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The topology of provability in complexity theory

Kenneth W. Regan
<span title="">1988</span> <i title="Elsevier BV"> <a target="_blank" rel="noopener" href="https://fatcat.wiki/container/p6ovb2qpkfenhmb7mcksobrcxq" style="color: black;">Journal of computer and system sciences (Print)</a> </i> &nbsp;
In conclusion we argue that the type of independence result presented here forms an obstacle for day-to-day work in complexity theory, but does not bear directly on the possible independence of the P g  ...  We explain the phenomenon of unprovability in terms of both recursive properties of the formal systems 5 under consideration, and topological properties of complexity classes in a natural space which we  ...  INTRODUCTION Which assertions in complexity theory are provable in the strong logical theories that underpin most mathematical work?  ... 
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<a target="_blank" rel="noopener" href="https://web.archive.org/web/20190319175706/https://core.ac.uk/download/pdf/82258082.pdf" title="fulltext PDF download" data-goatcounter-click="serp-fulltext" data-goatcounter-title="serp-fulltext"> <button class="ui simple right pointing dropdown compact black labeled icon button serp-button"> <i class="icon ia-icon"></i> Web Archive [PDF] <div class="menu fulltext-thumbnail"> <img src="https://blobs.fatcat.wiki/thumbnail/pdf/6d/16/6d1640e912fedd8ed9502137fa8fe9132abb5722.180px.jpg" alt="fulltext thumbnail" loading="lazy"> </div> </button> </a> <a target="_blank" rel="external noopener noreferrer" href="https://doi.org/10.1016/0022-0000(88)90036-0"> <button class="ui left aligned compact blue labeled icon button serp-button"> <i class="external alternate icon"></i> elsevier.com </button> </a>

Model theory and the cardinal numbers p and t

J. T. Moore
<span title="2013-07-26">2013</span> <i title="Proceedings of the National Academy of Sciences"> <a target="_blank" rel="noopener" href="https://fatcat.wiki/container/nvtuoas5pbdsllkntnhizy4f4q" style="color: black;">Proceedings of the National Academy of Sciences of the United States of America</a> </i> &nbsp;
These cardinals, and p in particular, arise in the study of more exotic topological spaces in topology and analysis.  ...  The proof proceeds by showing that, even in this generality, theories that satisfy the hypothesis of the theorem admit an abstract notion of a basis, much as in the setting of vector spaces.  ...  These cardinals, and p in particular, arise in the study of more exotic topological spaces in topology and analysis.  ... 
<span class="external-identifiers"> <a target="_blank" rel="external noopener noreferrer" href="https://doi.org/10.1073/pnas.1310920110">doi:10.1073/pnas.1310920110</a> <a target="_blank" rel="external noopener" href="https://www.ncbi.nlm.nih.gov/pubmed/23893298">pmid:23893298</a> <a target="_blank" rel="external noopener" href="https://pubmed.ncbi.nlm.nih.gov/PMC3746849/">pmcid:PMC3746849</a> <a target="_blank" rel="external noopener" href="https://fatcat.wiki/release/le3xblvrqfaj3nryor6gegghda">fatcat:le3xblvrqfaj3nryor6gegghda</a> </span>
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Page 6352 of Mathematical Reviews Vol. , Issue 89K [page]

<span title="">1989</span> <i title="American Mathematical Society"> <a target="_blank" rel="noopener" href="https://archive.org/details/pub_mathematical-reviews" style="color: black;">Mathematical Reviews </a> </i> &nbsp;
{For the entire collection see MR 88j:00017.} 89k:68050 68Q15 03D10 03D15 68Q05 Regan, Kenneth W. (1-CRNL) The topology of provability in complexity theory.  ...  Structure in Complexity Theory Conference (Berkeley, CA, 1986). J. Comput. System Sci. 36 (1988), no. 3, 384-432.  ... 
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Page 600 of Mathematical Reviews Vol. , Issue 87b [page]

<span title="">1987</span> <i title="American Mathematical Society"> <a target="_blank" rel="noopener" href="https://archive.org/details/pub_mathematical-reviews" style="color: black;">Mathematical Reviews </a> </i> &nbsp;
We check that the notion of a proof in the formal theory ¥ corresponds to that of a path in the topological space F.  ...  In 2.5.2 we give an alternative construction of the topological space.  ... 
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Page 873 of Mathematical Reviews Vol. 30, Issue 6 [page]

<span title="">1965</span> <i title="American Mathematical Society"> <a target="_blank" rel="noopener" href="https://archive.org/details/pub_mathematical-reviews" style="color: black;">Mathematical Reviews </a> </i> &nbsp;
Here is to be found Whitehead’s version of homotopy theory in terms of CW-complexes (originally ‘membrane’ complexes). “The papers in the last volume fall into two groups.  ...  the direction of algebraic topology.  ... 
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Page 7293 of Mathematical Reviews Vol. , Issue 2003i [page]

<span title="">2003</span> <i title="American Mathematical Society"> <a target="_blank" rel="noopener" href="https://archive.org/details/pub_mathematical-reviews" style="color: black;">Mathematical Reviews </a> </i> &nbsp;
and beyond Research in the history of astronomy, No. 25 Research in the history of astronomy, No. 26 .....  ...  SOR Oe oooh shee td cacti Symposium: * 68002 * 53001 * 01001 * 01002 * 70001 * 60001 * 68001 * 57001 * 65005 * NO008 Cybernetics in the 21st Century: Information and Complexity in Control Theory * 93001  ... 
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Page 6578 of Mathematical Reviews Vol. , Issue 97K [page]

