsection | 136 | 223 | Below [PR:BDN]BD-N | Since *{u_k} \geqslant L we must have k \geqslant K and therefore
f(\alpha _k) = f(\beta _k). But this is a contradiction, sincef(\alpha _k) = f(\overline{\alpha }L \ast w \ast 000\ldots ) \ne f(\overline{\alpha }L \ast w \ast 1 \ast 000\ldots ) = f(\beta _k) \ .Altogether L is a modulus of constancy and therefore for ... | {
"cite_spans": []
} | 1804.05495 | Constructive Reverse Mathematics | [
"Hannes Diener"
] | [
"math.LO"
] | 2,018 | en | Mathematics | [
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5ce186f00dd69a80f35df88d815f7cd9c74c371a | subsection | 137 | 223 | Introduction | Let us assume that K \subset 2^{\ast } is a decidable tree (i.e. it is closed under restriction) that does not admit infinite paths; that is\forall {\alpha \in 2^{\mathbb {N}}} : {\exists {n \in \mathbb {N}} : {\overline{\alpha }n \notin K}} \ .Then the complement B \subset 2^{\ast } has the following properties:B is d... | {
"cite_spans": []
} | 1804.05495 | Constructive Reverse Mathematics | [
"Hannes Diener"
] | [
"math.LO"
] | 2,018 | en | Mathematics | [
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e0144ec7588100dbf17fc340118ec68bbd7ec999 | subsection | 138 | 223 | Introduction | Therefore there are actually only two Anti-Fan principles to consider: Anti-[PR:FAN]FAN_{\Delta } and Anti-[PR:FAN]FAN_{c}.In a similar spirit we can also define Anti-WWKL for k \in (0,1):
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} | 1804.05495 | Constructive Reverse Mathematics | [
"Hannes Diener"
] | [
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ce309292759d04835ddaa2c7e0b7de2863f9a669 | subsection | 139 | 223 | Singular Covers | By slightly extending, in an obvious way, the observation at the start of Section we can see that Anti-[PR:WWKL]WWKL (k) is equivalent to
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} | 1804.05495 | Constructive Reverse Mathematics | [
"Hannes Diener"
] | [
"math.LO"
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8b24b8a915e869a59e86f8d479a50b3eb63defb6 | subsection | 140 | 223 | Singular Covers | Since T blocks \alpha there is n_{1} \in \mathbb {N} such that \alpha [ \colon \!n_{1}] \in B. Since also \alpha [n_{1}+1 \colon \!] gets blocked there exists n_{2} such that \alpha [n_{1}+1 \colon \!n_{2}] \in B, which means that \alpha [ \colon \!n_{2}] \notin S.Last, we need to count the nodes at a level n in S. The... | {
"cite_spans": []
} | 1804.05495 | Constructive Reverse Mathematics | [
"Hannes Diener"
] | [
"math.LO"
] | 2,018 | en | Mathematics | [
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b1e5ac502396260c0f67feea2a1b3662ef2e24d4 | subsection | 141 | 223 | Singular Covers | If we collect all of the finite sequences w_n that are just barely not in T, i.e. w_n \notin T but \overline{w_n}(*{w_n} -1) \in T, then these give us basic open sets U_n = U_{w_n} where U_{w} = \mbox{$\left\lbrace \,\alpha \in 2^{\mathbb {N}} \, | \,\overline{\alpha }*{w} = w \,\right\rbrace $}. These form a cover of ... | {
"cite_spans": []
} | 1804.05495 | Constructive Reverse Mathematics | [
"Hannes Diener"
] | [
"math.LO"
] | 2,018 | en | Mathematics | [
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69596f34761ac0b5dabcbe12be469a45edbfe008 | subsection | 142 | 223 | Singular Covers | Using dependent choice we can construct a sequence \alpha \in 2^{\mathbb {N}} such that x \in I_{\overline{\alpha }n} for all n \in \mathbb {N}. Now, since T does not admit infinite paths, there exists m such that w_m = \overline{\alpha }M where M = *{w_m}. This means thatx \in I_{w_m} \subset \bigcup _{n \in \mathbb {... | {
"cite_spans": []
} | 1804.05495 | Constructive Reverse Mathematics | [
"Hannes Diener"
] | [
"math.LO"
] | 2,018 | en | Mathematics | [
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39ed2e740f0227a1cb479f629b62dcdc10a0fbe6 | subsection | 143 | 223 | Singular Covers | Now choose rationals a^\prime _n and and b_n^\prime such that a_n^\prime \leqslant a_n, b_n \leqslant b_n^\prime , *{a_n^\prime - a_n} < \frac{\alpha }{2^{n+2}}, and *{b_n^\prime - b_n} < \frac{\alpha }{2^{n+2}}. Then I^\prime _n = (a_n^\prime , b_n^\prime ) are obviously still a cover of [0,1] and for any n \in \mathb... | {
"cite_spans": []
} | 1804.05495 | Constructive Reverse Mathematics | [
"Hannes Diener"
] | [
"math.LO"
] | 2,018 | en | Mathematics | [
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18e7f9428c8ab4be9b0af5017662c85d9c0816b8 | subsection | 144 | 223 | Singular Covers | Hence T does not admit infinite paths.So the only item left to consider is to show that T satisfies Equation (REF ). First notice that by definition of T for n \in \mathbb {N}\sum _{{u \notin T \\ *{u} = n}} *{I_u} \leqslant \sum _{i=1}^n *{J_i} \ .So\sum _{{u \in T \\ *{u} = n}} \frac{1}{2^{{u}}} = 1 - \sum _{{u \noti... | {
"cite_spans": []
} | 1804.05495 | Constructive Reverse Mathematics | [
"Hannes Diener"
] | [
"math.LO"
] | 2,018 | en | Mathematics | [
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e9c85414dae855eec8be311c1fdbc45d21fa1b88 | subsection | 145 | 223 | Kleene Trees | As already mentioned in the introduction of Chapter , in some schools of constructive mathematics there exists a Kleene tree. In this section we want to investigate equivalences of the existence of such an object.
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} | 1804.05495 | Constructive Reverse Mathematics | [
"Hannes Diener"
] | [
"math.LO"
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c3189b20920573e4e6cd670cb9145e507449b3ba | subsection | 146 | 223 | Kleene Trees | Assume that [PR:KT]KT holds, so assume that T is a decidable infinite tree that blocks ever infinite path. In particular we can find \left\lbrace u_{1}, u_{2}, \dots \right\rbrace an injective enumeration of all u \notin T such that \overline{u}({u}-1) \in T. In particular, for every \alpha \in 2^{\mathbb {N}} there ex... | {
"cite_spans": []
} | 1804.05495 | Constructive Reverse Mathematics | [
"Hannes Diener"
] | [
"math.LO"
] | 2,018 | en | Mathematics | [
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a7a3bce2180d1da6d81cd92d789af6fbd8f1d49c | subsection | 147 | 223 | Kleene Trees | Now defineT = \mbox{$\left\lbrace \,u \in 2^{\ast } \, | \,\forall {i \leqslant {u}} : { \mu (\overline{u}i \ast 000\ldots ,1) > *{i}} \,\right\rbrace $} \ .By definition T is decidable and closed under restriction. We claim that it does not admit infinite paths, but is infinite.
