In this article and other discussions of the Axiom of Choice the following abbreviations are common: AC — the Axiom of Choice. There are many other equivalent statements of the axiom of choice; these are equivalent in the sense that, in the presence of other basic axioms of set theory , they imply the axiom of choice and are implied by it. One variation avoids the use of choice functions by, in effect, replacing each choice function with its range.
Given any set X of pairwise disjoint non-empty sets, there exists at least one set C that contains one element in common with each of the sets in X; this guarantees for any partition of a set X the existence of a subset C of X containing one element from each part of the partition. Another equivalent axiom only considers collections X that are powersets of other sets: For any set A, the power set of A has a choice function. Authors who use this formulation speak of the choice function on A, but be advised that this is a different notion of choice function, its domain is the powerset of A, and.
While still in gymnasium in Halle , Bernstein heard the university seminar of Georg Cantor , a friend of Bernstein's father. He died of cancer in Zurich on 3 December Felix Bernstein. Mathematische Annalen. Jahresbericht der Deutschen Mathematiker-Vereinigung. Stieltjes und E. Nachrichten von der Mathematisch-Physikalische Klasse. Not all statements of this form are true.
The Early Development of Set Theory
For example, assume that objects are triangles, "a part" means a triangle inside the given triangle, "similar" is interpreted as usual in elementary geometry : triangles related by a dilation; the statement fails badly: every triangle X evidently is similar to some triangle inside Y, the other way round. Instead of the relation "be a part of" one may use a binary relation "be embeddable into" interpreted as "be similar to some part of". The relation "injects into" is a preorder, "be isomorphic" is an equivalence relation. Embeddability is a preorder, similarity is an equivalence relation.
Srivastava, S. Kadison, Richard V.. Gowers, W.
London Math. Casazza, P. D; when does Cantor Bernstein hold? After teaching school for a few years, he moved to the Technische Hochschule Darmstadt in Two years he took up a chair in mathematics at the Polytechnische Schule in Karlsruhe , where he spent the remainder of his life, he never married. To their work he subsequently added several important concepts due to Charles Sanders Peirce, including subsumption and quantification.
Felix Bernstein subsequently corrected the proof as part of his Ph. The Vorlesungen was a comprehensive and scholarly survey of "algebraic" logic up to the end of the 19th century, one that had a considerable influence on the emergence of mathematical logic in the 20th century; the Vorlesungen is a prolix affair, only a small part of, translated into English.
That part, along with an extended discussion of the entire Vorlesungen, is in Brady. See Grattan-Guinness. Peirce's work on quantification, is at least as great as that of Frege or Peano. Leipzig : B. Leipzig: B. Reprints: , Chelsea. Naturf — Both Primary and Secondary Brady, Geraldine, From Peirce to Skolem. North Holland. Includes an English translation of parts of the Vorlesungen. Secondary Anellis, I. Dipert, R. Frege, G. Blackwell: 86— Princeton University Press.
Clarence Irving Lewis, A Survey of Symbolic Logic. Peckhaus, V. Logik, Mathesis universalis und allgemeine Wissenschaft. Leibniz und die Wiederentdeckung der formalen Logik im Reprinted in Glen van Brummelen and Michael Kinyon, eds. Mathematics and the Historian's Craft; the Kenneth O.
May Lectures. Springer: — Online here or here. Handbook of the History of Logic. North Holland: — Hilary Putnam, International Congress of Mathematicians The International Congress of Mathematicians is the largest conference for the topic of mathematics. It meets once every four years, hosted by the International Mathematical Union ; the Fields Medals, the Nevanlinna Prize , the Gauss Prize , the Chern Medal are awarded during the congress's opening ceremony.
Each congress is memorialized by a printed set of Proceedings recording academic papers based on invited talks intended to be relevant to current topics of general interest. Being invited to talk at the ICM has been called "the equivalent of an induction to a hall of fame. The congress was attended by mathematicians from 16 countries, including 12 from Russia and 7 from the U. During the congress in Paris , David Hilbert announced his famous list of 23 unsolved mathematical problems , now termed Hilbert's problems.
Moritz Cantor and Vito Volterra gave. During the congress in Cambridge, Edmund Landau listed four basic problems about prime numbers, now called Landau's problems ; the congress in Toronto was organized by initiator of the Fields Medal. This resulted in a still unresolved controversy as to whether to count the Strasbourg and Toronto congresses as true ICMs. The congress was attended by 3, participants. The American Mathematical Society reported that more than 4, participants attended the conference in Madrid, Spain.
