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1. Some Consequences Of Martin S Axiom And The Negation Of The
Some consequences of Martin s axiom and the negation of the Continuum hypothesis. Juichi Shinoda. Source Nagoya Math. J. Volume 49 (1973), 117125.
http://projecteuclid.org/handle/euclid.nmj/1118798877
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##### Some consequences of Martin's axiom and the negation of the continuum hypothesis
Juichi Shinoda Source: Nagoya Math. J. Volume 49 (1973), 117-125. Primary Subjects: Full-text: Access granted (open access) PDF File (745 KB) Links and Identifiers Permanent link to this document: http://projecteuclid.org/euclid.nmj/1118798877 Mathematical Reviews number (MathSciNet): Zentralblatt Math identifier: back to Table of Contents
##### References
[1] K. Kunen, Inaccessibility properties of cardinals, Doctoral Dissertation, Stanford University, 1968. [2] D. A. Martin and R. M. Solovay, Internal Cohen extensions, Annals of Math. Logic vol. 2 (1970) 143-178. Mathematical Reviews (MathSciNet): Zentralblatt MATH: [3] W. Sierpinski, Hypothese du contnu, Second Edition (Chelsea, New York, 1956). Nagoya University Mathematical Reviews (MathSciNet): previous next
##### Nagoya Mathematical Journal

 2. JSTOR Higher Souslin Trees And The Generalized Continuum Hypothesis. Generalized Martin s axiom and Souslin s hypothesis for higher cardinals. axiom (MA) together with the negation of the Continuum hypothesis (CH) implieshttp://links.jstor.org/sici?sici=0022-4812(198406)49:2<663:HSTATG>2.0.CO;2-D

3. PlanetMath: A Shorter Proof: Martin's Axiom And The Continuum Hypothesis
This is version 8 of a shorter proof Martin s axiom and the Continuum hypothesis, born on 200308-24, modified 2004-03-15.
http://planetmath.org/encyclopedia/SomethingRelatedToMartinsAxiomAndTheContinuum
 (more info) Math for the people, by the people. Encyclopedia Requests Forums Docs ... RSS Login create new user name: pass: forget your password? Main Menu sections EncyclopÂ¦dia Papers Books Expositions meta Requests Orphanage Unclass'd Unproven ... Classification talkback Polls Forums Feedback Bug Reports downloads Snapshots PM Book information News Docs Wiki ChangeLog ... About a shorter proof: Martin's axiom and the continuum hypothesis (Proof) This is another, shorter, proof for the fact that always holds. Let be a partially ordered set and be a collection of subsets of . We remember that a filter on is -generic if for all which are dense in is dense in , then for every there's a such that Let be a partially ordered set and a countable collection of dense subsets of . Then there exists a -generic filter on . Moreover, it could be shown that for every there's such a -generic filter with Proof . Let be the dense subsets in . Furthermore let . Now we can choose for every an element such that and . If we now consider the set , then it is easy to check that is a -generic filter on and obviously. This completes the proof.

4. A Shorter Proof: Martin's Axiom And The Continuum Hypothesis
This is another, shorter, proof for the fact that always holds. Let be a partially ordered set and be a collection of subsets of P. We remember
http://nsdl.org/resource/2200/20061003062432262T

5. Book's Preface
The book is finished with Section 9.5 in which it is proved the simultaneous consistency of Martin s axiom and the negation of Continuum hypothesis.
http://www.math.wvu.edu/~kcies/Preface.html
##### Cambridge University Press, 1997.
Hardback ISBN 0-521-59441-3, price $59.95; paperback ISBN 0-521-59465-0, price$19.95.
To order call 1-800-872-7423 or link to Cambridge University Press order page.
##### Preface
The course is presented as a textbook that is appropriate for either a lower-level graduate course or an advanced undergraduate course. However, the potential readership should also include mathematicians whose expertise lies outside set theory but who would like to learn more about modern set-theoretic techniques that might be applicable in their field.
The reader of this text is assumed to have a good understanding of abstract proving techniques, and of the basic geometric and topological structure of the n-dimensional Euclidean space R n . In particular, a comfort in dealing with the continuous functions from R n into R n is assumed. A basic set-theoretic knowledge is also required. This includes a good understanding of the basic set operations (union, intersection, Cartesian product of arbitrary families of sets, and difference of two sets), abstract functions (the operations of taking images and preimages of sets with respect to functions), and elements of the theory of cardinal numbers (finite, countable, and uncountable sets.) Most of this knowledge is included in any course in analysis, topology, or algebra. These prerequisites are also discussed briefly in Part I of the text.

6. Uri Abraham
Assuming the Continuum hypothesis there is an inseparable sequence of length $\omega_1$ that contains no Lusin subsequence, while if Martin s axiom and
http://www.cs.bgu.ac.il/~abraham/math.html
##### Uri Abraham
This page contains papers in set theory with abstracts. It seems that html is not very well suited to math expressions, and so I've used the $latex$ notations.
I have a different homepage for CS papers: cs homepage
• ##### Infinite games on finite sets.
with Rene Schipperus. Israel J. of Math. 159 (2007) 205219. ( pdf.
• ##### On Jakovlev spaces.
with Isac Gorelic and Istvan Juhasz. Israel Journal of Mathematics 152 (2006) 205219. ( pdf.
• with Stefan Geschke. Uri Abraham, Stefan Geschke, Proc. Amer. Math. Soc. 132 (2004), 3367-3377. ( dvi, 10pp
• ##### Ladder gaps over stationary sets.
with S. Shelah. J. Symbolic Logic 69 (2004), no 2, 518-532. ( dvi 21 pp
• ##### Lusin sequences under CH and under Martin's Axiom.
with Saharon Shelah.
Fundamenta Mathematicae 169 (2001), pp 97-103. ps 8 pp
• ##### Hausdorff's theorem for posets that satisfy the finite antichain property.
with Robert Bonnet.
Fundamenta Mathematicae , 159 (1999) pp. 5169. ( ps 24 pp abstract
• with Saharon Shelah.
Archive for Mathematical Logic compressed ps 16 a link to Springer's Archive for Mathematical Logic version See also Coding with Ladders for an extension which does not require an inaccessible cardinal.
• 7. Springer Online Reference Works
The existence of a Luzin space on the real line follows from the Continuum hypothesis. From the negation of the Continuum hypothesis and Martin s axiom (cf.
http://eom.springer.de/l/l061110.htm