To see that T does not admit infinite p... | {
"cite_spans": []
} | 1804.05495 | Constructive Reverse Mathematics | [
"Hannes Diener"
] | [
"math.LO"
] | 2,018 | en | Mathematics | [
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9e98fe502f0b29618eb22692dcdfe54317b9c05c | subsection | 148 | 223 | Specker Sequences | In the seminal article E. Specker showed that in recursive mathematics there exists an increasing, computable sequence of rationals (r_n)_{n \geqslant 1} in [0,1] that does not converge to a computable number. More than that, he showed that it does not converge to a computable number in the strong sense that it is comp... | {
"cite_spans": []
} | 1804.05495 | Constructive Reverse Mathematics | [
"Hannes Diener"
] | [
"math.LO"
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ca996bc312d534553829b22a11c58463b8f45655 | subsection | 149 | 223 | Specker Sequences | In the second case by Lemma REF .2 we have that d(x,F(2^{\mathbb {N}})) > \delta for some \delta > 0. In both cases *{x_n - x} > \min \left\lbrace \delta , 3^{-(N+2)} \right\rbrace for all n \geqslant N.In the following we want to weaken the requirement of decidability on a Kleene tree. Just as a decidable tree is the ... | {
"cite_spans": []
} | 1804.05495 | Constructive Reverse Mathematics | [
"Hannes Diener"
] | [
"math.LO"
] | 2,018 | en | Mathematics | [
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3326ae1196fcd37518de7ef1c19215d04ef79bee | subsection | 150 | 223 | Specker Sequences | We want to show thatu \in T \iff \exists {w \in 2^{\ast }} : {uw \in \left\lbrace w_{1},w_{2},\dots \right\rbrace }is a c-Kleene tree. It is clear that it is a c-tree by definition, and that it is infinite, since w_n \in T and *{w_n} \geqslant n for all n \in \mathbb {N}. So it remains to show that T does not admit any... | {
"cite_spans": []
} | 1804.05495 | Constructive Reverse Mathematics | [
"Hannes Diener"
] | [
"math.LO"
] | 2,018 | en | Mathematics | [
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476ce718d261a4099ceb7f9c8587fe818298740e | subsection | 151 | 223 | Specker Sequences | Now for \beta \in 2^{\mathbb {N}} defined by\beta (i) = \alpha _{\pi _{1}(i)}(\pi _{2}(i))we have that \beta \circ ( \pi ^{-1}(m,\cdot )) = \alpha _{m} and therefore \Phi (\beta )(m) = \varphi (\alpha _{m}) = \gamma (m). Thus \Phi (\beta ) = \gamma , which means \Phi is surjective. It is also straightforward to show th... | {
"cite_spans": []
} | 1804.05495 | Constructive Reverse Mathematics | [
"Hannes Diener"
] | [
"math.LO"
] | 2,018 | en | Mathematics | [
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ce0bfcf46d0cbd6c9cc737ba4e7381dc4c521e9a | subsection | 152 | 223 | Basic Relations | It has long been known that [PR:WKL]WKL implies [PR:FAN]FAN_{\Delta } . In Berger has shown that it also implies [PR:FAN]FAN_{c}. This result in turn was again slightly improved upon in where we showed that it also implies [PR:UCT]UCT. For completeness' sake we will include the proof here.Lemma 1.1 [PR:LLPO]LLPO/ [PR:W... | {
"cite_spans": []
} | 1804.05495 | Constructive Reverse Mathematics | [
"Hannes Diener"
] | [
"math.LO"
] | 2,018 | en | Mathematics | [
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31b77a2841691f3e036f0d7fbc14a19b3a57f128 | subsection | 153 | 223 | Basic Relations | Now define a decidable set T byT = \mbox{$\left\lbrace \,u \in 2^{\ast } \, | \,\Lambda (u) \,\right\rbrace $} \cup \mbox{$\left\lbrace \,u \in 2^{\ast } \, | \,\exists {v} : {(F(v) \wedge u = v\ast 0 \ast \dots \ast 0)} \,\right\rbrace $} \ .Notice that for every n \in \mathbb {N} either \mu (u) = 0 for all u \in 2^{*... | {
"cite_spans": []
} | 1804.05495 | Constructive Reverse Mathematics | [
"Hannes Diener"
] | [
"math.LO"
] | 2,018 | en | Mathematics | [
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0922dd5b60c9675b1c2c5296731faf43ed477955 | subsection | 154 | 223 | Basic Relations | But this leads to the contradictionb \leqslant f(v \ast 000\ldots ) = f (\alpha ) < b \ .By sequential continuity, the fact that f(u \ast 000\ldots ) \leqslant b for all u \in 2^{*} now implies f(\alpha ) \leqslant b for all \alpha \in 2^{\mathbb {N}}.Lemma 1.2 [PR:FAN]FAN_{c} implies that an order located image of a p... | {
"cite_spans": []
} | 1804.05495 | Constructive Reverse Mathematics | [
"Hannes Diener"
] | [
"math.LO"
] | 2,018 | en | Mathematics | [
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5166b481301fb62241343eb0a6d07a3f36116c16 | subsection | 155 | 223 | Basic Relations | Moreover, this last statement was shown to be an equivalent of [PR:UCT]UCT in
.This result enables us to replace “uniformly continuous” by “point-wise continuous” thereby improving the well known characterisation of [PR:WKL]{\textrm {WKL}} .Corollary 1.4 [PR:WKL]WKL is equivalent to the statement that every point-wise ... | {
"cite_spans": []
} | 1804.05495 | Constructive Reverse Mathematics | [
"Hannes Diener"
] | [
"math.LO"
] | 2,018 | en | Mathematics | [
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2de02f64bcde1802c309eeaf2e6be1cbcc01440f | subsection | 156 | 223 | Kripke's Schema and the Principle of Finite Possibility | Kripke's Schema, which goes back to Kreisel (see historical notes *Ch apter 4, 10.6) and Myhill jM68, jM66, states that
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"cite_spans": []
} | 1804.05495 | Constructive Reverse Mathematics | [
"Hannes Diener"
] | [
"math.LO"
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b6d9b9ef7628c08876792f3fd01cd1cabf7c3cee | subsection | 157 | 223 | Kripke's Schema and the Principle of Finite Possibility | The “rejection” of [PR:MP]MP using [PR:WPFP]WPFP is of course only convincing if one agrees with the latter and disagrees with [PR:LPO]LPO.Proposition 2.1
[PR:WPFP]{\textrm {WPFP}} + [PR:MP]{\textrm {MP}} \iff [PR:LPO]{\textrm {LPO}}Note that a proof of this fact has already been sketched in *page 258. It will, howeve... | {
"cite_spans": []
} | 1804.05495 | Constructive Reverse Mathematics | [
"Hannes Diener"
] | [
"math.LO"
] | 2,018 | en | Mathematics | [
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e44d5e6db61604c6328f85afc41320eced64473e | subsection | 158 | 223 | Kripke's Schema and the Principle of Finite Possibility | We remind the reader that [PR:LLPO]LLPO is strictly stronger than [PR:MPv]{\textrm {MP}^{\vee }}.Proposition 2.3 [PR:WPFP]{\textrm {WPFP}} + [PR:MPv]{\textrm {MP}^{\vee }} \iff [PR:WLPO]{\textrm {WLPO}} .This has also been proven in and was also pointed out in *Proposition 25.Let (a_n)_{n \geqslant 1} be a binary seque... | {
"cite_spans": []
} | 1804.05495 | Constructive Reverse Mathematics | [
"Hannes Diener"
] | [
"math.LO"
] | 2,018 | en | Mathematics | [
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4a1b8a6f85180c294d5129c7ea8aaa039de75021 | subsection | 159 | 223 | Kripke's Schema and the Principle of Finite Possibility | Every open subset of a separable metric space is a countable union of open balls.There are also some equivalences of [PR:KS]KS involving the countability of subsets of \mathbb {N} in .Finally, even though it might seem like [PR:PFP]PFP and [PR:WPFP]WPFP are rather “ad hoc” principles among many others, the following re... | {
"cite_spans": []
} | 1804.05495 | Constructive Reverse Mathematics | [
"Hannes Diener"
] | [
"math.LO"
] | 2,018 | en | Mathematics | [
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42103cdd7c779e05902ec873b84d41f33b16e1f3 | subsection | 160 | 223 | Kripke's Schema and the Principle of Finite Possibility | The rest is pretty much the well known argument from recursion theory that shows that if a set A \subset \mathbb {N} and its complement \overline{A} are recursively enumerable, then it is decidable.