The King of Spain presided over the conference opening ceremony; the Congress took place in Hyderabad, India , on August 19—27, The organizing committees of the early ICMs were formed in large part on an ad hoc basis and there was no single body continuously overseeing the ICMs. At the IRC's instructions, in the Union Mathematique Internationale was created; this was the immediate predecessor of the current International Mathematical Union. Under the IRC's pressure, UMI reassigned the congress from Stockholm to Strasbourg and insisted on the rule which excluded from the congress mathematicians representing the former Central Powers.
The exclusion rule, which applied to the ICM, turned out to be quite unpopular among mathematicians from the U. The ICM was scheduled to be held in New York, but had to be moved to Toronto after the American Mathematical Society withdrew its invitation to host the congress, in protest against the exclusion rule; as a result of the exclusion rule and the protests it generated, the and the ICMs were smaller than the previous ones.
In the face of the protests against the exclusion rule and the possibility of a boycott of the congress by the American Mathematical Society and the London Mathematical Society , the congress's organizers decided to hold the ICM under the auspices of the University of Bologna rather than of the UMI; the congress and all the subsequent congresses have been open for participation by mathematicians of all countries. At the ICM the participants voted to reconstitute the International Mathematical Union, formally established in Starting with the congress. Felix Hausdorff Felix Hausdorff was a German mathematician , considered to be one of the founders of modern topology and who contributed to set theory, descriptive set theory, measure theory, function theory, functional analysis.
Life became difficult for Hausdorff and his family after Kristallnacht in ; the next year he initiated efforts to emigrate to the United States , but was unable to make arrangements to receive a research fellowship. On 26 January , Felix Hausdorff, along with his wife and his sister-in-law, committed suicide by taking an overdose of veronal , rather than comply with German orders to move to the Endenich camp, there suffer the implications, about which he held no illusions.
Hausdorff's father, the Jewish merchant Louis Hausdorff, moved in the autumn of with his young family to Leipzig and worked over time at various companies, including a linen-and cotton goods factory, he was an educated man and had become a Morenu at the age of There are several treatises from his pen, including a long work on the Aramaic translations of the Bible from the perspective of Talmudic law.
Hausdorff's mother, referred to in various documents as Johanna, came from the Jewish Tietz family. From another branch of this family came Hermann Tietz , founder of the first department store, co-owner of the department store chain called "Hermann Tietz". During the period of Nazi dictatorship the name was "Aryanised" to Hertie.
From to Felix Hausdorff attended the Nicolai School in Leipzig, a facility that had a reputation as a hotbed of humanistic education, he was an excellent student, class leader for many years and recited self-written Latin or German poems at school celebrations. In his graduation in , he was the only one; the choice of subject was not easy for Hausdorff. Magda Dierkesmann, a guest in the home of Hausdorff as a student in Bonn in the years —, reported in that: His versatile musical talent was so great that only the insistence of his father made him give up his plan to study music and become a composer; the decision was made to study the sciences in high school.
From summer term to summer semester Hausdorff studied mathematics and astronomy in his native city of Leipzig, interrupted by one semester in Freiburg and Berlin. The surviving testimony of other students show him as versatile interested young man, who, in addition to the mathematical and astronomical lectures, attended lectures in physics and geography, lectures on philosophy and history of philosophy as well as on issues of language and social sciences. In Leipzig he heard lectures on the history of music from musicologist Paul, his early love of music lasted a lifetime.
As a student in Leipzig, he was an admirer and connoisseur of the music of Richard Wagner. In semesters of his studies, Hausdorff was close to Heinrich Bruns. Bruns was professor of director of the observatory at the University of Leipzig. Under him, Hausdorff graduated in with a work on the theory of astronomical refraction of light in the atmosphere.
Two publications on the same subject followed, in his habilitation followed with a thesis on the absorbance of light in the atmosphere. These early astronomical works of Hausdorff have—despite their excellent mathematical working through—not gained importance. Firstly, the underlying idea of Bruns has not proved viable. On the other hand, the progress in the direct measurement of atmospheric data has since made the painstaking accuracy of this data from refraction observations unnecessary.