Encyclopaedia of Mathematics
L
Article referred from
Article refers to
Luzin space
An uncountable topological -space without isolated points in which every nowhere-dense subset is countable. The existence of a Luzin space on the real line follows from the continuum hypothesis . From the negation of the continuum hypothesis and Martin's axiom (cf. Suslin hypothesis axiom of choice . The existence of metrizable Luzin spaces has been proved under very general assumptions about the place of the cardinality of the continuum in the scale of alephs. Any Luzin space that lies in a separable metric space has the following property: For any sequence of positive numbers there is a sequence of sets such that and , where is the diameter of the set . This property is invariant under continuous mappings. Any continuous image of a Luzin space lying in -weight and with the cardinality of the continuum.
##### References
B.A. Efimov
Three slightly different definitions of Luzin space are still in use (apart from whether they must be or ): An uncountable space all of whose nowhere-dense sets are countable, with 1) no isolated points; or 2) at most

8. Hilbert's First And Second Problems And The Foundations Of Mathematics By Peter
D.H. Fremlin, Consequences of Martin s axiom, Cambridge University Press, 1984. inter alia, why GÃ¶del believed the Continuum hypothesis to be dubious
http://at.yorku.ca/t/a/i/c/52.htm
Topology Atlas Document # taic-52 Topology Atlas Invited Contributions vol. 9, no. 3 (2004) 6 pp.
##### Peter J. Nyikos
Department of Mathematics
University of South Carolina
Columbia, SC 29208 USA
http://www.math.sc.edu/~nyikos
In 1900, David Hilbert gave a seminal lecture in which he spoke about a list of unsolved problems in mathematics that he deemed to be of outstanding importance. The first of these was Cantor's continuum problem, which has to do with infinite numbers with which Cantor revolutionised set theory. The smallest infinite number, , aleph-nought,' gives the number of positive whole numbers. A set is of this cardinality if it is possible to list its members in an arrangement such that each one is encountered after a finite number (however large) of steps. Cantor's revolutionary discovery was that the points on a line cannot be so listed, and so the number of points on a line is a strictly higher infinite number ( c , the cardinality of the continuum') than . Hilbert's First Problem asks whether any infinite subset of the real line is of one of these two cardinalities. The axiom that this is indeed the case is known as the Continuum Hypothesis ( CH This problem had unexpected connections with Hilbert's Second Problem (and even with the Tenth, see the article by M. Davis and the comments on the book edited by F. Browder). The Second Problem asked for a proof of the consistency of the foundations of mathematics. Some of the flavor of the urgency of that problem is provided by the following passage from an article by S.G. Simpson in the same volume of JSL as the article by P. Maddy:

9. CiteULike: Axioms Of Symmetry: Throwing Darts At The Real Number Line
We will also show why Martin s axiom must be false, and we will prove the extension of proof of the negation of Cantor s Continuum hypothesis (CH).
http://www.citeulike.org/user/rwebb/article/2088308
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##### Copy-and-Pasteable Citation
The Journal of Symbolic Logic , Vol. 51, No. 1. (1986), pp. 190-200. Citation format: Plain APA Chicago Elsevier Harvard MLA Nature Oxford Science Turabian Vancouver
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All tags in rwebb's library Filter: distance distributed esn hierarchical ... sensory CiteULike organises scholarly (or academic) papers or literature and provides bibliographic (which means it makes bibliographies) for universities and higher education establishments. It helps undergraduates and postgraduates. People studying for PhDs or in postdoctoral (postdoc) positions. The service is similar in scope to EndNote or RefWorks or any other reference manager like BibTeX, but it is a social bookmarking service for scientists and humanities researchers.

10. Topics In Discovering Modern Set Theory. II.
Applications of the Continuum hypothesis. Luzin sets and Sierpinski sets. Martin s axiom. Consistency of the Suslin hypothesis. Solovay s Lemma.
http://www.math.ohiou.edu/~just/anno2.html
##### From the Preface:
"Our aim is to present the most important set-theoretic techniques that have found applications outside of set theory. We think of Volume II as a natural continuation of Volume I of the same text, but it is sufficiently self-contained to be studied separately. The main prerequisite is a knowledge of basic naive and axiomatic set theory. Moreover, some knowledge of mathematical logic and general topology is indispensible for reading this volume. A minicourse in mathematical logic was given in Chapters 5 and 6 of Volume I, and we include an appendix on general topology at the end of this volume. Our terminology is fairly standard. For the benefit of those readers who learned their basic set theory from a different source than our Volume I we include a short section on somewhat idiosyncratic notations introduced in Volume I. In particular, some of the material on mathematical logic covered in Chapter 5 is briefly reviewed. The book can be used as a text in the classroom as well as for self-study."
##### Chapter 13. Filters and Ideals in Partial Orders

11. Axiomatic Set Theory: Encyclopedia II - Axiomatic Set Theory - Independence In Z
The constructible universe satisfies the Generalized Continuum hypothesis, the Diamond Principle, Martin s axiom and the Kurepa hypothesis.
http://www.experiencefestival.com/a/Axiomatic_set_theory_-_Independence_in_ZFC/i
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##### Axiomatic set theory - Independence in ZFC: Encyclopedia II - Axiomatic set theory - Independence in ZFC
Many important statements are independent of ZFC, see the list of statements undecidable in ZFC. The independence is usually proved by forcing, that is, it is shown that every countable transitive model of ZFC (plus, occasionally, large cardinal axioms) can be expanded to satisfy the statement in question, and (through a different expansion) its negation. An independence proof by forcing automatically proves independence from arithmetical statements, other concrete statements, and large cardinal axioms. Some statements independent of ZFC can ...
Axiomatic set theory, Axiomatic set theory - The origins of rigorous set theory, Axiomatic set theory - Axioms for set theory, Axiomatic set theory - Independence in ZFC, Axiomatic set theory - Set theory ZFC foundations for mathematics, Axiomatic set theory - Well-foundedness and hypersets, Axiomatic set theory - Objections to set theory

