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Reflections on reflections in explicit mathematics
 Ann. Pure Appl. Logic
, 2005
"... We give a broad discussion of reflection principles in explicit mathematics, thereby addressing various kinds of universe existence principles. The prooftheoretic strength of the relevant systems of explicit mathematics is couched in terms of suitable extensions of KripkePlatek set theory. 1 ..."
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We give a broad discussion of reflection principles in explicit mathematics, thereby addressing various kinds of universe existence principles. The prooftheoretic strength of the relevant systems of explicit mathematics is couched in terms of suitable extensions of KripkePlatek set theory. 1
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"... This article appeared in a journal published by Elsevier. The attached copy is furnished to the author for internal noncommercial research and education use, including for instruction at the authors institution and sharing with colleagues. Other uses, including reproduction and distribution, or sel ..."
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This article appeared in a journal published by Elsevier. The attached copy is furnished to the author for internal noncommercial research and education use, including for instruction at the authors institution and sharing with colleagues. Other uses, including reproduction and distribution, or selling or licensing copies, or posting to personal, institutional or third party websites are prohibited. In most cases authors are permitted to post their version of the article (e.g. in Word or Tex form) to their personal website or institutional repository. Authors requiring further information regarding Elsevier’s archiving and manuscript policies are encouraged to visit:
Universes in Type Theory Part II Autonomous Mahlo and \Pi 3Reflection
, 2005
"... Abstract We introduce an extension of MartinL"of type theory, which we conjecture to have the same proof theoretic strength as KripkePlatek set theory (KP) extended by one \Pi 3reflecting ordinal and finitely many admissibles above it. That would mean that the proof theoretic strength of ..."
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Abstract We introduce an extension of MartinL&quot;of type theory, which we conjecture to have the same proof theoretic strength as KripkePlatek set theory (KP) extended by one \Pi 3reflecting ordinal and finitely many admissibles above it. That would mean that the proof theoretic strength of this type theory is substantially bigger than that of any previous predicatively justified extensions of MartinL&quot;of type theory, including the Mahlo universe. The universe is constructed following the principles of ordinal notation systems of strength KP plus one \Pi 3reflecting ordinal, therefore extracting key ideas of these notation systems. We introduce a model for this type theory, and determine an upper bound for its proof theoretic strength. This article only presents the main ideas of this model construction, full details will be given in a future article. 1 Introduction This article is a step in a research programme of the author with the goal ofintroducing proof theoretically as strong as possible extensions of MartinL&quot;of type theory, which still can be regarded as predicatively justified. (However,because of our lack of expertise in philosophy, we refrain from giving any meaning explanations.) We have three main reasons for following such a researchprogramme: (1) We hope that this approach gives more insights into the development ofordinal theoretic proof theory. Results in the area of proof theory of
Universes in Type Theory Part II – Autonomous Mahlo
, 2009
"... We introduce the autonomous Mahlo universe which is an extension of MartinLöf type theory which we consider as predicatively justified and which has a strength which goes substantially beyond that of the Mahlo universe, which is before writing this paper the strongest predicatively justified publis ..."
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We introduce the autonomous Mahlo universe which is an extension of MartinLöf type theory which we consider as predicatively justified and which has a strength which goes substantially beyond that of the Mahlo universe, which is before writing this paper the strongest predicatively justified published extension of MartinLöf type theory. We conjecture it to have the same proof theoretic strength as KripkePlatek set theory extended by one recursively autonomous Mahlo ordinal and finitely many admissibles above it. Here a recursively autonomous Mahlo universe ordinal is an ordinal κ which is recursively hyper αMahlo for all α < κ. We introduce as well as intermediate steps the hyperMahlo and hyper αMahlo universes, and give meaning explanations for these theories as well as for the super and the Mahlo universe. We introduce a model for the autonomous Mahlo universe, and determine an upper bound for its proof theoretic strength, therefore establishing one half of the conjecture mentioned before. The autonomous Mahlo universe is the crucial intermediate step for understanding the Π3reflecting universe, which will be published in a successor of this article and which is even stronger and will slightly exceed the strength of KripkePlatek set theory plus the principle of Π3reflection. 1
A Π 1 1UNIFORMIZATION PRINCIPLE FOR REALS
"... Abstract. We introduce a Π 1 1uniformization principle and establish its equivalence with the settheoretic hypothesis (ω1) L = ω1. This principle is then applied to derive the equivalence, to suitable settheoretic hypotheses, of the existence of Π 1 1 maximal chains and thin maximal antichains in ..."
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Abstract. We introduce a Π 1 1uniformization principle and establish its equivalence with the settheoretic hypothesis (ω1) L = ω1. This principle is then applied to derive the equivalence, to suitable settheoretic hypotheses, of the existence of Π 1 1 maximal chains and thin maximal antichains in the Turing degrees. We also use the Π 1 1uniformization principle to study Martin’s conjecture on cones of Turing degrees, and show that under V = L the conjecture fails for uniformly degree invariant Π 1 1 functions. 1.
Presentation to the panel, “Does mathematics need new axioms?”
"... The point of departure for this panel is a somewhat controversial paper that I published in the American Mathematical Monthly under the title “Does mathematics need new axioms? ” [4]. The paper itself was based on a lecture that I gave in 1997 to a joint session of the American Mathematical Society ..."
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The point of departure for this panel is a somewhat controversial paper that I published in the American Mathematical Monthly under the title “Does mathematics need new axioms? ” [4]. The paper itself was based on a lecture that I gave in 1997 to a joint session of the American Mathematical Society and the Mathematical Association of America, and it was thus written for a general mathematical audience. Basically, it was intended as an assessment of Gödel’s program for new axioms that he had advanced most prominently in his 1947 paper for the Monthly, entitled “What is Cantor’s continuum problem? ” [7]. My paper aimed to be an assessment of that program in the light of research in mathematical logic in the intervening years, beginning in the 1960s, but especially in more recent years. In my presentation here I shall be following [4] in its main points, though enlarging on some of them. Some passages are even taken almost verbatim from that paper where convenient, though of course all expository background material that was necessary there for a general audience is omitted. 1 For a logical audience I have written before about