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ZFK := ZFC with a Complement, or: Hegel and the Synto-Set-Theory [article]

Werner DePauli-Schimanovich
<span title="2008-07-30">2008</span> <i > arXiv </i> &nbsp; <span class="release-stage" >pre-print</span>
What is the slightest modification of ZF to add a complement-axiom? The answer in my Ph.D. thesis 1971 was ZF": Zermelo-Fraenkel with replacement for only well-founded domains and an omega-axiom.  ...  In this article we argue that ZF" is the slightest modification of ZF among all these systems and that its successor ZFK is the simplest set theory with a universal set at all.  ...  This is the only "really restriction" of ZFC. The second prime stands for the Omega axiom instead of infinity axiom, because we need a well-founded set for induction and recursion.  ... 
<span class="external-identifiers"> <a target="_blank" rel="external noopener" href="https://arxiv.org/abs/0807.4938v1">arXiv:0807.4938v1</a> <a target="_blank" rel="external noopener" href="https://fatcat.wiki/release/krihidcn45b37fxi5imhvu5wvu">fatcat:krihidcn45b37fxi5imhvu5wvu</a> </span>
<a target="_blank" rel="noopener" href="https://web.archive.org/web/20200826092032/https://arxiv.org/ftp/arxiv/papers/0807/0807.4938.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/d7/1a/d71ab7ed820dc6d463e753f8075fdda1aae5b19b.180px.jpg" alt="fulltext thumbnail" loading="lazy"> </div> </button> </a> <a target="_blank" rel="external noopener" href="https://arxiv.org/abs/0807.4938v1" title="arxiv.org access"> <button class="ui compact blue labeled icon button serp-button"> <i class="file alternate outline icon"></i> arxiv.org </button> </a>

Scientific Intuition of Genii Against Mytho-'Logic' of Cantor's Transfinite 'Paradise'

Alexander A. Zenkin
<span title="2005-11-01">2005</span> <i title="OpenEdition"> <a target="_blank" rel="noopener" href="https://fatcat.wiki/container/nzrfqkv6m5ellc6srdfdwoke4q" style="color: black;">Philosophia Scientiæ</a> </i> &nbsp;
-AZ> in that to denote the series ( * ) as a whole with a name (or symbol ), say, 'ω', to call the name 'omega' an integer and then to go on further a 'count': 'omega', 'omega'+1, 'omega'+2, 'omega'+3,  ...  From the earliest times, classical mathematics is, by right, called the "Science on Infinity", so the understanding of a genuine nature of mathematical infinity has a more closed relation to the foundations  ... 
<span class="external-identifiers"> <a target="_blank" rel="external noopener noreferrer" href="https://doi.org/10.4000/philosophiascientiae.532">doi:10.4000/philosophiascientiae.532</a> <a target="_blank" rel="external noopener" href="https://fatcat.wiki/release/wulicnc4gjg53jkqiq26ngnxzm">fatcat:wulicnc4gjg53jkqiq26ngnxzm</a> </span>
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Naive Axiomatic Class Theory: A Solution for the Antinomies of Naive Mengenlehre [article]

Werner DePauli-Schimanovich
<span title="2008-07-27">2008</span> <i > arXiv </i> &nbsp; <span class="release-stage" >pre-print</span>
Therefore we construct a series of partial systems which form a recursive axiom system too.  ...  After that we discuss the medium classes of these systems. Finally we present the inconsistent NSA-systems based on Not-SelfApplicability and explain their help for computational set theory.  ...  instead of the axiom of infinity], plus the Complement-Existence-Axiom, and therefore more important than ZFC.  ... 
<span class="external-identifiers"> <a target="_blank" rel="external noopener" href="https://arxiv.org/abs/0807.4324v1">arXiv:0807.4324v1</a> <a target="_blank" rel="external noopener" href="https://fatcat.wiki/release/szqeystwgvf4rcmaurafsdidde">fatcat:szqeystwgvf4rcmaurafsdidde</a> </span>
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RCF4: Inconsistent Quantification [article]