In the time between PhD and habilitation Hausdorff completed the yearlong-volunteer military requirement and worked for two years as a human computer at the observatory in Leipzig. With his habilitation, Hausdorff became a lecturer at the University of Leipzig and began an extensive teaching in a variety of mathematical areas. In addition to teaching and research in mathematics, he went with his literary and philosophical inclinations. A man of varied interests, educated sensitive and sophisticated in thinking and experiencing, he frequented in his Leipzig period with a number of famous writers and publishers such as Hermann Conradi , Richard Dehmel , Otto Erich Hartleben , Gustav Kirstein, Max Klinger , Max Reger and Frank Wedekind.
The years to about mark the high point of his literary and philosophical creativity, during which time 18 of his 22 pseudonymous works were published, including a book of poetry, a play, an epistemological book and a volume of aphorisms. Hausdorff married Charlotte Goldschmidt in daughter of Jewish doctor Siegismund Goldschmidt, her stepmother was preschool teacher Henriette Goldschmidt. Hausdorff's only child, daughter Lenore, was born in When applying in Leipzig , Dean Kirchner had been led to positive vote of his colleagues, written by Hei.
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Revision History. Walther von Dyck. Related Images. After agreeing on definitions, Toby Bartels gave the answer, which I am taking the liberty to adapt a bit and present here. I am probably just reinventing the wheel, so if someone knows an original reference, please provide it in the comments. The theorem holds constructively, but for a bizarre reason: if there exists a complete ordered field, then the law of excluded middle holds, and the standard proof is valid! Published as arXiv Abstract: This note recapitulates and expands the contents of a tutorial on the mathematical theory of algebraic effects and handlers which I gave at the Dagstuhl seminar "Algebraic effect handlers go mainstream".
It is targeted roughly at the level of a doctoral student with some amount of mathematical training, or at anyone already familiar with algebraic effects and handlers as programming concepts who would like to know what they have to do with algebra. We draw an uninterrupted line of thought between algebra and computational effects. We begin on the mathematical side of things, by reviewing the classic notions of universal algebra: signatures, algebraic theories, and their models.
We then generalize and adapt the theory so that it applies to computational effects. In the last step we replace traditional mathematical notation with one that is closer to programming languages. You may have noticed that lately I have had trouble with the blog. It was dying periodically because the backend database kept crashing.
It was high time I moved away from Wordpress anyway, so I bit the bullet and ported the blog. It was the "outsider" talk, where they invite someone to tell them something outside of their area. So how does one sell homotopy type theory to people who are interested in combinatorics? That is a tough sell. I used my MathOverflow question "What is an explicit bijection? I am told I plunged a little too hard. For instance, people asked "why are we doing this" because I did not make it clear enough that we are trying to make a distinction between "abstractly exists" and "concretely constructed".
Anyhow, I hope you can get something useful from the slides. Download slides: what-is-an-explicit-bijection. Video recording of the lecture is now available. This semester my colleague Jaka Smrekar and I are teaching a graduate course on homotopy theory and homotopy type theory.
The first part was taught by Jaka and was a nice review of classical homotopy theory leading up to Quillen model categories.
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In the second part I am covering basic homotopy type theory. The course materials are available at the GitHub repository homotopy-type-theory-course. The homotopy type theory lectures are also recorded on video. I was purging the disk on my laptop of large files and found a video lecture which I forgot to publish. Here it is with some delay. You may watch the video and visit the accompanying GitHub repository spartan-type-theory.
The notes, materials and the lectures are available online:. I gave four lectures which started with the mathematics of algebraic theories, explained how they can be used to model computational effects, how we make a programming language out of them, and how to program with handlers.
The abstract for the talk is available online. It describes a complete formalization of dependent type theory which allows you to turn various features of type theory on and off, and it proves several basic formal theorems. The programme starts in October and lasts three years. The positions will be fully funded subject to approval by the funding agency. No knowledge of Slovene is required. The first PhD student will be advised by dr. Andrej Bauer. The topic of research is foundations of type theory. The candidate should have interest in mathematical aspects of type theory, and familiarity with proof assistants is desirable.
The second PhD student will be advised by dr. Matija Pretnar. The topic of research is the theory of programming languages with a focus on computational effects. The candidate should have interest in both the mathematical foundations and practical implementation of programming languages.
Candidates should send their applications as soon as possible, but no later than the end of April, to Andrej Bauer andrej. Please include a short CV, academic record, and a statement of interest. I turned the talk into a paper, polished it up a bit, added things here and there, and finally it has now been published in the Bulletin of the American Mathematical Society. It is not quite a survey paper, but it is not very technical either.