<span title="">1997</span> <i title="American Mathematical Society"> <a target="_blank" rel="noopener" href="https://archive.org/details/pub_mathematical-reviews" style="color: black;">Mathematical Reviews </a> </i> &nbsp;
An important and interesting part of the paper is devoted to the study of some topological aspects of the theory.  ...  The result of this paper is that the Sacks density theorem (the re. degrees are dense) is provable in BY.  ... 
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Page 2299 of Mathematical Reviews Vol. , Issue 99d [page]

<span title="">1991</span> <i title="American Mathematical Society"> <a target="_blank" rel="noopener" href="https://archive.org/details/pub_mathematical-reviews" style="color: black;">Mathematical Reviews </a> </i> &nbsp;
The main result obtained in this article is an exact correspondence between the rank of a proof in an arithmetic system with respect to the occurrences of induction (not the complexity of the induction  ...  Wainer, Hierarchies of provably recursive functions (149-207); Wolfram Pohlers, Subsys- tems of set theory and second order number theory (209-335); Jeremy Avigad and Solomon Feferman, Gédel’s functional  ... 
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Page 345 of American Journal of Mathematics Vol. 58, Issue 2 [page]

<span title="">1936</span> <i title="Johns Hopkins University Press"> <a target="_blank" rel="noopener" href="https://archive.org/details/pub_american-journal-of-mathematics" style="color: black;">American Journal of Mathematics </a> </i> &nbsp;
This problem can be interpreted as a problem of elementary number theory in view of the fact that topological complexes are representable by matrices of incidence.  ...  that f(m,n) is equal to 2 if and only if the m-th set of incidence matrices and the n-th set of incidence matrices in the enumeration represent homeomorphic complexes.  ... 
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Page 1698 of Mathematical Reviews Vol. , Issue 88d [page]

<span title="">1988</span> <i title="American Mathematical Society"> <a target="_blank" rel="noopener" href="https://archive.org/details/pub_mathematical-reviews" style="color: black;">Mathematical Reviews </a> </i> &nbsp;
Axiomatic results in the paper exhibit subsystems of second-order arithmetic Z, in which HT is provable; e.g., by analysing Baumgartner’s proof of HT by using £-models the authors show that HT is provable  ...  of combinatorics and topological dynamics.  ... 
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Page 627 of Mathematical Reviews Vol. , Issue 96b [page]

<span title="">1996</span> <i title="American Mathematical Society"> <a target="_blank" rel="noopener" href="https://archive.org/details/pub_mathematical-reviews" style="color: black;">Mathematical Reviews </a> </i> &nbsp;
The bulk of the paper consists of proving by permutation models that the following implications are not provable in the theory ZFU: LFC > CF, DS > LFC, DF = F=> DS, CF => DF =F, PDF > LDF = F.  ...  A Borel subset A of X x Y is potentially of class T if it is possible to make A of class I by replacing the original topologies on X and Y by suitable finer Polish topologies.  ... 
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The unification of Mathematics via Topos Theory [article]

Olivia Caramello
<span title="2010-06-20">2010</span> <i > arXiv </i> &nbsp; <span class="release-stage" >pre-print</span>
We present a set of principles and methodologies which may serve as foundations of a unifying theory of Mathematics.  ...  These principles are based on a new view of Grothendieck toposes as unifying spaces being able to act as 'bridges' for transferring information, ideas and results between distinct mathematical theories  ...  Acknowledgements: I am grateful to Marco Benini for his useful comments on a draft of this paper.  ... 
<span class="external-identifiers"> <a target="_blank" rel="external noopener" href="https://arxiv.org/abs/1006.3930v1">arXiv:1006.3930v1</a> <a target="_blank" rel="external noopener" href="https://fatcat.wiki/release/msligcyek5ebjipegmqlqjdppm">fatcat:msligcyek5ebjipegmqlqjdppm</a> </span>
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Page 1685 of Mathematical Reviews Vol. 55, Issue 6 [page]

<span title="">1978</span> <i title="American Mathematical Society"> <a target="_blank" rel="noopener" href="https://archive.org/details/pub_mathematical-reviews" style="color: black;">Mathematical Reviews </a> </i> &nbsp;
He then shows by a simple argument that in a large number of sublogics of intuitionist logic (including positive relevant logics) a formula is provable in the logic if and only if it is true on the canonical  ...  If the conclusion VxA(x), '—A of the rule +V is cut-free provable, then there exists a sequence of cut-free provable sequents At;, ',; ~A,; Ata, ',—>Ag; «+ such that a cut- free derivation of I’, [—A,  ... 
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Page 492 of Mathematical Reviews Vol. , Issue 80B [page]

<span title="">1980</span> <i title="American Mathematical Society"> <a target="_blank" rel="noopener" href="https://archive.org/details/pub_mathematical-reviews" style="color: black;">Mathematical Reviews </a> </i> &nbsp;
In the remainder of the paper the author investigates provable complexity classes introduced by Hartmanis, For @ a class of functions, we say that a TM M works in time © if M works in time f for some f  ...  occur in the study of computational complexity.  ... 
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Proving Gödel's completeness theorem (in restricted cases) using Sd - Topology over the theory of a Model

Soham Dasgupta
<span title="2021-01-23">2021</span> <i title="Zenodo"> Zenodo </i> &nbsp;
Abstract: This is a modification as well as a model theoretic application of my previous paper Sd- Topology over the theory of a Model.  ...  The main intention of this article is to proof and visualize the celebrated Gödel's completeness theorem in a different and lucid approach.  ...  Here as we are using Sd Topology so the conditions will be restricted as the cardinality of the universe of the working model should be card ≥ℵ0 and also the cardinality of the working theory should be  ... 
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