So let (a_n)_{n \geqslant 1} be arbitrary. By co-[PR:PFP]PFP there exists (b_n)_{n \geqslant 1}, such that
\exists {n \i... | {
"cite_spans": []
} | 1804.05495 | Constructive Reverse Mathematics | [
"Hannes Diener"
] | [
"math.LO"
] | 2,018 | en | Mathematics | [
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d1e606ef2037e7458e4a087c97b54e83f7801815 | subsection | 161 | 223 | Collapsing the Fan Theorems | As we have seen in Proposition REF under the assumption of [PR:BDN]BD-N we have
[PR:FAN]{\textrm {FAN}_{c}} \Rightarrow [PR:FAN]{\textrm {FAN}_{\smash{\Pi ^{0}_{1}}}} . This result was first proved in . It is still interesting to give a direct proof, in order to understand the subtle difference between [PR:FAN]FAN_{c} ... | {
"cite_spans": []
} | 1804.05495 | Constructive Reverse Mathematics | [
"Hannes Diener"
] | [
"math.LO"
] | 2,018 | en | Mathematics | [
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5d6e5b12dec718fac21c7f54e68c580605f9edfd | subsection | 162 | 223 | Collapsing the Fan Theorems | Now we can use [PR:BDN]BD-N to conclude that A is bounded, which immediately translates into B being uniform.Remark 3.2 Notice that the previous proof relies on the assumption that B is closed under extension and therefore does not work for [PR:FANst]FAN_{\textrm {stable}}.Since [PR:BDN]BD-N is a very weak principle al... | {
"cite_spans": []
} | 1804.05495 | Constructive Reverse Mathematics | [
"Hannes Diener"
] | [
"math.LO"
] | 2,018 | en | Mathematics | [
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70c2d324638f997aaa4c96d4ec6a1f576751e373 | subsection | 163 | 223 | Collapsing the Fan Theorems | What is the relationship to [PR:WCN]WCN[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:CC]CC) Continuous choice.CC(1)
Any function... | {
"cite_spans": []
} | 1804.05495 | Constructive Reverse Mathematics | [
"Hannes Diener"
] | [
"math.LO"
] | 2,018 | en | Mathematics | [
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24677ca880aafc89498867c88166dca09e0fecfa | subsection | 164 | 223 | Collapsing the Fan Theorems | It is easy to see that this bound K is also a uniform bound for B.Proposition 3.5 Under the assumption of [PR:KS]KS we have [PR:FAN]{\textrm {FAN}_{\Delta }} \Rightarrow [PR:FAN]{\textrm {FAN}_{\textrm {full}}} .If B is an arbitrary bar, then, using [PR:KS]KS, there exists, for every u \in 2^{\ast } a sequence (a^u_n)_... | {
"cite_spans": []
} | 1804.05495 | Constructive Reverse Mathematics | [
"Hannes Diener"
] | [
"math.LO"
] | 2,018 | en | Mathematics | [
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d077d21e68ff3681b83d06063fb9b5b038ba97a6 | subsection | 165 | 223 | Other Implications | Proposition 4.1 [PR:WMP]WMP implies that [PR:LPO]LPO and [PR:WLPO]WLPO are equivalent.Consider x \in \mathbb {R}, such that x \geqslant 0. By [PR:WLPO]WLPO we know that either x=0 or \lnot (x=0); or with the notation introduced in Chapter whether x=0 or 0 \lessdot x. In the second case we can, using [PR:WLPO]WLPO again... | {
"cite_spans": []
} | 1804.05495 | Constructive Reverse Mathematics | [
"Hannes Diener"
] | [
"math.LO"
] | 2,018 | en | Mathematics | [
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0... | |
aa0bd2cc87fbc264c4c82dac36525016ae11574a | subsection | 166 | 223 | Other Implications | In the second case \lnot (\forall {n \in \mathbb {N}} : {a_n=0}) and hence [PR:WLPO]WLPO holds.Proposition 4.4 [PR:CC]CC (1) implies [PR:BDN]BD-N.A simple consequence of Proposition REF . | {
"cite_spans": []
} | 1804.05495 | Constructive Reverse Mathematics | [
"Hannes Diener"
] | [
"math.LO"
] | 2,018 | en | Mathematics | [
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57f6ca2161fc12ce93e83bfcf3030320a279b033 | subsection | 167 | 223 | The Big Picture | As a handy overview of the relationship between most of the principles discussed we include the following diagram. Dotted lines indicate contradictions.
[Figure: NO_CAPTION] | {
"cite_spans": []
} | 1804.05495 | Constructive Reverse Mathematics | [
"Hannes Diener"
] | [
"math.LO"
] | 2,018 | en | Mathematics | [
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2324a7ffb6424f4c8cee82b79c521849382f8177 | subsection | 168 | 223 | The Big Three | The easiest and most convenient way to see that principle A does not imply principle B, or more general, that theorem T is not provable in BISH is to show that theorem T is false in classical mathematics (CLASS), Brouwer's intuitionism (INT), or in Markov's recursive school of mathematics (RUSS).
The view that all thre... | {
"cite_spans": []
} | 1804.05495 | Constructive Reverse Mathematics | [
"Hannes Diener"
] | [
"math.LO"
] | 2,018 | en | Mathematics | [
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47c13bbc52cd0af2a0ca83d30098a79cc77472f9 | subsection | 169 | 223 | Topological and Heyting-valued Models | Topological models are a natural setting to interpret formalised intuitionistic theories. By “intuitionistic” we mean theories using intuitionistic logic; it is worth noting though, that topological models also have a distinct intuitionistic flavour a'la Brouwer. For example they all validate [PR:FAN]FAN_{\textrm {full... | {
"cite_spans": []
} | 1804.05495 | Constructive Reverse Mathematics | [
"Hannes Diener"
] | [
"math.LO"
] | 2,018 | en | Mathematics | [
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f2667dfe9364dc5e427bfc76e4e8ea43eb81c586 | subsection | 170 | 223 | Propositional Logic | The basic idea of topological models is to use open sets the truth values. As usual, the propositional case is a lot easier and cleaner to deal with than the predicate case.A topological model for propositional intuitionistic logic consists of a topological space (T,\tau ), and a function \llbracket \cdot \rrbracket th... | {
"cite_spans": []
} | 1804.05495 | Constructive Reverse Mathematics | [
"Hannes Diener"
] | [
"math.LO"
] | 2,018 | en | Mathematics | [
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7e61b0cd0c7fed0787d19dcfcd4fb00c944ba9e2 | subsection | 171 | 223 | Propositional Logic | Then \llbracket \lnot P \rrbracket = (0,\infty ) and therefore \llbracket \lnot \lnot P \rrbracket = (-\infty ,0). However \mathbb {R}\ne \llbracket \lnot P \vee \lnot \lnot P \rrbracket = (-\infty ,0) \cup (0,\infty ).This space T is actually not the simplest model showing that [PR:WLEM]WLEM is not derivable in intuit... | {
"cite_spans": []
} | 1804.05495 | Constructive Reverse Mathematics | [
"Hannes Diener"
] | [
"math.LO"
] | 2,018 | en | Mathematics | [
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0c6b1ff97bf112a5b15ef6f32771c1a0af07774a | subsection | 172 | 223 | Predicate Logic | To extend the topological interpretation to predicate logic we also need some universe \mathcal {U} to interpret constants and variables. A predicate \llbracket P(x_1, \dots , x_k) \rrbracket should be mapped to a function \mathcal {U}^k \rightarrow \tau .
It is convenient for d \in \mathcal {U} and a formula \varphi (... | {
"cite_spans": []
} | 1804.05495 | Constructive Reverse Mathematics | [
"Hannes Diener"
] | [
"math.LO"
] | 2,018 | en | Mathematics | [
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c7ea783a5ce09d193f80bbc43cd7ef46ed5cb96b | subsection | 173 | 223 | Predicate Logic | Since it will not affect the following, we will not distinguish between having a logic with different types or having some sort of set theory.Again one can easily show soundness, by induction over deductions.