I hope you will enjoy reading it. It is my pleasure to announce the new and improved Programming languages Zoo , a potpourri of miniature but fully functioning programming language implementations. Many thanks to Matija Pretnar for all the work. The purpose of the zoo is to demonstrate design and implementation techniques, from dirty practical details to lofty theoretical considerations:. I have participated in a couple of lengthy discussions about formal proofs.
I realized that an old misconception is creeping in. Let me expose it. I much agree with what he says, but I would like to offer my own perspective. Here are the slides of my talk, with speaker notes, as well as the Andromeda examples that I am planning to cover. Here are the slides, with extensive speaker notes, comment and questions are welcome.
Thinking about what they did I realized that their conditions allow a self-interpreter for practically any total language expressive enough to encode numbers and pairs. It is clear from the construction that I abused the definition given by Brown and Palsberg. Their self-interpreter has good structural properties which mine obviously lacks. So what we really need is a better definition of self-interpreters, one that captures the desired structural properties. Can someone suggest a good definition?
A postdoc position in the Effmath research project is available at the University of Ljubljana, Faculty of Mathematics and Physics. The precise topic is flexible, but should generally be aligned with the project see project description.
Pacific Journal of Mathematics
Possible topics include:. The candidate should have a PhD degree in mathematics or computer science, with background knowledge relevant to the project area. The position is available for a period of one year with possibility of extension, preferably starting in early No knowledge of the Slovene language is required. The candidates should contact Andrej Bauer by email as soon as possible, but no later than January 8th Please include a short CV and a statement of interest. Agda Writer is open source. Everybody is welcome to help out and participate on the Agda Writer repository.
Who is Agda Writer for? Obviously for students, mathematicians, and other potential users who were not born with Emacs hard-wired into their brains. It is great for teaching Agda as you do not have to spend two weeks explaining Emacs. The only drawback is that it is limited to OS X. Someone should write equivalent Windows and Linux applications.
Then perhaps proof assistants will have a chance of being more widely adopted. Stop it. Stop it! I have not written a blog post in a while, so I decided to write up a short observation about truth values in intuitionistic logic which sometimes seems a bit puzzling. A position is available for a PhD student at the University of Ljubljana in the general research area of modelling and reasoning about computational effects.
The precise topic is somewhat flexible, and will be decided in discussion with the student. The position will be funded by the Effmath project see project description. The student will officially enrol in October at the University of Ljubljana. Simpson fmf. I am looking for a PhD student in mathematics. The topic of research is somewhat flexible and varies from constructive models of homotopy type theory to development of a programming language for a proof assistant based on dependent type theory, see the short summary of the Effmath project for a more detailed description.
However, it is possible, and even desirable, to start with the actual work and stipend earlier, as soon as in the spring of The candidates should contact me by email as soon as possible. The topic was how programming influences various aspects of life. I showed the audence how a bit of simple programming can reveal the beauty of mathematics.http://gon337trungvan.dev3.develag.com/recalled-landscapes.php
Mathematics and Computation | All posts
Who can give me 30 degrees? Here are the slides from my Logic Coloquium talk in Vienna. For instance, it is well known that the Limited principle of omniscience implies that equality of real numbers is decidable. Most such reductions proceed by reducing an instance of the consequent to an instance of the antecedent. We may therefore define a notion of instance reducibility , which turns out to have a very rich structure.
We may also ask about a constructive treatment of other reducibilities in computability theory. In the post Seemingly impossible functional programs , I wrote increasingly efficient Haskell programs to realize the mathematical statement. I also look at ways of constructing new omniscient sets from given ones. Such sets include, in particular, ordinals , for which we can find minimal witnesses if any witness exists. By the Curry-Howard correspondence , Agda is also a language for formulating mathematical theorems types and writing down their proofs programs.
Agda acts as a thorough referee, only accepting correct theorems and proofs. Moreover, Agda can run your proofs. Here is a graph of the main Agda modules for this post, and here is a full graph with all modules. I spoke about a new proof checker Chris Stone and I are working on. It is based on a homotopy type system proposed by Vladimir Voevodsky. Download slides: brazilian-type-checking. Abstract: Proof assistants verify that inputs are correct up to judgmental equality.
Proofs are easier and smaller if equalities without computational content are verified by an oracle, because proof terms for these equations can be omitted. In order to keep judgmental equality decidable, though, typical proof assistants use a limited definition implemented by a fixed equivalence algorithm.