Michael Pfender
<span title="2009-09-08">2009</span> <i > arXiv </i> &nbsp; <span class="release-stage" >pre-print</span>
Appropriate diagonal argument then shows inconsistency of this subsystem and (hence) of its extensions Q + wo(omega^omega) and ZF.  ...  We strengthen Theory Q by axiomatically forcing the lexicographical order on its omega^omega to become a well-order: "finite descent".  ...  RCF 1 and quote section 2 of RCF 2 0 legend of logo: AC : Axiom of Choice, ∀∃!  ... 
<span class="external-identifiers"> <a target="_blank" rel="external noopener" href="https://arxiv.org/abs/0901.4865v2">arXiv:0901.4865v2</a> <a target="_blank" rel="external noopener" href="https://fatcat.wiki/release/rcik7krm2nc7bf3lnsmuk5tgyi">fatcat:rcik7krm2nc7bf3lnsmuk5tgyi</a> </span>
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A Century of Controversy over the Foundations of Mathematics [article]

G. J. Chaitin
<span title="1999-08-27">1999</span> <i > arXiv </i> &nbsp; <span class="release-stage" >pre-print</span>
This is the transcript of a lecture given at UMass-Lowell in which I compare and contrast the work of Godel and of Turing and my own work on incompleteness.  ...  So there are an infinity of elegant programs out there.  ...  Omega plus one, omega plus two, then you go on for an infinite amount of time. And what do you put afterwards? Well, two omega?  ... 
<span class="external-identifiers"> <a target="_blank" rel="external noopener" href="https://arxiv.org/abs/chao-dyn/9909001v1">arXiv:chao-dyn/9909001v1</a> <a target="_blank" rel="external noopener" href="https://fatcat.wiki/release/bdh22p2pzfhtbfywxwondl2m2y">fatcat:bdh22p2pzfhtbfywxwondl2m2y</a> </span>
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The formal roots of Platonism [article]

Bhupinder Singh Anand
<span title="2003-05-17">2003</span> <i > arXiv </i> &nbsp; <span class="release-stage" >pre-print</span>
as an assumption that such truth cannot be expressed in a constructive, and effectively verifiable, manner.  ...  We present some arguments for the thesis that a set-theoretic inspired faith, in the ability of intuitive truth to faithfully reflect relationships between elements of a Platonic universe, may be as misplaced  ...  Of course the integers and rational numbers are also considered reals so one could argue that the cardinality of the reals is aleph 0 or the same as omega.  ... 
<span class="external-identifiers"> <a target="_blank" rel="external noopener" href="https://arxiv.org/abs/math/0305055v2">arXiv:math/0305055v2</a> <a target="_blank" rel="external noopener" href="https://fatcat.wiki/release/dsdgmxq6ajcbbebzy6hmb5aplm">fatcat:dsdgmxq6ajcbbebzy6hmb5aplm</a> </span>
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What is Absolute Undecidability?†

Justin Clarke-Doane
<span title="2012-06-21">2012</span> <i title="Wiley"> <a target="_blank" rel="noopener" href="https://fatcat.wiki/container/pyrfyab2zva6lmpkx2bzn2vz2y" style="color: black;">Noûs</a> </i> &nbsp;
This is a corrected version of an article which appears in Noûs, Vol. 47.]  ...  imaginative, and so on.  ...  Friedman writes, "I have seen some…go so far as to challenge the existence of 2^100 as a natural number… [2002, p. 4] ." 26 See Nelson [1986] and Buss [1996] . imaginative, and so on.  ... 
<span class="external-identifiers"> <a target="_blank" rel="external noopener noreferrer" href="https://doi.org/10.1111/j.1468-0068.2012.00861.x">doi:10.1111/j.1468-0068.2012.00861.x</a> <a target="_blank" rel="external noopener" href="https://fatcat.wiki/release/2wrwtyhenjbkdmjuu5gbu2gbty">fatcat:2wrwtyhenjbkdmjuu5gbu2gbty</a> </span>
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How are Concepts of Infinity Acquired?