 12. Topology And Forcing V I Malykhin RUSS MATH SURV, 1983, 38 (1), 77 5, Cohen P J, Set theory and the Continuum hypothesis, (W. A. Benjamin, 9, Malykhin V I, The equivalence of Martin s axiom and a purely topologicalhttp://www.turpion.org/php/reference.phtml?journal_id=rm&paper_id=3382&volume=38

13. Paul Joseph Cohen (American Mathematician) --Â Britannica Online Encyclopedia
axiom of choice and Continuum hypothesis. Two phenomenological studies by major philosophers are Martin More results
http://www.britannica.com/eb/topic-124547/article-9085726
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##### Paul Joseph Cohen (American mathematician)
A selection of articles discussing this topic.
##### Main article: Paul Joseph Cohen
American mathematician, who was awarded the Fields Medal in 1966 for his proof of the independence of the continuum hypothesis from the other axioms of set theory.
axiom of choice and continuum hypothesis
• ##### axiom of choice and continuum hypothesis (in logic, history of: 20th-century set theory)
...set theories with Urelementen very early; the independence of the axiom of choice in NBG or ZF set theories was one of the major outstanding problems in 20th-century mathematical logic until Paul Cohen showed in 1963 that the axiom of choice was indeed independent of the other standard axioms for set theory.
##### axiom of choice and continuum hypothesis (in axiom of choice)
...are consistent, then they do not disprove the axiom of choice. That is, the result of adding the axiom of choice to the other axioms (ZFC) remains consistent. Then in 1963 the American mathematician Paul Cohen completed the picture by showing, again under the assumption that ZF is consistent, that ZF does not yield a proof of the axiom of choice; that is, the axiom of choice is independent.

14. Reflexive Subgroups Of The Baer-Specker Group And Martin's Axiom
models of set theory, respectively of the special Continuum hypothesis (CH). We will use Martin s axiom to find reflexive modules with the above
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15. PlanetMath 2007-06-18 Snapshot: G
generic (defined in a shorter proof Martin s axiom and the Continuum hypothesis). generic generically (defined in generic). generic manifold
http://202.41.85.103/manuals/planetmath/G.html
 G Gabor frame (defined in Gabor frame Gabor frame Gabor super-frame (defined in Gabor frame Gabriel Filter (defined in multiplicative filter Gagliardo Nirenberg inequality Sobolev inequality gale (defined in gale gale Galois closure Galois cohomology (defined in Galois representation Galois conjugate Galois connection Galois connexion Galois connection Galois correspondence Galois connection Galois correspondence fundamental theorem of Galois theory Galois covering regular covering Galois criterion for solvability of a polynomial by radicals Galois extension Galois field finite field Galois group Galois group of a cover (defined in deck transformation Galois group of the compositum of two Galois extensions Galois is not transitive Galois representation Galois-theoretic derivation of the cubic formula ... Galois theory fundamental theorem of Galois theory game game (defined in game Game of Fifteen 15 Puzzle Game of Life game-theoretical quantifier game-theoretic quantifier (defined in game-theoretical quantifier game theory gamma distribution gamma random variable gamma function gamma function gamma-function gamma function gamma function (multivariate complex) (defined in multivariate gamma function (complex-valued) gamma function (multivariate real) (defined in multivariate gamma function (real-valued) Gamma L (defined in semilinear transformation gamma random variable Gao Dena Donald Knuth Garfield's proof of Pythagorean theorem Gaston Julia Gaston Maurice Julia Gaston Julia gauge (defined in generalized Riemann integral gauge group gauge integral generalized Riemann integral

 16. \documentclass{article} \usepackage{amssymb} \begin{document} Both \documentclass{article} \usepackage{amssymb} \begin{document} Both the Continuum hypothesis and Martin s axiom allow inductive constructions to continue inhttp://basilo.kaist.ac.kr/API/?MIval=db_jour_tex&con=110666

 17. JSTOR Martin S Axiom And The Continuum The Continuum hypothesis (CH) says that every infinite set of reals .. MARTIN S axiom and THE Continuum 379 is finite, while if kVA then X n Cfl+k ishttp://dx.doi.org/10.2307/2275837

18. Re: Continuum Hypothesis
If one DENIES the Continuum hypothesis, and ZF+MA is consistent with not AC Since the proof given uses Martin s axiom and not the stronger CH,
http://sci.tech-archive.net/Archive/sci.logic/2007-08/msg00397.html
##### Re: Continuum hypothesis
• From (Daryl McCullough) Date : 20 Aug 2007 13:28:53 -0700
Alan Smaill says...
wrote:
Bell's Theorem proves that no measurable function f can possible
satisfy this constraint. However, Pitowsky proved that if one
assumes the continuum hypothesis, one can construct a nonmeasurable
function that satisfies this constraint.
One line of the truth table still has not been completed here.
If one DENIES the continuum hypothesis, can there still
exist a NON-constructible nonmeasurable function that
satisfies the constraint? Or is the truth of the CH necessary
to the existence of the non-measurable function at all (regardless of whether it can be proven to exist)? Since the proof given uses Martin's Axiom and not the stronger CH, and ZF+MA is consistent with not AC (assuming ZF consistent, presumably), the existence of such functions is consistent with not CH. As I understand Pitowsky's construction, what is needed for the construction to go through is something along the lines of this axiom: The continuum hypothesis of course implies this (because if you well-order the reals with order type omega-1, then the

19. Front: [math.LO/9807178] Lusin Sequences Under CH And Under Martin's Axiom
Abstract Assuming the Continuum hypothesis there is an inseparable sequence of length omega_1 that contains no Lusin subsequence, while if Martin s axiom
http://front.math.ucdavis.edu/9807.5178
 Front for the arXiv Mon, 24 Dec 2007 Front math LO math.LO/9807178 search register submit journals ... iFAQ math.LO/9807178 Title: Lusin sequences under CH and under Martin's Axiom Authors: Uri Abraham , Saharon Shelah Categories: math.LO Logic Report number: Shelah [Sh:537] Abstract: Assuming the continuum hypothesis there is an inseparable sequence of length omega_1 that contains no Lusin subsequence, while if Martin's Axiom and the negation of CH is assumed then every inseparable sequence (of length omega_1) is a union of countably many Lusin subsequences. Owner: Shelah Office Version 1: Wed, 15 Jul 1998 00:00:00 GMT - for questions or comments about the Front arXiv contact page - for questions about downloading and submitting e-prints