While other equalities can be expressed using propositional identity types and explicit equality proofs and coercions, in some situations these create prohibitive levels of overhead in the proof. Voevodsky has proposed a type theory with two identity types, one propositional and one judgmental. This lets us hypothesize new judgmental equalities for use during type checking, but generally renders the equational theory undecidable without help from the user. As a special case, we retain a simple form of handlers even in the final proof terms, small proof-specific hints that extend the trusted verifier in sound ways.
Consider it a teaser for the rest of the book, which contains papers by an impressive list of authors. Abstract: Intuitionistic mathematics perceives subtle variations in meaning where classical mathematics asserts equivalence, and permits geometrically and computationally motivated axioms that classical mathematics prohibits. It is therefore well-suited as a logical foundation on which questions about computability in the real world are studied.
The realizability interpretation explains the computational content of intuitionistic mathematics, and relates it to classical models of computation, as well as to more speculative ones that push the laws of physics to their limits. Through the realizability interpretation Brouwerian continuity principles and Markovian computability axioms become statements about the computational nature of the physical world. A discussion on the homotopytypetheory mailing list prompted me to write this short note. Apparently a mistaken belief has gone viral among certain mathematicians that Univalent foundations is somehow limited to constructive mathematics.
This is false. Let me be perfectly clear:. Univalent foundations subsume classical mathematics! The next time you hear someone having doubts about this point, please refer them to this post. A more detailed explanation follows. For this purpose she needed to triangulate the and compute normals to it at the vertices.
It is not too hard to do the latter part, and the Internet offers several ways of doing it, but the normals are a bit tricky. I went back to my undergraduate days when I actually did differential geometry and churned out the normals with Mathematica. The code is released into public domain. Here is mine deliberately non-fancy :. Apparently the following belief is widely spread, and I admit to holding it a couple of years ago:. An inductive type contains exactly those elements that we obtain by repeatedly using the constructors. If you believe the above statement you should keep reading.
I am going to convince you that the statement is unfounded, or that at the very least it is preventing you from understanding type theory. Recently I reviewed a paper in which most proofs were done in a proof assistant. Yes, the machine guaranteed that the proofs were correct, but I still had to make sure that the authors correctly formulated their definitions and theorems, that the code did not contain hidden assumptions, that there were no unfinished proofs, and so on.
In a typical situation an author submits a paper accompanied with some source code which contains the formalized parts of the work. Sometimes the code is enclosed with the paper, and sometimes it is available for download somewhere. It is easy to ignore the code! The journal finds it difficult to archive the code, the editor naturally focuses on the paper itself, the reviewer trusts the authors and the proof assistant, and the authors are tempted not to mention dirty little secrets about their code.
If the proponents of formalized mathematics want to avert a disaster that could destroy their efforts in a single blow, they must adopt a set of rules that will ensure high standards. There is much more to trusting a piece of formalized mathematics than just running it through a proof checker.
Since spring, and even before that, I have participated in a great collaborative effort on writing a book on Homotopy Type Theory. It is finally finished and ready for public consumption. Mike Shulman has written about the contents of the book , so I am not going to repeat that here. Instead, I would like to comment on the socio-technological aspects of making the book, and in particular about what we learned from open-source community about collaborative research. Mathematicians are often confused about the meaning of variables.
I am addressing this post to those who share this opinion. I spent a week trying to implement higher-order pattern unification.
I looked at couple of PhD dissertations, talked to lots of smart people, and failed because the substitutions were just getting in the way all the time. So today we are going to bite the bullet and implement de Bruijn indices and explicit substitutions. I am spending a semester at the Institute for Advanced Study where we have a special year on Univalent foundations. We are doing all sorts of things, among others experimenting with type theories. In the meanwhile I have been thinking how one might implement dependent type theories with undecidable type checking.
This is a tricky subject and I am certainly not the first one to think about it. Anyhow, if I want to experiment with type theories, I need a small prototype first. Today I will present a very minimal one, and build on it in future posts. Make a guess, how many lines of code does it take to implement a dependent type theory with universes, dependent products, a parser, lexer, pretty-printer, and a toplevel which uses line-editing when available? It seems to me that people think I am a constructive mathematician, or worse a constructivist a word which carries a certain amount of philosophical stigma.