Kazimierz Trzęsicki
<span title="2015-03-01">2015</span> <i title="Walter de Gruyter GmbH"> <a target="_blank" rel="noopener" href="https://fatcat.wiki/container/qwzusus4sbb45ib6ar7bifd6ie" style="color: black;">Studies in Logic, Grammar and Rhetoric</a> </i> &nbsp;
Concepts of infinity have been subjects of dispute since antiquity. The main problems of this paper are: is the mind able to acquire a concept of infinity? and: how are concepts of infinity acquired?  ...  The aim of this paper is neither to say what the meanings of the word "infinity" are nor what infinity is and whether it exists.  ...  Though the concept of (potential) infinity is common and clear in such ordinary phrases as "and so on" or "and so forth", it is capable of giving rise to vertiginous bewilderment: which may lead, on the  ... 
<span class="external-identifiers"> <a target="_blank" rel="external noopener noreferrer" href="https://doi.org/10.1515/slgr-2015-0010">doi:10.1515/slgr-2015-0010</a> <a target="_blank" rel="external noopener" href="https://fatcat.wiki/release/ickkncv7xnaynmn5di3eym5gdu">fatcat:ickkncv7xnaynmn5di3eym5gdu</a> </span>
<a target="_blank" rel="noopener" href="https://web.archive.org/web/20190501233404/https://content.sciendo.com/downloadpdf/journals/slgr/40/1/article-p179.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/5f/f6/5ff61586c72fc4db3980a20a651273e8723db671.180px.jpg" alt="fulltext thumbnail" loading="lazy"> </div> </button> </a> <a target="_blank" rel="external noopener noreferrer" href="https://doi.org/10.1515/slgr-2015-0010"> <button class="ui left aligned compact blue labeled icon button serp-button"> <i class="unlock alternate icon" style="background-color: #fb971f;"></i> degruyter.com </button> </a>

Naïve Infinitesimal Analysis: Its Construction and Its Properties [article]

Anggha Nugraha, Maarten McKubre-Jordens, Hannes Diener
<span title="2020-09-24">2020</span> <i > arXiv </i> &nbsp; <span class="release-stage" >pre-print</span>
The main contribution of this paper is the construction of a new set of numbers, ℝ^ℤ_<, which includes infinities and infinitesimals.  ...  The construction of this new set is done naïvely in the sense that it does not require any heavy mathematical machinery, and so it will be much less problematic in a long term.  ...  We would also like to thank Professor Elemér Rosinger and Dr. Josef Berger for their invaluable inputs.  ... 
<span class="external-identifiers"> <a target="_blank" rel="external noopener" href="https://arxiv.org/abs/2009.11424v1">arXiv:2009.11424v1</a> <a target="_blank" rel="external noopener" href="https://fatcat.wiki/release/aezfthdrd5gz3p3gv7welwbbmu">fatcat:aezfthdrd5gz3p3gv7welwbbmu</a> </span>
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Foundations of Transmathematics: Set Membership

James Anderson
<span title="2020-03-11">2020</span> <i title="figshare"> Figshare </i> &nbsp;
Thanks are due both to Walter Gomide and to Tiago dos Reis for many helpful discussions on set theories.  ...  This question finds technical answers in the body of this text but we recommend the following definition to him. Nullity is the order type of a banana.  ...  Firstly, we know that infinity itself, ∞ = O, and every member of infinity, mi ∈ O, is a member of the universal set, mi, ∞ ∈ V, so the cardinality of infinity is not greater than the cardinality of the  ... 
<span class="external-identifiers"> <a target="_blank" rel="external noopener noreferrer" href="https://doi.org/10.6084/m9.figshare.11967642.v1">doi:10.6084/m9.figshare.11967642.v1</a> <a target="_blank" rel="external noopener" href="https://fatcat.wiki/release/ohkcibxkibgn3pdqcby23ww5my">fatcat:ohkcibxkibgn3pdqcby23ww5my</a> </span>
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Incompleteness, Complexity, Randomness and Beyond [article]

Cristian S. Calude
<span title="2001-12-17">2001</span> <i > arXiv </i> &nbsp; <span class="release-stage" >pre-print</span>
Goedel's Incompleteness Theorems have the same scientific status as Einstein's principle of relativity, Heisenberg's uncertainty principle, and Watson and Crick's double helix model of DNA.  ...  Our aim is to discuss some new faces of the incompleteness phenomenon unveiled by an information-theoretic approach to randomness and recent developments in quantum computing.  ...  Acknowledgment We thank Greg Chaitin, Jack Copeland, Fred Kroon, Sergiu Rudeanu, Jerry Seligman and Karl Svozil for useful comments and criticism. Bibliography  ... 
<span class="external-identifiers"> <a target="_blank" rel="external noopener" href="https://arxiv.org/abs/quant-ph/0111118v2">arXiv:quant-ph/0111118v2</a> <a target="_blank" rel="external noopener" href="https://fatcat.wiki/release/5mejnxnnjjhb7jbkylh33eo7xq">fatcat:5mejnxnnjjhb7jbkylh33eo7xq</a> </span>
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Determinacy Maximum [article]