20. Seminari De LÃ²gica De Barcelona. Sesions Anteriors
Omegalogic and the Continuum hypothesis; 17.03.99 Josep Maria Font The bounded Martin s axiom; 21.10.98 Joan Bagaria (Universitat de Barcelona)
http://www.ub.es/slb/Past.html
 Sessions anteriors/Sesiones anteriores/Previous sessions Panteleimon Eleftheriou (Universidad de Barcelona) Groups definable in linear o-minimal structures. Alexander Berenstein Pares bellos de estructuras o-minimales. Thomas Johnstone (The City University of New York) Indestructible cardinals and forcing axioms. Martin Koerwien Complexity of isomorphism for countable models of omega-stable theories. Matt Foreman (University of California at Irvine) Generic elementary embeddings. Pseudofinite fields and random hypergraphs. Joerg Brendle (Kobe University) Mad families. John T. Baldwin (University of Illinois at Chicago) Abstract elementary classes. Anuj Dawar (University of Cambridge) Finite model theory of well-behaved finite structures. Martin D. Davis (New York University) The myth of hypercomputation. Dugald Macpherson (University of Leeds) Asymptotics of definable sets in finite structures. (University of Auckland (New Zealand)) Automata presentable Boolean algebras. Hans Adler Forking and thorn-forking. Piotr Koszmider Boolean algebras and projections in Banach spaces c(K). Set-theoretic aspects.

21. Sachgebiete Der AMS-Klassifikation: 00-09
ordinal definability, and related notions 03E47 Other notions of settheoretic definability 03E50 Continuum hypothesis and Martin s axiom,
http://www.math.fu-berlin.de/litrech/Class/ams-00-09.html
##### nach 90-99 Weiter nach 10-19 Suche in allen Klassifikationen
01-XX 03-XX 04-XX 05-XX 06-XX 08-XX
##### nach 90-99 Weiter nach 10-19 Suche in allen Klassifikationen

22. IngentaConnect Residual Measures In Locally Compact Spaces
Results on the influence of Martin s axiom and the Continuum hypothesis on the existence of residual and normal measures in locally compact spaces are
http://www.ingentaconnect.com/content/els/01668641/2000/00000108/00000003/art001
var tcdacmd="dt";

23. 03Exx
03E50, Continuum hypothesis and Martin s axiom. 03E55, Large cardinals. 03E60, Determinacy principles. 03E65, Other hypotheses and axioms
http://www.impan.gov.pl/MSC2000/03Exx.html
 Set theory Partition relations Ordered sets and their cofinalities; pcf theory Other combinatorial set theory Ordinal and cardinal numbers Descriptive set theory [See also Cardinal characteristics of the continuum Other classical set theory (including functions, relations, and set algebra) Axiom of choice and related propositions Axiomatics of classical set theory and its fragments Consistency and independence results Other aspects of forcing and Boolean-valued models Inner models, including constructibility, ordinal definability, and core models Other notions of set-theoretic definability Continuum hypothesis and Martin's axiom Large cardinals Determinacy principles Other hypotheses and axioms Nonclassical and second-order set theories Fuzzy set theory Applications of set theory None of the above, but in this section

 24. Citebase - Lusin Sequences Under CH And Under Martin's Axiom Assuming the Continuum hypothesis there is an inseparable sequence of length omega1 that contains no Lusin subsequence, while if Martin s axiom and thehttp://www.citebase.org/cgi-bin/citations?id=oai:arXiv.org:math/9807178

 25. CMS - Search Journals Todorcevic, Stevo , Remarks on Martin s axiom and the Continuum hypothesis , Canadian Journal of Mathematics, Vol 43, No , 832-851.http://www.journals.cms.math.ca/cgi-bin/vault/search?_query=martin&journal=C

26. The Continuum Hypothesis
Cohen used a technique called forcing to prove the independence in set theory of the axiom of choice and of the generalised Continuum hypothesis.
http://www.humboldt.edu/~mef2/Courses/m446s02n2.html
 The Continuum Hypothesis Notes for Math 446 M. Flashman Spring, 2002 I. Background: Cantor 1845-1918: Investigation of discontinuities with Fourier series and Set Theory Beginnings. Any infinite subset of the natural numbers or the integers is countable. The rational numbers are a countable set. "Godel counting" argument. The algebraic numbers are countable. [ Another first type of diagonal argument.] 1874 Cantor's proof that the number of points on a line segment are uncountable. (1874) A decimal based proof that there is an uncountable set of real numbers.(similar to 1891 proof) There is no onto function from R, the set of real numbers, to P(R), the set of all subsets of the real number. There are sets which are larger than the reals. The rational numbers between and 1 have "measure" zero. Any countable set of real numbers has "measure" zero. II. The XXth Century: An Age of Exploration and Discovery. Hilbert: (Finitistic Formalization of Arithmetic) The continuum hypothesis problem was the first of Hilbert's famous 23 problems delivered to the Second International Congress of Mathematicians in Paris in 1900. Hilbert's famous speech The Problems of Mathematics challenged (and still today challenge) mathematicians to solve these fundamental questions Brouwer: (1881-1966) (Rejection of the law of excluded middle for infinite sets) He rejected in mathematical proofs the Principle of the Excluded Middle, which states that any mathematical statement is either true or false. In 1918 he published a set theory, in 1919 a measure theory and in 1923 a theory of functions all developed without using the Principle of the Excluded Middle.