Let me be perfectly clear: it is not decidable whether I am a constructive mathematician. I am sitting on a tutorial on categorical semantics of dependent type theory given by Peter Lumsdaine. He is talking about categories with attributes and other variants of categories that come up in the semantics of dependent type theory. He is amazingly good at fielding questions about definitional equality from the type theorists.
And it looks like some people are puzzling over pullbacks showing up, which Peter is about to explain using syntactic categories. Here is a pedestrian explanation of a very important fact:. Egbert Rijke successfully defended his master thesis in Utrecht a couple of weeks ago. He published it on the Homotopy type theory blog here is a direct link to the PDF file revised.
The thesis is well written and it contains several new results, but most importantly, it is a gentle yet non-trivial introduction to homotopy type theory. What I find most amazing about the work is that Egbert does not have to pretend to be a homotopy type theorist, like us old folks. If we perform enough such experiments on young bright students, strange things will happen. This is an advertisement for two great meetings we are organizing in Ljubljana from June 15 to June 20, Not to mention that the schedule is fairly light, everything is within walking distance, and we are organizing dinners at some excellent restaurants.
If you decide to come, make sure to book a hotel early and register today! I remember how hard it was to assimilate category theory when I was a student. A beginning student on math. It really is the sort of thing one should do by oneself. Nevertheless, here it is, in gory details. It is well known that, both in constructive mathematics and in programming languages, types are secretly topological spaces and functions are secretly continuous. I have previously exploited this in the posts Seemingly impossible functional programs and A Haskell monad for infinite search in finite time , using the language Haskell.
What is it? There are a number of ways of approaching topology. The most popular one is via open sets. For some spaces, one can instead use convergent sequences , and this approach is more convenient in our situation. The Agda pages can be navigated be clicking at any defined symbol or word, in particular by clicking at the imported module names.
Matija and I are pleased to announce a new major release of the eff programming language. Abstract: Eff is a programming language based on the algebraic approach to computational effects, in which effects are viewed as algebraic operations and effect handlers as homomorphisms from free algebras. Eff supports first-class effects and handlers through which we may easily define new computational effects, seamlessly combine existing ones, and handle them in novel ways. We give a denotational semantics of eff and discuss a prototype implementation based on it.
Through examples we demonstrate how the standard effects are treated in eff, and how eff supports programming techniques that use various forms of delimited continuations, such as backtracking, breadth-first search, selection functionals, cooperative multi-threading, and others. ArXiv version: arXiv If I am not mistaken, all such terms can be typed. For example:. For example, the Ackermann function can be typed as follows, although the type prevents it from doing the right thing in a typed setting:.
It is provable classically, but not intuitionistically. We study this and related principles in an intuitionistic setting. Among other things, we show that Bourbaki-Witt fails exactly when the trichotomous ordinals form a set, but does not imply that fixed points can always be found by transfinite iteration. Meanwhile, on the side of models, we see that the principle fails in realisability toposes, and does not hold in the free topos, but does hold in all cocomplete toposes. Download paper: bw. This paper is an extension of my previous paper on the Bourbaki-Witt and Knaster-Tarski fixed-point theorems in the effective topos arXiv On December 6th I gave a talk about homotopy equivalences in the context of homotopy type theory at our seminar for foundations of mathematics and theoretical computer science.
I discuss the differences and relations between isomorphism in the sense of type theory , an adjoint equivalence, and a homotopy equivalence. Even though the talk itself was not super-well prepared, I hope the recording will be interesting to some people. I was going fairly slowly, so it should be possible to follow the talk. I apologize for such a long video, but I really did not see how to chop it up into smaller pieces. Also, I need to figure out why I cannot fast forward the video beyond what has been downloaded. I describe a realizability model based on infinite-time Turing machines in which it is possible to embed the Baire space infinite sequences of numbers into the space of numbers.
Also see the post Constructive gem: an injection from Baire space to natural numbers for written notes on this topic. Last year I participated in a project whose goal was to record at low cost my lectures on video and put them on-line. Since the most expensive parts of recording are having a camera man and manual post production, we set up a static camera and just uploaded raw video online at videolectures. As you can see for yourself, the sound is good I wore a microphone but the whiteboard is mostly illegible.