Notice that in general we do not have an existence property, that is we do not have that if T \Vdash \exists {x... | {
"cite_spans": []
} | 1804.05495 | Constructive Reverse Mathematics | [
"Hannes Diener"
] | [
"math.LO"
] | 2,018 | en | Mathematics | [
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9149825c7a61382c6b897fdbb19463d677ef5cb5 | subsection | 174 | 223 | Topological Models of Arithmetic and of Analysis | Naturally, we want to consider models that validate, at least, Heyting arithmetic. To be precise, we assume that our language contains a constant 0 and function symbols s, +, \cdot and that a model validates the axioms
2\forall {x} : {\lnot (x = s(0))}
\forall {x,y} : {s(x)=s(y) \rightarrow x=y}
\forall {x} : {x+0 = ... | {
"cite_spans": []
} | 1804.05495 | Constructive Reverse Mathematics | [
"Hannes Diener"
] | [
"math.LO"
] | 2,018 | en | Mathematics | [
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c5f8760d0da1cd47bf774157ddff2c4e01fb2202 | subsection | 175 | 223 | Topological Models of Arithmetic and of Analysis | In other words, we can export natural numbers locally.It is enough to
use natural induction (H7) to show thatT \Vdash x \in \mathbb {N}\rightarrow N(x) \ .The case x = 0 is obviously fine. So let x be an arbitrary variable symbol of the natural number type, and let t \in T be arbitrary.By the induction hypothesis there... | {
"cite_spans": []
} | 1804.05495 | Constructive Reverse Mathematics | [
"Hannes Diener"
] | [
"math.LO"
] | 2,018 | en | Mathematics | [
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7355e3749973fbf1daf8db8a57d2b83f29554995 | subsection | 176 | 223 | Topological Models of Arithmetic and of Analysis | Therefore also, in particular,\llbracket x>0 \rrbracket & = \mbox{$\left\lbrace \,t \, | \,f(t) > 0 \,\right\rbrace $} \ , \\
\llbracket x=0 \rrbracket & = \mathrm {Int}\left(\mbox{$\left\lbrace \,t \, | \,f(t) = 0 \,\right\rbrace $}\right) \ , \\
\llbracket x \geqslant 0 \rrbracket & = \mathrm {Int}\left(\mbox{$\left\... | {
"cite_spans": []
} | 1804.05495 | Constructive Reverse Mathematics | [
"Hannes Diener"
] | [
"math.LO"
] | 2,018 | en | Mathematics | [
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8b74bb373c59fc42af2d1923df071477fdafe97a | subsection | 177 | 223 | Topological Models of Arithmetic and of Analysis | It suffices to show that, given \varepsilon > 0, we can find an open neighbourhood U of t_0 such that\forall {t,t^\prime \in U, q \in f(t)} : {\exists {p\in f(t^\prime ) } : { *{q-p} < \varepsilon }} \ .So let \varepsilon >0 and t_0 \in T be arbitrary. We may, without loss of generality, assume that \varepsilon \in \ma... | {
"cite_spans": []
} | 1804.05495 | Constructive Reverse Mathematics | [
"Hannes Diener"
] | [
"math.LO"
] | 2,018 | en | Mathematics | [
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2cb51f4ac1c9e9c0a6649144d75c2ce1e215cd95 | subsection | 178 | 223 | Topological Models of Arithmetic and of Analysis | ThusU \subset \llbracket \lnot ( \breve{q} < x) \rrbracket \subset {\llbracket \breve{q} < x \rrbracket }^\prime \subset \llbracket x < \breve{s} \rrbracket ;because we have assumed that T is a model for the constructive reals and thereforeT = \llbracket \breve{q}< x \ \vee \ x < \breve{s} \rrbracket = \llbracket \brev... | {
"cite_spans": []
} | 1804.05495 | Constructive Reverse Mathematics | [
"Hannes Diener"
] | [
"math.LO"
] | 2,018 | en | Mathematics | [
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eeb2e89ec92b0c6a5a07e5ebeff0bc3cafa13dd3 | subsection | 179 | 223 | The Full Model and Countable Choice | The commonly used models are the “full” ones, whose existence is guaranteed by the following.Proposition 2.6 For any topological space (T,\tau ) there exists a model such thatIt is a model of IZF (therefore also of CZF, HA, and the real numbers).
For every V \in \tau there is a proposition P_V such that \llbracket P_V... | {
"cite_spans": []
} | 1804.05495 | Constructive Reverse Mathematics | [
"Hannes Diener"
] | [
"math.LO"
] | 2,018 | en | Mathematics | [
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ce7f04de0fa529f5e13b0814695eb13ce4de4650 | subsection | 180 | 223 | The Full Model and Countable Choice | Thus [PR:LPO]{\textrm {LPO}} _\sigma holds, since we are working with a classical metatheory.Together we have\mathbb {R}\Vdash [PR:LPO]{\textrm {LPO}} _\sigma \text{ but } \mathbb {R}\nVdash [PR:LPO]{\textrm {LPO}} _\mathbb {R}\ ,and therefore\mathbb {R}\nVdash [Sec:Choice]{\textrm {ACC}}\ .For the rest of the section ... | {
"cite_spans": []
} | 1804.05495 | Constructive Reverse Mathematics | [
"Hannes Diener"
] | [
"math.LO"
] | 2,018 | en | Mathematics | [
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d5bb459c608bb54ce0ae0fda608fc27bce70d2b1 | subsection | 181 | 223 | Reverse Reverse Mathematics | In the following we give some characterisation of properties of topological spaces, such that the full model satisfies certain principles. This is actually quite a natural question to consider, and these kind of results are very helpful to find custom separations of principles. We have—not entirely seriously—named this... | {
"cite_spans": []
} | 1804.05495 | Constructive Reverse Mathematics | [
"Hannes Diener"
] | [
"math.LO"
] | 2,018 | en | Mathematics | [
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8f651a30d7114af958414f4c230b09cd5fad9188 | subsection | 182 | 223 | Reverse Reverse Mathematics | But also in the case that \llbracket \varphi \rrbracket = \left\lbrace 1 \right\rbrace we have that \llbracket \lnot \varphi \rrbracket = \mathrm {Int}\left(\left\lbrace 0 \right\rbrace \right) = \emptyset and therefore X \Vdash \lnot \lnot \varphi .We remind the reader that a topological space X is called functionally... | {
"cite_spans": []
} | 1804.05495 | Constructive Reverse Mathematics | [
"Hannes Diener"
] | [
"math.LO"
] | 2,018 | en | Mathematics | [
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85e450e6ffb472b77dfa5cb625fbc6476a3ee97b | subsection | 183 | 223 | Reverse Reverse Mathematics | Therefore we have that for every neighbourhood U of t_{0}U \Vdash \lnot (x_{g}=0) \ .However t_{0} \notin \mathrm {Int}\left(\mbox{$\left\lbrace \,t \in T \, | \,g(t) \geqslant 0 \,\right\rbrace $}\right) and t_{0} \notin \mathrm {Int}\left(\mbox{$\left\lbrace \,t \in T \, | \,g(t) \leqslant 0 \,\right\rbrace $}\right)... | {
"cite_spans": []
} | 1804.05495 | Constructive Reverse Mathematics | [
"Hannes Diener"
] | [
"math.LO"
] | 2,018 | en | Mathematics | [
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5b782af9d0a8c72a1690ec55ddb28fe3e3308c3f | subsection | 184 | 223 | Reverse Reverse Mathematics | By using an appropriate subsequence we may assume, without loss of generality, that f(t_n) > f(t_{n+1}).