 27. HeiDOK 03E50 Continuum hypothesis and Martin s axiom ( 0 Dok. ) 03E55 Large cardinals ( 0 Dok. ) 03E60 Determinacy principles ( 0 Dok.http://archiv.ub.uni-heidelberg.de/volltextserver/msc_ebene3.php?zahl=03E&anzahl

 28. News | TimesDaily.com | TimesDaily | Florence, Alabama (AL) The Continuum hypothesis and the axiom of choice were among the first Martin, D. (1976). Hilbert s first problem the Continuum hypothesis, inhttp://www.timesdaily.com/apps/pbcs.dll/section?category=NEWS&template=wiki&text

29. Infinite Ink: The Continuum Hypothesis By Nancy McGough
4.3.2.1 Martin s axiom A Weak Version of CH; 4.3.2.3 The Singular Cardinal hypothesis . AC, The axiom of Choice. CH, The Continuum hypothesis
http://www.ii.com/math/ch/

mathematics
##### T HE C ONTINUUM H YPOTHESIS
• Overview
• 1.1 What is the Continuum Hypothesis?
• 1.2 Current Status of CH
• Alternate Overview
• Assumptions, Style, and Terminology
• 2.1 Assumptions
• 2.1.1 Audience Assumptions
• 2.1.2 Mathematical Assumptions
• 2.2 Style
• 2.3 Terminology
• 2.3.1 The Word "continuum"
• 2.3.2 Ordered Sets
• 2.3.3 More Terms and Notation
• Mathematics of the Continuum and CH
• 3.1 Sizes of Sets: Cardinal Numbers
• aleph c aleph
• 3.1.2 CH and GCH
• 3.1.3 Sample Cardinalities
• 3.2 Ordering Sets: Ordinal Numbers
• 3.3 Analysis of the Continuum
• 3.3.1 Decomposing the Reals
• 3.3.2 Characterizing the Reals
• 3.3.3 Characterizing Continuity
• 3.4 What ZFC Does and Does Not Tell Us About c
• Metamathematics and CH
• 4.1 Consistency, Completeness, and Compactness of ...
• 4.1.1 a Logical System
• 4.1.2 an Axiomatic Theory
• 4.2 Models of ...
• 4.2.1 Real Numbers
• 4.2.2 Set Theory
• 4.2.2.1 Inner Models
• 4.2.2.2 Forcing and Outer Models
• 4.3 Adding Axioms to Zermelo Fraenkel Set Theory
• 4.3.1 Axioms that Imply CH or GCH
• 4.3.1.1 Explicitly Adding CH or GCH
• 4.3.1.2 V=L: Shrinking the Set Theoretic Universe
• 30. Sci.math FAQ: The Continuum Hypothesis
sci.math FAQ The Continuum hypothesis Various forcing axioms (e.g. Martin s axiom), which are maximality principles (in the sense of (2) above),
http://www.faqs.org/faqs/sci-math-faq/AC/ContinuumHyp/
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##### sci.math FAQ: The Continuum Hypothesis
Newsgroups: sci.math alopez-o@neumann.uwaterloo.ca alopez-o@barrow.uwaterloo.ca Tue Apr 04 17:26:57 EDT 1995 Rate this FAQ N/A Worst Weak OK Good Great Related questions and answers
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31. Martin's Axiom - Wikipedia, The Free Encyclopedia
In the mathematical field of set theory, Martin s axiom, named after Donald A. Martin MA together with the negation of the Continuum hypothesis implies
http://en.wikipedia.org/wiki/Martin's_axiom
var wgNotice = ""; var wgNoticeLocal = ""; var wgNoticeLang = "en"; var wgNoticeProject = "wikipedia";
##### Martin's axiom
Jump to: navigation search In the mathematical field of set theory Martin's axiom , named after Donald A. Martin , is a statement which is independent of the usual axioms of ZFC set theory . It is implied by the continuum hypothesis cardinality of the continuum , behave roughly like . The intuition behind this can be understood by studying the proof of the Rasiowa-Sikorski lemma . More formally it is a principle that is used to control certain forcing arguments. The various statements of Martin's axiom typically take two parts. MA( k ) is the assertion that for any partial order P satisfying the countable chain condition (hereafter ccc ) and any family D of dense sets in P D k , there is a filter F on P such that F Ã¢ÂÂ© d is non- empty for every d ÃÂµ D . MA is then the statement that MA( k ) holds for every k less than the continuum. (It is a theorem of ZFC that MA( ) fails.) Note that, in this case (for application of ccc), an antichain is a subset A of P such that any two distinct members of A are incompatible (two elements are said to be compatible if there exists a common element below both of them in the partial order). This differs from, for example, the notion of antichain in the context of

32. The Continuum Hypothesis
Various forcing axioms (e.g. Martin s axiom), which are maximality principles (in the Nancy McGough s *Continuum hypothesis article* or its *mirror*.
http://www.cs.uwaterloo.ca/~alopez-o/math-faq/node71.html
Next: Formulas of General Interest Up: Axiom of Choice and Previous: Cutting a sphere into
##### The Continuum Hypothesis
A basic reference is Godel's What is Cantor's Continuum Problem?", from 1947 with a 1963 supplement, reprinted in Benacerraf and Putnam's collection Philosophy of Mathematics. This outlines Godel's generally anti-CH views, giving some implausible" consequences of CH. "I believe that adding up all that has been said one has good reason to suspect that the role of the continuum problem in set theory will be to lead to the discovery of new axioms which will make it possible to disprove Cantor's conjecture." At one stage he believed he had a proof that from some new axioms, but this turned out to be fallacious. (See Ellentuck, Godel's Square Axioms for the Continuum", Mathematische Annalen 1975.) Maddy's Believing the Axioms", Journal of Symbolic Logic 1988 (in 2 parts) is an extremely interesting paper and a lot of fun to read. A bonus is that it gives a non-set-theorist who knows the basics a good feeling for a lot of issues in contemporary set theory. Most of the first part is devoted to plausible arguments" for or against CH: how it stands relative to both other possible axioms and to various set-theoretic rules of thumb". One gets the feeling that the weight of the arguments is against CH, although Maddy says that many younger members" of the set-theoretic community are becoming more sympathetic to CH than their elders. There's far too much here for me to be able to go into it in much detail.