In addition, it took about two weeks for the lectures to show up on-line because there were men-in-the-middle. So that got me thinking whether there was a better way. In an earlier post I talked about the modulus of continuity functional, where I stated that it cannot be defined without using some form of computational effects. It is a bit hard to find the proof of this fact so I am posting it on my blog in two parts, for Google and everyone else to find more easily. In the first part I show that there is no extensional modulus of continuity. I am not sure whether to call this one a constructive gem or stone.
I suppose it is a matter of personal taste. Classical Stone duality tells us that Boolean algebras are dual to Stone spaces zero-dimensional compact Hausdorff spaces , and that the generalized Boolean algebras which are like Boolean algebras without a top element are dual to Boolean spaces zero-dimensional locally-compact Hausdorff spaces. We use the duality to construct a right-handed skew Boolean algebra with intersections which does not have a lattice section. It has been an open question whether such skew lattices exist.
Abstract: We extend Stone duality between generalized Boolean algebras and Boolean spaces, which are the zero-dimensional locally-compact Hausdorff spaces, to a non-commutative setting. We then extend the duality to skew Boolean algebras with intersections, and consider several variations in which the morphisms are restricted. Finally, we use the duality to construct a right-handed skew Boolean algebra without a lattice section. This we can run, and it runs fast enough.
The point is to illustrate in Agda how we can get witnesses from classical proofs that use countable choice. The finite pigeonhole principle has a simple constructive proof, of course, and hence this is really for illustration only. These are Agda files converted to html so that you can navigate them by clicking at words to go to their definitions.
A zip file with all Agda files is available.
MSC Classification Codes
Not much more information is available here. The three little modules that implement the Berardi-Bezem-Coquand, Berger-Oliva and Escardo-Oliva functionals disable the termination checker, but no other module does. The type of these functionals in Agda is the J-shift principle , which generalizes the double-negation shift. He is a leading expert in Programming Languages. I remember being deeply inspired the first time I heard him talk.
His posts are fun to read, unreserved and very educational. Highly recommended! Next week I am going to a meeting where I am supposed to give a tutorial on the Coq proof assistant. Inspired by the Catsters , I decided to prepare the material in the form of screencasts.
So far I have:. You should turn on the high quality HD stream when you watch these. Feedback is welcome and easy to provide on Youtube. I find it very, very difficult to listen to my own voice. I hope to have many more lectures soon, but I am starting to feel out of my depth, so if anyone wants to help they are welcome! I usually just post the abstract, but this time I would like to explain the general idea informally, the way one can do it on a blog. But first, here is the abstract:.
Abstract: We prove general theorems about unique existence of effective subalgebras of classical algebras. The theorems are consequences of standard facts about completions of metric spaces within the framework of constructive mathematics, suitably interpreted in realizability models. We work with general realizability models rather than with a particular model of computation. Consequently, all the results are applicable in various established schools of computability, such as type 1 and type 2 effectivity, domain representations, equilogical spaces, and others.
Download paper: effalg. Alg is a program for enumeration of finite models of single-sorted first-order theories. These include groups, rings, fields, lattices, posets, graphs, and many more. I joined the effort, added bells and whistles, as well as an alternative algorithm that works well for relational structures. Instructions for downloading alg are included at the end of this post. Alex Simpson , Matija Pretnar and I are organizing a workshop on computational effects. It will take place in Ljubljana on March 17th and 18th More information is available at the workshop web page.
I think I am getting addicted to Coq , or more generally to doing mathematics, including the proofs, with computers. There does not seem to be one available already, which is a good opportunity for a blog post. I would like to record the following fact, which is hard to find on the internet: every subgroup is an equalizer, constructively. In other words, all monos in the category of groups are regular, constructively. Here are delimited continuations in eff, and a bunch of questions I do not know the answers to. From some of the responses we have been getting it looks like people think that the io effect in eff is like unsafePerformIO in Haskell, namely that it causes an effect but pretends to be pure.
This is not the case. Let me explain how eff handles built-in effects. We covered the theory behind it in a previous post. Now we turn to the programming language itself. Please bear in mind that eff is an academic experiment. It is not meant to take over the world.
We just wanted to show that the theoretical ideas about the algebraic nature of computational effects can be put into practice. Eff has many superficial similarities with Haskell. This is no surprise because there is a precise connection between algebras and monads. The main advantage of eff over Haskell is supposed to be the ease with which computational effects can be combined.
For updated information, please visit the eff page. It was a great visit, everyone was very hospitable, the food was great, and the weather was nice. I spoke at their seminar where I presented a new programming language eff which is based on the idea that computational effects are algebras.