Now choose reals s_n,r_n such thatf(t_{n+1}) < s_n < r_n < f(t_n) \ .Let h: \mathbb {R}\rightarrow \mathbb {R} be the piece-wise linear function that is 0 for x \leqslant 0 and is otherwise such that
h (x) = x for x... | {
"cite_spans": []
} | 1804.05495 | Constructive Reverse Mathematics | [
"Hannes Diener"
] | [
"math.LO"
] | 2,018 | en | Mathematics | [
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9c609518821b86061c3bbbf7ba37ad78259533a6 | subsection | 185 | 223 | Reverse Reverse Mathematics | Together we have the desired contradiction to (REF ).This nicely fits in with a result of Lubarsky and Hendtlass who showed in that there is a topological model satisfying [PR:LLPO]LLPO, but not [PR:LPO]LPO, which means that that model also does not satisfy [PR:WMP]WMP.Their space X_{U} is defined to consist of \mathbb... | {
"cite_spans": []
} | 1804.05495 | Constructive Reverse Mathematics | [
"Hannes Diener"
] | [
"math.LO"
] | 2,018 | en | Mathematics | [
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05cc531e5681a6f878d3c640e7a9b43081f840a4 | subsection | 186 | 223 | Reverse Reverse Mathematics | In the second case \llbracket \lnot \varphi \rrbracket = \mathrm {Int}\left({\llbracket \varphi \rrbracket }^\prime \right) = \left\lbrace \omega \right\rbrace \cup (\mathbb {N}\setminus \llbracket \lnot \varphi \rrbracket ) and \llbracket \lnot \lnot \varphi \rrbracket = \llbracket \varphi \rrbracket . In both cases w... | {
"cite_spans": []
} | 1804.05495 | Constructive Reverse Mathematics | [
"Hannes Diener"
] | [
"math.LO"
] | 2,018 | en | Mathematics | [
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6595ace9c27335d104fe3c39ee85c667be3c6ce5 | subsection | 187 | 223 | Reverse Reverse Mathematics | Then \left\lbrace \omega \right\rbrace \cup A is open and should hence contain U_{N} for some N \in \mathbb {N}. Consider M such that f(2M+1) > N. Since y_{2M+1} \in U_{f(2M+1)} \subset U_{N} \subset A, we get a contradiction.The following is actually a special case of *Theorem 3.2, where it is shown that in any spatia... | {
"cite_spans": []
} | 1804.05495 | Constructive Reverse Mathematics | [
"Hannes Diener"
] | [
"math.LO"
] | 2,018 | en | Mathematics | [
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6efc90671ef001e3b1e25c41b02720dcf1abfa5b | subsection | 188 | 223 | Reverse Reverse Mathematics | This provides an alternative way of seeing that \mathbb {N}^{\mathbb {N}} satisfies [PR:KS]KS.Corollary 2.19
\mathbb {N}^{\mathbb {N}}\nVdash [PR:IIIa]{\textrm {III}_{a}} .Since [PR:PFP]{\textrm {PFP}} \vdash [PR:IIIa]{\textrm {III}_{a}} \Rightarrow [PR:WLPO]{\textrm {WLPO}} , and \mathbb {N}^{\mathbb {N}}\Vdash [PR:K... | {
"cite_spans": []
} | 1804.05495 | Constructive Reverse Mathematics | [
"Hannes Diener"
] | [
"math.LO"
] | 2,018 | en | Mathematics | [
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4f99e2f2dfdaf0b6bd5e753a5d01cbdf541cc677 | subsection | 189 | 223 | Overview | The following quick overview might be useful to compare the big three varieties and three topological models, discussed above.1mylightgray
[Table: NO_CAPTION]We conclude this section by pointing out that topological models have been used to give models thatdo not satisfy [PR:BDN]BD-N, and [PR:BD]BD ,
separate [PR:LPO]... | {
"cite_spans": []
} | 1804.05495 | Constructive Reverse Mathematics | [
"Hannes Diener"
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"math.LO"
] | 2,018 | en | Mathematics | [
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01d5a4d4e855bd0a4bf4a858c6aa7cb41b5b5e6b | subsection | 190 | 223 | Realizability and Other Methods | In a realizability model we extend intuitionistic logic by allowing witnesses of statements to be attached to statements. This is, in particular, interesting in a constructive context, where we want to attach computable objects (“realizers”) that describe the computational content of a formula. For example, we would li... | {
"cite_spans": []
} | 1804.05495 | Constructive Reverse Mathematics | [
"Hannes Diener"
] | [
"math.LO"
] | 2,018 | en | Mathematics | [
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97325fdcac831acbbd4eb00775ca0da23b917429 | subsection | 191 | 223 | [PR:LLPOn] | In Richman introduced a natural weakening of [PR:LLPO]LLPO.
[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:LLPOn]\textrm {LLPO}_{n}) ... | {
"cite_spans": []
} | 1804.05495 | Constructive Reverse Mathematics | [
"Hannes Diener"
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"math.LO"
] | 2,018 | en | Mathematics | [
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359583dcd1b8d35cc364417ee35e40bea3fc66be | subsection | 192 | 223 | [PR:LLPOn] | The reader has probably already noticed that [PR:LLPOn]\textrm {LLPO}_{2} is simply [PR:LLPO]LLPO.Proposition 1.2
[PR:LLPOn]\textrm {LLPO}_{n} is equivalent to the statement that if x_{1}, \dots , x_n are real numbers such that x_{i}x_{j}=0 for i \ne j, then there is m such that x_{m}=0.Assume [PR:LLPOn]\textrm {LLPO}... | {
"cite_spans": []
} | 1804.05495 | Constructive Reverse Mathematics | [
"Hannes Diener"
] | [
"math.LO"
] | 2,018 | en | Mathematics | [
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a0ea49421608c95f47186b9a58f16eee0281b368 | subsection | 193 | 223 | [PR:LLPOn] | Hence b^{(p)}_{m} = 0 for all m \in \mathbb {N} and therefore a^{(p)}_{m} = 0 for all m \in \mathbb {N}.Conversely let (a_{m})_{m \geqslant 1} and for 1 \leqslant i \leqslant n define a real numberx_{i} = \sum _{m \geqslant 1} \frac{a_{in+m}}{2^{m}} \ .It is easy to check that these numbers have the desired property. N... | {
"cite_spans": []
} | 1804.05495 | Constructive Reverse Mathematics | [
"Hannes Diener"
] | [
"math.LO"
] | 2,018 | en | Mathematics | [
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cd31a8f743abdcc75b5521544961c8af284c71ce | subsection | 194 | 223 | [PR:LLPOn] | Is [PR:LLPOn]\textrm {LLPO}_{n} equivalent to the statement that whenever x_1, \dots , x_n are reals such that \prod _{i=1}^n x_i = 0, there exists j such that\prod _{i=1, \ i \ne j}^n x_i = 0 \ ?The forward direction is trivial. However the converse does not generalise straightforwardlyThe reason that [PR:LLPOn]\textr... | {
"cite_spans": []
} | 1804.05495 | Constructive Reverse Mathematics | [
"Hannes Diener"
] | [
"math.LO"
] | 2,018 | en | Mathematics | [
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6c2165705b9ba5a05f29e8d3ecaaf3115cf4d7d9 | subsection | 195 | 223 | Open Induction | Coquand , U. Berger , Schuster , and others have all investigated aspects of open induction. This principle is interesting, since, heuristically speaking, invocations of Zorn's Lemma in a classical proof can be replaced by using open induction. Now open induction is constructive at least for some sets, which also depen... | {
"cite_spans": []
} | 1804.05495 | Constructive Reverse Mathematics | [
"Hannes Diener"
] | [
"math.LO"
] | 2,018 | en | Mathematics | [
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1ac7d55ca4014b7c5921b6626b3193aa3054f7f5 | subsection | 196 | 223 | Open Induction | Furthermore there exists u_{1}, \dots , u_n such that\forall {\beta < \alpha } : {\beta \in \bigcup _{i =1}^n B_{u_{i}}} \ ,and therefore, with u_{n+1} = \overline{\gamma }m also\forall {\beta < \gamma } : {\beta \in \bigcup _{i =1}^{n+1} B_{u_{i}}} \ ,that is \gamma \in A.Along similar lines assume that \alpha is such... | {
"cite_spans": []
} | 1804.05495 | Constructive Reverse Mathematics | [
"Hannes Diener"
] | [
"math.LO"
] | 2,018 | en | Mathematics | [
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4740e45701474c1843380376ecca7a1d006bf61d | subsection | 197 | 223 | Open Induction | Then [PR:OI]{\textrm {OI}{_{\mathbb {N}_{\infty }}}} holds.Let A \subset X progressive (since \mathbb {N} has the discrete topology openness is irrelevant here). We need to show that X \subset A. So let x \in X be arbitrary.