33. Math Forum - Ask Dr. Math
THE Continuum hypothesis A basic reference is Godel s What is Cantor s Continuum Various forcing axioms (e.g. Martin s axiom), which are maximality
http://mathforum.org/library/drmath/view/51437.html

Associated Topics
Dr. Math Home Search Dr. Math
##### The Continuum Hypothesis
Date: Wed, 24 May 1995 09:04:05 +0800 From: SheparD Subject: Math Problem Although I am not a K12 type person my daughter is. She is the one with the math problem, but I am the one with the internet connection. But really it IS me with the problem... I volunteered to assist her with an essay assignment and I thought to retrieve some information from the net. But, alas, I can find no information on the net. I would only like to have you point me in the right direction, if you would. The problem: (or question as it may be) "The continuum theory, what is it and has it been resolved?" I would be grateful if you could provide any assistance to me. Thanks for your time, David Date: 9 Jun 1995 10:25:29 -0400 From: Dr. Ken Subject: Re: Math Problem Hello there! I'm sorry it's taken us so long to get back to you. If you're still interested, here's something I found in the Frequently-Asked-Questions for the sci.math newsgroup. If you want to look in the site yourself sometime, the site name is ftp.belnet.be (you can log in with the user name "anonymous") and this file's name is /pub/usenet-faqs/usenet-by-hierarchy/sci/math/ sci.math_FAQ:_The_Continuum_Hypothesis I found it by searching FAQs at the site http://mailserv.cc.kuleuven.ac.be/faq/faq.html

34. [math/0009062] Reflexive Subgroups Of The Baer-Specker Group And Martin's Axiom
455, Problem 12) under the set theoretical hypothesis of We will use Martin s axiom to find reflexive modules with the above decomposition which are
http://arxiv.org/abs/math/0009062
##### arXiv.orgmath
Search or Article-id Help Advanced search All papers Titles Authors Abstracts Full text Help pages
Citations p revious n ... ext
##### Title:
Authors: Saharon Shelah (Submitted on 6 Sep 2000) Abstract: In two recent papers ( math.LO/0003164 and math.LO/0003165 Subjects: Logic (math.LO) ; Commutative Algebra (math.AC); Rings and Algebras (math.RA) MSC classes: Report number: Shelah [GbSh:727] Cite as: arXiv:math/0009062v1 [math.LO]
##### Submission history
From: Saharon Shelah's Office [ view email
Wed, 6 Sep 2000 18:33:24 GMT (43kb,S)
Which authors of this paper are endorsers?
Link back to: arXiv form interface contact

35. [FOM] Continuum Hypothesis
1) Do you believe that the Continuum hypothesis is true, or false? . of Martin s Maximum or axiom (*), but the arguments themselves use distinct
http://cs.nyu.edu/pipermail/fom/2003-May/006611.html
##### [FOM] Continuum Hypothesis
Paul Larson larson at ime.usp.br
Mon May 19 18:02:41 EDT 2003 1) Do you believe that the continuum hypothesis is true, or false? 2) Is there any general consensus amongst the mathematical/FOM community regarding the truth or falsity of CH? 3) What are the most important recent developments post-Cohen which have contributed to this consensus (or lack thereof)? Have set-theorists proposed any plausible axioms which might decide CH? Have any consequences of CH been discovered which either strongly support or strongly undermine it?

36. 03E: Set Theory
Natural axioms which imply the negation of Continuum hypothesis; How does (Set Theory axiom) V=L prove the Continuum hypothesis?
http://www.math.niu.edu/~rusin/known-math/index/03EXX.html
Search Subject Index MathMap Tour ... Help! ABOUT: Introduction History Related areas Subfields
POINTERS: Texts Software Web links Selected topics here
##### Introduction
Naive set theory considers elementary properties of the union and intersection operators Venn diagrams, the DeMorgan laws, elementary counting techniques such as the inclusion-exclusion principle, partially ordered sets, and so on. This is perhaps as much of set theory as the typical mathematician uses. Indeed, one may "construct" the natural numbers, real numbers, and so on in this framework. However, situations such as Russell's paradox show that some care must be taken to define what, precisely, is a set. However, results in mathematical logic imply it is impossible to determine whether or not these axioms are consistent using only proofs expressed in this language. Assuming they are indeed consistent, there are also statements whose truth or falsity cannot be determined from them. These statements (or their negations!) can be taken as axioms for set theory as well. For example, Cohen's technique of forcing showed that the Axiom of Choice is independent of the other axioms of ZF. (That axiom states that for every collection of nonempty sets, there is a set containing one element from each set in the collection.) This axiom is equivalent to a number of other statements (e.g. Zorn's Lemma) whose assumption allows the proof of surprising even paradoxical results such as the Banach-Tarski sphere decomposition. Thus, some authors are careful to distinguish results which depend on this or other non-ZF axioms; most assume it (that is, they work in ZFC Set Theory).

37. Wiki Martin S Axiom
Martin, is a statement which is independent of the usual axioms of ZFC set theory. It is implied by the Continuum hypothesis, so certainly consistent with
http://wapedia.mobi/en/Martin's_axiom
 Wiki: Martin's axiom In the mathematical field of set theory , named after Donald A. Martin , is a statement which is independent of the usual axioms of ZFC set theory . It is implied by the continuum hypothesis , so certainly consistent with ZFC, but is also known to be consistent with ZF + not CH. Indeed, it is only really of interest when the continuum hypothesis fails (otherwise it adds nothing to ZFC). It can informally be considered to say that all cardinals less than the cardinality of the continuum Rasiowa-Sikorski lemma . More formally it is a principle that is used to control certain forcing arguments. Home Licensing Wapedia: For Wikipedia on mobile phones