Dmytro Taranovsky
<span title="2016-12-15">2016</span> <i > arXiv </i> &nbsp; <span class="release-stage" >pre-print</span>
of determinacy, and provide a philosophical justification for determinacy.  ...  We propose a new determinacy hypothesis for transfinite games, use the hypothesis to extend the perfect set theorem, prove relationships between various determinacy hypotheses, expose inconsistent versions  ...  A Strengthening of AD There is also a strong determinacy hypothesis for ZF + DC (inconsistent with AC): Every game of length omega is weakly quasi-determined.  ... 
<span class="external-identifiers"> <a target="_blank" rel="external noopener" href="https://arxiv.org/abs/math/0311095v3">arXiv:math/0311095v3</a> <a target="_blank" rel="external noopener" href="https://fatcat.wiki/release/ibgekbkcbfgsbiydk56ib2nv5q">fatcat:ibgekbkcbfgsbiydk56ib2nv5q</a> </span>
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Matheme and mathematics. On the main concepts of the philosophy of Alain Badiou [article]

Maciej Malicki
<span title="2016-07-27">2016</span> <i > arXiv </i> &nbsp; <span class="release-stage" >pre-print</span>
The focus of my analysis is both on their mathematical consistency, and their philosophical consequences.  ...  On the basis of "Being and Event" as well as his other writings, I analyze the main notions of his philosophy such as the indiscernible, the undecidable, and the unnameable.  ...  The mathematical form of the undecidable implies that a truth never overrules the fundamental conceptual framework of a situation from before the occurrence of an event.  ... 
<span class="external-identifiers"> <a target="_blank" rel="external noopener" href="https://arxiv.org/abs/1406.0059v2">arXiv:1406.0059v2</a> <a target="_blank" rel="external noopener" href="https://fatcat.wiki/release/r766sauz2rcwficf33gdtt5aem">fatcat:r766sauz2rcwficf33gdtt5aem</a> </span>
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To the End: Exposing the Absolute

Frank Ruda
<span title="2020-12-31">2020</span> <i title="The Research Center of the Slovenian Academy of Sciences and Arts / Znanstvenoraziskovalni center Slovenske akademije znanosti in umetnosti (ZRC SAZU)"> <a target="_blank" rel="noopener" href="https://fatcat.wiki/container/acqqsz3pjnbj3ipvgls67kjp3a" style="color: black;">Filozofski Vestnik</a> </i> &nbsp;
This allows to argue that Badiou's is and always was a philosophy of freedom.  ...  It determines the perspective and task of The Immanence of Truths and demonstrates in what way an "absolute ontology" is the result of an act of forcing that serves as the measure for a truth's truthness  ...  (Too) brief addendum on the end of infinity Based on a given infinite set, one can easily construct a new sort of infinity by means of what is called orthogonal operation. 107 To do so, we measure it  ... 
<span class="external-identifiers"> <a target="_blank" rel="external noopener noreferrer" href="https://doi.org/10.3986/fv.41.2.12">doi:10.3986/fv.41.2.12</a> <a target="_blank" rel="external noopener" href="https://fatcat.wiki/release/4kip3v3smbhengxopj64vz3yc4">fatcat:4kip3v3smbhengxopj64vz3yc4</a> </span>
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Paradigms-Shift in Set Theory [article]

Werner DePauli-Schimanovich
<span title="2008-07-30">2008</span> <i > arXiv </i> &nbsp; <span class="release-stage" >pre-print</span>
In this article the author claims that there is a paradigm shift from ZFC to NFUM and further to NACT - due to philosophical reasons, not mathematical ones.  ...  With help of Non-Monotonic Logic, the consistent systems NACT-MoonW, NACT*W, and NACT-SunW are producing "largest possible universes" of sets.  ...  (ZF" 21 is a slight modification and restriction of ZF [with the axioms of separation and replacement restricted to small (here: well-founded) sets and an ω-axiom instead of the axiom of infinity.]  ... 
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