Since X is decidable we can find natural numbers a_{1} < \dots < a_n such that \mbox{$\left\lbr... | {
"cite_spans": []
} | 1804.05495 | Constructive Reverse Mathematics | [
"Hannes Diener"
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"math.LO"
] | 2,018 | en | Mathematics | [
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ea8efee7932d9a2f5f3af7467c9fbf5b104e9373 | subsection | 198 | 223 | Open Induction | Consider the setE = \mbox{$\left\lbrace \,\alpha \in 2^{\mathbb {N}} \, | \,F^{{2}{3}}(\alpha ) \in A \,\right\rbrace $} \ .Now 0 \in E, since F^{{2}{3}}(0)=0. Since E is the preimage of an open set under a continuous function it is also open. To see that it is progressive let \alpha \in 2^{\mathbb {N}} be such that \f... | {
"cite_spans": []
} | 1804.05495 | Constructive Reverse Mathematics | [
"Hannes Diener"
] | [
"math.LO"
] | 2,018 | en | Mathematics | [
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9e09bd8f1bbaab0c702e46c7f675e093577f11f4 | subsection | 199 | 223 | Open Induction | Since E is open there exists n \in \mathbb {N} such that \gamma \in E whenever \overline{\alpha }n = \overline{\gamma }n. If \overline{\alpha }n = 1^n we are done, since then A=[0,1] and thus contains an open neighbourhood of x. In the case that \overline{\alpha }n = u 0 1^{m} for some u \in 2^{\ast } and some m \in \m... | {
"cite_spans": []
} | 1804.05495 | Constructive Reverse Mathematics | [
"Hannes Diener"
] | [
"math.LO"
] | 2,018 | en | Mathematics | [
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5396ce7367b43f183c79b03c7f404ddb25e71fec | subsection | 200 | 223 | The Limited Anti-Specker Property | Douglas Bridges, James Dent, and Maarten McKubre-Jordens dB14,jD13 have considered various weakenings of the Anti-Specker property. One of them is the so-called Limited Anti-Specker property.[leftmargin=2em,rightmargin=3em,skipabove=1em,skipbelow=1em, innerleftmargin=-1em,innerrightmargin=0em, nobreak=true, linecolor=b... | {
"cite_spans": []
} | 1804.05495 | Constructive Reverse Mathematics | [
"Hannes Diener"
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"math.LO"
] | 2,018 | en | Mathematics | [
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0.02103346213698387,
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-0... | |
3a346f1371372276e0cb5446763f14857b203c6c | subsection | 201 | 223 | The Limited Anti-Specker Property | It is an easy calculationIf \overline{\beta }K \ne \overline{\gamma }K, then *{F^p(\beta ) - F^p(\gamma )} \geqslant (1-2p)p^{M-1} to see that for all such n we have*{F^{{1}{3}}(\alpha )-F^{{1}{3}}(\alpha _n)} \geqslant 3^{-M} = 2\delta .Now either *{x-F^{{1}{3}}(\alpha )}< \delta or *{x-F^{{1}{3}}(\alpha )} >0. In the... | {
"cite_spans": []
} | 1804.05495 | Constructive Reverse Mathematics | [
"Hannes Diener"
] | [
"math.LO"
] | 2,018 | en | Mathematics | [
-0.007202798034995794,
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... | |
a941c70e6b79bc65928668af0ed4f9f68614cb03 | subsection | 202 | 223 | The Limited Anti-Specker Property | Thus AS_{[0,1]}^{L} \Rightarrow AS_{2^{\mathbb {N}}}^{L}.To see that the converse holds we can use *Lemma 0.3 and the fact that there exists a point-wise continuous surjection 2^{\mathbb {N}}\rightarrow [0,1] (see Section ).As was shown in [PR:ASL]\textrm {AS}_{X}^{L} is equivalent to the following version of [PR:POS]P... | {
"cite_spans": []
} | 1804.05495 | Constructive Reverse Mathematics | [
"Hannes Diener"
] | [
"math.LO"
] | 2,018 | en | Mathematics | [
-0.03192882239818573,
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f3b9b2f45135728efc6fae75626590744da4c7c5 | subsection | 203 | 223 | Increasing Specker Sequences | As mentioned above Specker's original sequence is increasing. So it is natural to follow Dent and consider the following principle.[leftmargin=2em,rightmargin=3em,skipabove=1em,skipbelow=1em, innerleftmargin=-1em,innerrightmargin=0em, nobreak=true, linecolor=blueish, linewidth=5pt, bottomline=false, topline=false, righ... | {
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} | 1804.05495 | Constructive Reverse Mathematics | [
"Hannes Diener"
] | [
"math.LO"
] | 2,018 | en | Mathematics | [
-0.05043485388159752,
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18a3653742d5ba6912a105a672fd743e319da675 | subsection | 204 | 223 | Increasing Specker Sequences | Thus we have shown that[PR:FAN]{\textrm {FAN}_{c}} \Rightarrow [PR:ASL]{\textrm {AS}_{[0,1]}^{L}} \Rightarrow [PR:iAS]{\textrm {iAS}} \Rightarrow [PR:FAN]{\textrm {FAN}_{\Delta }}\ ,which of course raises the following question.
quQuestion 13Are all of the implications [PR:FAN]{\textrm {FAN}_{c}} \Rightarrow [PR:ASL]{\... | {
"cite_spans": []
} | 1804.05495 | Constructive Reverse Mathematics | [
"Hannes Diener"
] | [
"math.LO"
] | 2,018 | en | Mathematics | [
-0.04044010862708092,
0.004471302963793278,
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0.0010691831121221185,
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0.0034431321546435356,
-0.02447771281003952,... | |
b912b221db612ee56c3c0e2a5a70d31e069c9d8a | subsection | 205 | 223 | Increasing Specker Sequences | \end{array}\right.}Given x \in [0,1] there exists N and \varepsilon >0 such that for all n \geqslant N*{x_{k_{\ell _n}} -x}, *{x_{k_{\ell _n}} - (1-x)} > \varepsilon \ ,which means that also *{ (1-x_{k_{\ell _n}}) -x} > \varepsilon . In both cases *{y_n - x} > \varepsilon , and so (y_n)_{n \geqslant 1} is eventually bo... | {
"cite_spans": []
} | 1804.05495 | Constructive Reverse Mathematics | [
"Hannes Diener"
] | [
"math.LO"
] | 2,018 | en | Mathematics | [
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958dc4f5ad0e80745e5ef9a13ae42d6e476002a9 | subsection | 206 | 223 | Dirk Gently's Principle: | One statement that is notably missing from the list of the “paradoxes of material implication” equivalent to [PR:LEM]LEM (see Proposition REF ) is Dirk Gently's PrincipleThe name is based on the guiding principle of the protagonist of Douglas Adam's novel Dirk Gently's Holistic Detective Agency who believes in the “fun... | {
"cite_spans": []
} | 1804.05495 | Constructive Reverse Mathematics | [
"Hannes Diener"
] | [
"math.LO"
] | 2,018 | en | Mathematics | [
-0.03296154737472534,
0.0036852839402854443,
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0.024049721658229828,
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0.03338882699608803,
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bd3b552136c17e3540dcd89688c119ec46029dad | subsection | 207 | 223 | Dirk Gently's Principle: | Now if [Pr:DGP]DGP holds, then either \varphi \Rightarrow \psi or \psi \Rightarrow \varphi .
In the first case, if \varphi holds, then also \varphi \wedge \psi , and hence \vartheta holds. Together that means that in the first case we have \varphi \Rightarrow \vartheta . In the second case, similarly, we can show that ... | {
"cite_spans": []
} | 1804.05495 | Constructive Reverse Mathematics | [
"Hannes Diener"
] | [
"math.LO"
] | 2,018 | en | Mathematics | [
0.015039962716400623,
0.0360775962471962,
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0.019702274352312088,
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0.03204... | |
7996c284ed418914b6c7a3c8abb34172071efe0d | subsection | 208 | 223 | Dirk Gently's Principle: | So we have that(\varphi \wedge \psi \Rightarrow \varphi ) \vee (\varphi \wedge \psi \Rightarrow \psi ) \ ,which is equivalent to the desired(\psi \Rightarrow \varphi ) \vee (\varphi \Rightarrow \psi ) \ .Hence [Pr:DGP]DGP holds.What makes the principle [Pr:DGP]DGP interesting for our point of view is that we can actual... | {
"cite_spans": []
} | 1804.05495 | Constructive Reverse Mathematics | [
"Hannes Diener"
] | [
"math.LO"
] | 2,018 | en | Mathematics | [
0.020779822021722794,
0.01861335150897503,
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0.036433346569538116,
0.011595201678574085,
... | |
0ff6403ea3386480b504cde1f63b7c6fa398836f | subsection | 209 | 223 | Dirk Gently's Principle: | To see that [Pr:DGP]DGP fails consider \varphi and \psi such that \llbracket \varphi \rrbracket = \left\lbrace 1,2 \right\rbrace and \llbracket \psi \rrbracket = \left\lbrace 1,3 \right\rbrace . Then\llbracket \varphi \Rightarrow \psi \rrbracket = \mathrm {Int}\left(({\llbracket \varphi \rrbracket }^\prime \cup \llbrac... | {
"cite_spans": []
} | 1804.05495 | Constructive Reverse Mathematics | [
"Hannes Diener"
] | [
"math.LO"
] | 2,018 | en | Mathematics | [
-0.01176154799759388,
0.016780419275164604,
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0.04198155924677849,
0... | |
f4e7e7aabf4561d96f72d6d03d9520424ae4a5f0 | subsection | 210 | 223 | List of Open Questions | qu | {
"cite_spans": []
} | 1804.05495 | Constructive Reverse Mathematics | [
"Hannes Diener"
] | [
"math.LO"
] | 2,018 | en | Mathematics | [
-0.037482477724552155,
0.03513219207525253,
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0.02029792219400406,
0.027165640145540237,
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... | |