38. Logic Seminar In Semester I AY 2007/2008
Martin s axiom arose from the consistency problem for Souslin s hypothesis and it relates closely with iterated forcing. Souslin s hypothesis (SH) states
http://www.comp.nus.edu.sg/~fstephan/logicseminar.html
##### Logic Seminar in Semester I AY 2007/2008
Talks are Wednesday afternoon in Math Studio (S14#03-01)
If there is a Colloquium Talk
Then the Logic Seminar is from 14:00 hrs to 15:00 hrs (to avoid timetable clash)
Else the Logic Seminar is from 15:00 hrs to 16:00 hrs
• 15/08/2007, Week 1, 16.30 hrs: Organizational Meeting
• 22/08/2007, Week 2, 15.00 hrs: Frank Stephan (joint work with Sanjay Jain)
Learning in Friedberg Numberings
• 29/08/2007, Week 3, 15:00 hrs: Wu Guohua
Cuppable degrees and noncuppable pairs
In this talk, we will outline the proof on the following result: There is a minimal pair a, b such that both are cuppable and no incomplete computably enumerable degree cups both of them to 0'. Several possible projects will be proposed. We will also show that the dual of Lempp's conjecture is true: Given a, b with b not above a, if a is not complete, then there is an incomplete cuppable computably enumerable degree c above b, but not above a.
• 05/09/2007, Week 4, 15:00 hrs: Johanna Franklin
Schnorr triviality and genericity The Turing degrees of Schnorr trivial reals seem to have no simple characterization. Therefore, we may examine other properties in connection with Schnorr triviality to determine the extent to which they are compatible in the Turing degrees. In this talk, we will discuss the relationship between Schnorr trviality and genericity. We find that no Schnorr trivial real is Turing equivalent to a 2-generic real. However, we also show that while no Schnorr trivial real is truth-table equivalent to a 1-generic real, such reals may be Turing equivalent.
• 39. Piotr Koszmider's Web Page
Making a special settheoretic assumption like the Continuum hypothesis or Martin s axiom we construct an example of MrÃ³wka s space (i.e., obtained from an
http://www.ime.usp.br/~piotr/cv/ativpas12004e.html
 FORCING INFINITY TOPOLOGY BANACH SPACES C(K) Home Contact Recent preprints Publications ... Previous teaching Curiculum Vitae academic activities invited lectures contributed lectures profissional services ... Resources Pages of matematicians Conferences Papers on-line Bulletin Boards Funding sources Institutes Popular resources Previous activities by semester Personal page PIOTR KOSZMIDER, ASSOCIATED PROFESSOR, MAT - IME - USP A. J. Heschel Who is man? The Raymond Fred West Memorial Lectures at Stanford University, 1963. ACTIVITIES DURING THE FIRST SEMESTER 2004 RESEARCH PROJECT 2001 - 2006 Applications of Infinitary Combinatorics in Undecidable Problems on Boolean algebras, compact spaces and Banach spaces. Supported by National Reserach Council (Brazil): Produtividade em Pesquisa (B2) fellowship. There is a classical chain of mathematical constructions: infinite families of sets which form Boolean algebras which induce, by the Stone duality

40. Andy's Math/CS Page
Assuming the Continuum hypothesis is false, where does the threshold lie? On the other hand, Martin s axiom, which is independent of CH,
http://andysresearch.blogspot.com/
@import url("http://www.blogger.com/css/blog_controls.css"); @import url("http://www.blogger.com/dyn-css/authorization.css?targetBlogID=19908808");
##### Cardinal Rules
Pick your favorite countable set S . Let F be a 'nested' family of distinct subsets of S ; that is, if A, B are members of F , then either A is contained in B or B is contained in A
Then clearly F can be at most countable... right?
A puzzle from Bollobas' recent book
Given a collection C of functions from to N , say that C is unbounded if for any function f (in C or not), there exists a function g in C such that g(i) > f(i) for infinitely many i
Clearly the class C of all functions from N to N is unbounded. Also, standard diagonalization techniques tell us that no countable collection C can be unbounded (exercise).
The question then becomes: what is the smallest cardinality of any unbounded collection C ? Assuming the Continuum Hypothesis is false, where does the threshold lie?
Fortunately or unfortunately, there seems to be little we can say about this issue within the standard axioms of set theory. Assuming CH is false, the threshold could be as low as the first uncountable cardinal, or as large as the continuum, or in betweenthis I learned from Jech's encyclopedic book Set Theory . There are many questions of this flavor, where the key construction (in our case, constructing a 'bounding function' for a given 'small' collection

 41. EULER Record Details Subject, Lusin sequence; Martin s axiom; Continuum hypothesis. MSC, 03E05; 03E50; 03E35. Type, Text.Article. Record Source, Zentralblatt MATHhttp://www.emis.de/projects/EULER/detail?ide=2001abralusisequunde&matchno=2&matc

42. ScienceDirect - Expositiones Mathematicae : On Extensions Of Partial Functions
Assume the Continuum hypothesis (or, more generally, Martin s axiom). Let fR R be a partial function. The following two assertions are equivalent
 Athens/Institution Login Not Registered? User Name: Password: Remember me on this computer Forgotten password? Home Browse My Settings ... Help Quick Search Title, abstract, keywords Author e.g. j s smith Journal/book title Volume Issue Page Expositiones Mathematicae Volume 25, Issue 4 , 1 November 2007, Pages 345-353 Abstract Full Text + Links PDF (132 K) Related Articles in ScienceDirect Characterization of realizable space complexities Annals of Pure and Applied Logic Characterization of realizable space complexities Annals of Pure and Applied Logic Volume 73, Issue 2 1 June 1995 Pages 171-190 Joel I. Seiferas and Albert R. Meyer Abstract does characterize the complexity of some partial function, even one that assumes only the values and 1. Abstract Abstract + References PDF (1392 K) Partial functions and logics: A warning ... Information Processing Letters Partial functions and logics: A warning Information Processing Letters Volume 54, Issue 2 28 April 1995 Pages 65-67 C. B. Jones Abstract One approach to handling partial functions when specifying and reasoning about programs is to say that application outside their domain yields an indeterminate element of their range. This paper puts forward a counter example which suggests that this approach might be problematic in a specification language. Abstract Abstract + References PDF (229 K) Sum of Sierpiski ... Topology and its Applications Sum of Sierpi Topology and its Applications Volume 122, Issue 3

The Continuum hypothesis A basic reference is Godel s What is Cantor s Continuum Various forcing axioms (e.g. Martin s axiom), which are maximality
http://linas.org/mirrors/nntp/sci.math/faq.continuum.html