7db7d6ab1013e86688382df8f9ec98dc5c58b4ec | subsection | 211 | 223 | Topological Models | The PythonVersion 2.7 or 3.x program used to check all possibilities in the proof of Proposition REF is[linenos,
frame=lines,
framesep=2mm, breaklines]pythontopology.pyoA80book
author=Aberth, Oliver,
title=Computable analysis,
type=Book,
language=English,
publisher=McGraw-Hill, New York,
date=1980,
ISBN=0070000794,pA01... | {
"cite_spans": []
} | 1804.05495 | Constructive Reverse Mathematics | [
"Hannes Diener"
] | [
"math.LO"
] | 2,018 | en | Mathematics | [
-0.0488472580909729,
0.006906806956976652,
-0.033622536808252335,
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0.016262078657746315,
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0.036124393343925476,
0.010350676253437996,
0.03493448346853256,
0.02617798186838627,
-0.05424761027097702,
0.019602974876761436,
-0.04274516552686691,
0.01... | |
acd445731ce5acb7e09577fa20b48255ffc89e35 | subsection | 212 | 223 | Topological Models | V.,
series=Lecture Notes in Computer Science,
publisher=Springer Berlin / Heidelberg,
pages=3539,jB07bconference
author=Berger, Josef,
title=Weak König's lemma implies the fan theorem for c-bars,
organization=University of Siena,
date=2007,
booktitle=Computation and logic in the real world, cie 2007, quaderni del
dipar... | {
"cite_spans": []
} | 1804.05495 | Constructive Reverse Mathematics | [
"Hannes Diener"
] | [
"math.LO"
] | 2,018 | en | Mathematics | [
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0.03149747475981712,
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0.035068411380052567,
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0.023226335644721985,
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0.04101996868848801,
0.01599290408194065,
0.003446943825110793,
0.012177802622318268,
-0.03183320537209511,
-... | |
cc3f4f870188fd0f78c3721f9e153834895d8749 | subsection | 213 | 223 | Topological Models | (N.S.),
volume=18,
number=2,
pages=195202,jB08barticle
author=Berger, Josef,
author=Bridges, Douglas S.,
title=The fan theorem and positive-valued uniformly continuous
functions on compact intervals,
date=2008,
journal=New Zealand Journal of Mathematics,
volume=38,
pages=129135,dB10article
author=Berger, Josef,
author=... | {
"cite_spans": []
} | 1804.05495 | Constructive Reverse Mathematics | [
"Hannes Diener"
] | [
"math.LO"
] | 2,018 | en | Mathematics | [
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0.012062936089932919,
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0.023195018991827965,
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0.026582714170217514,
0.03259510546922684,
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0.027986623346805573,
-0.022523585706949234,
0... | |
c9052215b675653549ccd9709d25e94560e2be63 | subsection | 214 | 223 | Topological Models | Soc.,
volume=8,
number=2,
pages=179182,dB99barticle
author=Bridges, Douglas S.,
title=Constructive mathematics: a foundation for computable analysis,
date=1999,
ISSN=0304-3975,
journal=Theoretical Computer Science,
volume=219,
number=1–2,
pages=95 109,
url=http://www.sciencedirect.com/science/article/pii/S0304397598002... | {
"cite_spans": []
} | 1804.05495 | Constructive Reverse Mathematics | [
"Hannes Diener"
] | [
"math.LO"
] | 2,018 | en | Mathematics | [
-0.04650755971670151,
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0.0011863394174724817,
0.004180797375738621,
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0.02384885586798191,
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0.039885420352220535,
0.04229624569416046,
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0.047117892652750015,
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1b772e63a5760a41d16ba1664e76f3f487ab0ee0 | subsection | 215 | 223 | Topological Models | Thesis,
date=2013,rD87book
author=Devaney, Robert,
title=An introduction to chaotic dynamical systems,
publisher=Addison–Wesley Publishing,
date=1987,rD75article
author=Diaconescu, Radu,
title=Axiom of choice and complementation,
date=1975,
journal=Proceedings of the American Mathematical Society,
volume=51,
number=1,
... | {
"cite_spans": []
} | 1804.05495 | Constructive Reverse Mathematics | [
"Hannes Diener"
] | [
"math.LO"
] | 2,018 | en | Mathematics | [
-0.02585761807858944,
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0.021326813846826553,
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0.00040259509114548564,
0.057298652827739716,
0.0002481358824297786,
0.0057588499039411545,
0.04536905884742737,
-0.020411500707268715,
0.009694700129330158,
-0.04301975294947624,... | |
d01581bd9cb957da817287d76297378e6829ea36 | subsection | 216 | 223 | Topological Models | Dolores Jiménez,
editor=Loukanova, Roussanka,
editor=Moss, Larry,
publisher=Cambridge Scholars Publishing,rD05book
author=Diestel, R.,
title=Graph theory,
series=Graduate Texts in Mathematics,
publisher=Springer,
date=2005,
ISBN=9783540261827,sE07book
author=Elaydi, S.N.,
title=Discrete chaos, second edition: With appl... | {
"cite_spans": []
} | 1804.05495 | Constructive Reverse Mathematics | [
"Hannes Diener"
] | [
"math.LO"
] | 2,018 | en | Mathematics | [
-0.03169965371489525,
0.006868512835353613,
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-0.006757914554327726,
-0.005930337123572826,
0.018671981990337372,
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0.01112081203609705,
0.019434725865721703,
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0.02980804815888405,
-0.03325565159320831,
-0.0... | |
4040edd35fe99bbbe55f03e586205ba6b5374282 | subsection | 217 | 223 | Topological Models | Thesis,
date=2013,mH15article
author=Hendtlass, Matthew,
author=Lubarsky, Robert,
title=Separating fragments of WLEM, LPO, and MP,
date=2016,
journal=The Journal of Symbolic Logic,
volume=81,
number=4,
pages=13151343,bH76article
author=Hillam, Bruce,
title=A characterization of the convergence of successive
approximati... | {
"cite_spans": []
} | 1804.05495 | Constructive Reverse Mathematics | [
"Hannes Diener"
] | [
"math.LO"
] | 2,018 | en | Mathematics | [
-0.026924649253487587,
0.017832813784480095,
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0.0096333809196949,
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0.0320960134267807,
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0.02044137753546238,
0.02027357555925846,
0.006658702623099089,
0.04152345284819603,
-0.014957291074097157,
0.010... | |
950da9ebc480d09846f597d93a726fafe00433e4 | subsection | 218 | 223 | Topological Models | Thesis,
date=2004,iL12article
author=Loeb, Iris,
title=Questioning constructive reverse mathematics,
date=2012,
journal=Constructivist Foundations,
volume=7,
number=2,
pages=131140,
url=http://www.univie.ac.at/constructivism/journal/7/2/131.loeb,rL12article
author=Lubarsky, Robert,
title=On the failure of BD-n and BD, ... | {
"cite_spans": []
} | 1804.05495 | Constructive Reverse Mathematics | [
"Hannes Diener"
] | [
"math.LO"
] | 2,018 | en | Mathematics | [
-0.030724667012691498,
0.016765844076871872,
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0.006117473356425762,
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0.05177731812000275,
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0.008291388861835003,
0.04112894833087921,
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0.03316555544734001,
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-... | |
cf658d00e50f47892f6a721c3242263276516734 | subsection | 219 | 223 | Topological Models | Soc.,
volume=20,
pages=319320,mM89article
author=Mandelkern, Mark,
title=Brouwerian counterexamples,
date=1989/02/01,
journal=Mathematics Magazine,
volume=62,
number=1,
pages=327,
url=http://www.jstor.org/stable/2689939,pML98incollection
author=Martin-Löf, Per,
title=An intuitionistic theory of types,
date=1998,
bookti... | {
"cite_spans": []
} | 1804.05495 | Constructive Reverse Mathematics | [
"Hannes Diener"
] | [
"math.LO"
] | 2,018 | en | Mathematics | [
-0.04946225881576538,
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0.017362138256430626,
0.018109716475009918,
0.002774814609438181,
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0.02830120176076889,
-0.041894931346178055,
0.007029530126601458,
0.014089573174715042,
-0.014181113801896572,
0.026394112035632133,
-0.02900300920009613,
... | |
8018b1045f26a465f5aeec79b8261064731220ca | subsection | 220 | 223 | Topological Models | Van,
editor=Staal, J.F.,
series=Studies in Logic and the Foundations of Mathematics,
volume=52,
publisher=Elsevier,
pages=161 178,
url=http://www.sciencedirect.com/science/article/pii/S0049237X08711935,jM66article
author=Myhill, John,
title=Notes towards an axiomatization of intuitionistic analysis,
date=196611,
volume... | {
"cite_spans": []
} | 1804.05495 | Constructive Reverse Mathematics | [
"Hannes Diener"
] | [
"math.LO"
] | 2,018 | en | Mathematics | [
-0.03417404741048813,
0.020000971853733063,
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0.013051740825176239,
0.0025039357133209705,
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0.04012399539351463,
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0.010267470963299274,
0.027568083256483078,
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0.03850683197379112,
-0.036065828055143356,
-... | |
aea86463c4d2ad5d9e435b428b5b1653a01eab42 | subsection | 221 | 223 | Topological Models | 1968,
editor=A. Kino, J. Myhill,
editor=Vesley, R.E.,
series=Studies in Logic and the Foundations of Mathematics,
volume=60,
publisher=Elsevier,
pages=235 255,
url=http://www.sciencedirect.com/science/article/pii/S0049237X08707559,Simpson:1999lrbook
author=Simpson, S.G.,
title=Subsystems of second order arithmetic,
pub... | {
"cite_spans": []
} | 1804.05495 | Constructive Reverse Mathematics | [
"Hannes Diener"
] | [
"math.LO"
] | 2,018 | en | Mathematics | [
-0.021780217066407204,
0.015201888978481293,
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-0.002142535289749503,
0.03412793576717377,
0.011744833551347256,
0.006711878348141909,
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0.03299847990274429,
0.02144443243741989,
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0.024970171973109245,
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... | |
c7aa65ec7c3a3f5df6171a06dfd2e11af84d5908 | subsection | 222 | 223 | Topological Models | II,
series=Studies in Logic and the Foundations of Mathematics,
publisher=North-Holland Publishing Co.,
address=Amsterdam,
date=1988,
volume=123,
ISBN=0-444-70358-6,jvO08book
author=van Oosten, Jaap,
title=Realizability: An introduction to its categorical side,
series=Studies in Logic and the Foundations of Mathematics... | {
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696c6dafdf5327a6995eee3c23aad8c237d4b638 | abstract | 0 | 319 | Abstract | In this thesis we discuss how one can derive the quantum spectral curve for
the $\eta$-deformed AdS$_5 \times S^5$ superstring, an integrable deformation
of the AdS$_5 \times $S$^5$ superstring with quantum group symmetry. This model
can be viewed as a trigonometric version of the AdS$_5 \times $S$^5$
superstring, like... | {
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} | 1804.06741 | Quantum spectral curve for the $\eta$-deformed AdS$_5 \times $S$^5$
superstring | [
"Rob Klabbers"
] | [
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55d939468ccc2959bc2f2733f827b6fb832275b5 | subsection | 1 | 319 | Abstract | Being able to solve an interacting quantum field theory exactly is by itself an exciting prospect, as having full control allows for the precise study of phenomena described by the theory. In the context of the AdS/CFT correspondence, which hypothesises a duality between certain string and gauge theories, planar \mathc... | {
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} | 1804.06741 | Quantum spectral curve for the $\eta$-deformed AdS$_5 \times $S$^5$
superstring | [
"Rob Klabbers"
] | [
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] | 2,018 | en | Physics | [
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c94a3a8e8cea0c04010fc4baa520af2611a172df | subsection | 2 | 319 | Zusammenfassung | ngerman
Die Möglichkeit eine exakte Lösung einer wechselwirkenden Quantenfeldtheorie zu finden ist isoliert betrachtet bereits eine interessante Aussicht, da sie uns unbeschränkte Kontrolle liefert. Sie ermöglicht es Phänomene, die von der Theorie beschrieben werden, sehr präzise zu analysieren. Im Kontext der AdS/CFT-... | {
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"raw": "R. Klabbers and S. J. van Tongeren, “Quantum Spectral Curve for the eta-deformed AdS_5\\times S^5 superstring”, arxiv:1708.02894.",
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superstring | [
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85a5d6726e395846cbdd08597593ab3651a3ce99 | subsection | 3 | 319 | Zusammenfassung | B925, 2017, 252,
arxiv:1708.02894 [hep-th].Other publications by the author:
Arutyunov, Gleb and Frolov, Sergey and Klabbers, Rob and Savin, Sergei, Towards 4-point correlation functions of 1/2-BPS-operators from supergravity, JHEP 1704, 2017, 5, arxiv:1701.00998 [hep-th].
Klabbers, Rob, Thermodynamics of Inozemtsev'... | {
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"raw": "G. Arutyunov, S. Frolov, R. Klabbers and S. Savin, “Towards 4-point correlation functions of any \\frac{1}{2} -BPS operators from supergravity”, JHEP 1704, 005 (2017), arxiv:1701.00998.",
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superstring | [
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e6e4fd8dc09fc6c33f547591e85d2211d272d98e | subsection | 4 | 319 | Introduction | The ultimate goal of physics is to understand nature. A daunting task, considering the huge amount of phenomena one needs to understand in order to reach this goal, ranging all the way from the interactions of elementary particles on ultra-short distance scales to colliding galaxies on ultra-long distance scales.
Never... | {
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ace63328ea188ab81e61e43199e845a706ead3fe | subsection | 5 | 319 | Introduction | Moreover, since we will compare our derivation of the \eta -deformed QSC with the undeformed construction it will be beneficial to have some of the background of the undeformed model available as well.We first discuss the construction of the {\rm AdS}_5\times {\rm S}^5 non-linear sigma model starting from its target sp... | {
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"raw": "G. Arutyunov and S. Frolov, “Foundations of the {\\rm AdS}_5\\times {\\rm S}^5 Superstring. Part I”, J.Phys. A42, 254003 (2009), arxiv:0901.4937.",
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superstring | [
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f29b7f5a267d52d657aec3f3db3fe5d120b3e212 | subsection | 6 | 319 | Introduction | A useful definition for these functions is the following: a function f: is \emph {real periodic} if there exists some R such that f(z+) = f(z) for all z. In particular, if we do not specify \omega it is understood that the function is defined on the standard cylinder with circumference \omega = 2\pi . The undeformed ca... | {
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superstring | [
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e5d7447b3c0cdf68c639895766e889508097b619 | subsection | 7 | 319 | Introduction | This derivation is inspired by . As it turns out this solution can be further simplified by noticing that it can be recast into a beautifully-simple-looking form as was derived for the undeformed case in .This allows us to perform the next step: the basic building blocks for one of the T gauges can be interpreted as th... | {
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7091c3dd1017d5efd8ba7a246d81fa2ecbf065fb | subsection | 8 | 319 | Introduction | Obviously the second option is ludicrous and as we will see the boundary conditions do not uniquely specify the solution, rendering option one false.It turns out that there are still some constraints coming from the analytic properties of the T system that we have not revisited in the \mathbf {P} language: even though ... | {
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superstring | [
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8d76cc86de018c0f563375daebb16d39013abc1a | subsection | 9 | 319 | Body | A prime example Making the first thing bold of a tractable model is the gauge theory known as \mathcal {N}=4 super Yang-Mills theory in four dimensions with gauge group SU(N) and gauge coupling g_{\textsc {YM}} (or \mathcal {N}=4 SYM for short). Its tractability is due to the large degree of symmetry present in the the... | {
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d2143763ee3d86b8417f3abfa145d368b15e5dd9 | subsection | 10 | 319 | Body | Therefore, even though the correspondence is believed to hold for all values of N, almost all of the evidence has been collected in the planar limit, which on the string theory side leads to free string theory: for large N the string coupling constant g_s tends to zero since it relates to the 't Hooft coupling \lambda ... | {
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f54ce83d0f3b9582f036fe799d09b0aa6c602f98 | subsection | 11 | 319 | Body | By analysing eqn. (REF ) we find that the allowed matrices M_1,M_2 span \mathfrak {su}(2,2) and \mathfrak {su}(4) respectively, leaving a one-parameter freedom generated by the central element i\mathbb {I}_8, which is left fixed under the Cartan involution and has vanishing supertrace. Together this forms the bosonic s... | {
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} | 1804.06741 | Quantum spectral curve for the $\eta$-deformed AdS$_5 \times $S$^5$
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ea279d879a54a88fd939497b6f8e01c49363ebf4 | subsection | 12 | 319 | Body | This \mathbb {Z}_4 grading is induced by the fourth-order automorphism \Omega : \mathfrak {sl}(4|4) \rightarrow \mathfrak {sl}(4|4) defined by\Omega (\mathbf {M}) = -\mathbf {K} \mathbf {M}^{st} \mathbf {K}^{-1}, \quad \text{ with } \mathbf {K} = \left( \begin{array}{c|c}
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