44. List
We show in particular that under Martin s axiom there is compact $X$ with $X =c$ carrying Abstract Assuming the Continuum hypothesis, we show that
http://www.math.uni.wroc.pl/~grzes/abs.htm
 My research interests are: measure theory, set-theoretic topology and analysis. This page offers the list of my recent papers - their abstracts appear here after clicking the corresponding title, links to coauthors, and in most cases DVI and PS files of the preprints. Last modified: October 2007 On compactness of measures on Polish spaces Abstract: A construction of a Banach space C(K) with few operators Abstract: We present a construction, requiring no additional axioms, of a compact connected space K such that every bounded operator T: C(K) -> C(K) can be written as T=gI+S, where g is in C(K) and S is a weakly compact operator. This extends a result due to Koszmider, who proved such a result assuming the continuum hypothesis. Large families of mutually singular Radon measures Abstract: On compact spaces carrying Radon measures of large Maharam type Abstract: The paper surveys results related to Pelczynski's conjecture on Banach spaces containing $l_1$ spaces, and Haydon's problem on compact spaces carrying Radon measures on uncountable type and mappings onto Tychonoff cubes. We discuss results due to R. Haydon, S. Argyros, D.H. Fremlin, G. Plebanek and others. In some cases we present new proofs based on an unpublished measure-theoretic lemma due to Haydon. This survey was presented in the series of invited lectures during Winter School in Abstract Analysis (Section Analysis), Lhota, Czech Republic 2002. Precalibre pairs of measure algebras Abstract: Generalisations of epsilon-density

45. Papers.html
The second paper introduces a maximally strong forcing axiom Martin s Maximum Counterexamples to the Continuum hypothesis, (joint with M. Magidor),
http://www.math.uci.edu/personnel/Foreman/homepage/newpapers.html
 Annotated List of Matt Foreman's Papers: The papers below are organized by subject matter and annotated with comments. If you prefer a simple chronological listing go here. Click to jump to one of the Topics A.) Paradoxical decompositions and the Banach-Tarski Paradox with nice pieces B.) A descriptive view of ergodic theory C.) Amenable group actions, the Rusiewicz problem, the Hahn-Banach theorem and Lebesgue measurability. D.) Foundations of mathematics. E.) Consistency results of foundational interest F.) Model Theory: (Non-regular ultrafilters, Chang's Conjecture, Lowenheim-Skolem Theorems). G.) Saturated Ideals H.) Natural structure in set theory I.) Applications of set theory to Abelian groups and modules. J.) Aronzahn Trees K.) Hungarian Combinatorics L.) Pot Pouri A.) Paradoxical decompositions and the Banach-Tarski Paradox with nice pieces: The main result also gives paradoxical decompositions without the Axiom of Choice of open dense subsets of S^n. Banach-Tarski Decompositions Using Sets with the Property of Baire , (Joint with R. Dougherty.)

46. Sci.math FAQ: The Continuum Hypothesis
The Continuum hypothesis A basic reference is Godel s What is Cantor s Continuum Various forcing axioms (eg Martin s axiom), which are maximality
http://faqs.cs.uu.nl/na-dir/sci-math-faq/continuum.html
Note from archiver cs.uu.nl: This page is part of a big collection of Usenet postings, archived here for your convenience. For matters concerning the content of this page , please contact its author(s); use the source , if all else fails. For matters concerning the archive as a whole, please refer to the archive description or contact the archiver.
##### All FAQs in Directory: sci-math-faq All FAQs posted in: sci.math Source: Usenet Version
http://www.jazzie.com/ii/math/ch/ http://www.best.com/ ii/math/ch/ Alex Lopez-Ortiz alopez-o@unb.ca http://www.cs.unb.ca/~alopez-o Assistant Professor Faculty of Computer Science University of New Brunswick

47. Tree Structure Of LoLaLi Concept Hierarchy Updated On 2004624
393 Martin s axiom . . . . 382 Continuum hypothesis g . . . . Not 379 cardinal number . . . Par 216 abstract model theory + . . . . 254 quantifier (5) + g .
http://remote.science.uva.nl/~caterina/LoLaLi/soft/ch-data/tree.txt

48. Sci.math FAQ: The Continuum Hypothesis
Subject sci.math FAQ The Continuum hypothesis Various forcing axioms (eg Martin s axiom), which are maximality principles (in the sense of (2)
http://www.uni-giessen.de/faq/archiv/sci-math-faq.ac.continuumhyp/msg00000.html
Index
##### sci.math FAQ: The Continuum Hypothesis

 49. Todorcevic, Velickovic: Martin's Axiom And Partitions KT K. Kunen and F. Tall Between Martin s axiom and Souslin s hypothesis. Fund. Math. 102 (1979) 173181. MS D. Martin and R. Solovay Internal Cohenhttp://www.numdam.org/numdam-bin/fitem?id=CM_1987__63_3_391_0

50. Sci.math FAQ: The Continuum Hypothesis
sci.math FAQ The Continuum hypothesis Various forcing axioms (eg Martin s axiom), which are maximality principles (in the sense of (2) above),
http://doc.rz.ifi.lmu.de/FAQ/sci-math-faq/AC/ContinuumHyp/index.html
Usenet FAQs Search Web FAQs Documents ... RFC Index
##### sci.math FAQ: The Continuum Hypothesis
Newsgroups: sci.math Tue Apr 04 17:26:57 EDT 1995 Rate this FAQ N/A Worst Weak OK Good Great
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 51. Solution To Continuum Hypothesis - Common Ascent Cohen showed Continuum hypothesis is independent of axioms of ZFC. . Martin s Maximum or axiom (*), but the arguments themselves use distincthttp://www.forum.commonascent.org/showthread.php?t=1701

52. Set Theory (Stanford Encyclopedia Of Philosophy)
The Continuum hypothesis; 4. Axiomatic Set Theory; 5. .. developed by D. A. Martin, Robert Solovay and others, brought together methods of, among others,
http://plato.stanford.edu/entries/set-theory/
Cite this entry Search the SEP Advanced Search Tools ...
##### Set Theory
First published Thu 11 Jul, 2002
##### 1. The Essence of Set Theory
The objects of study of Set Theory are sets . As sets are fundamental objects that can be used to define all other concepts in mathematics, they are not defined in terms of more fundamental concepts. Rather, sets are introduced either informally, and are understood as something self-evident, or, as is now standard in modern mathematics, axiomatically, and their properties are postulated by the appropriate formal axioms. The language of set theory is based on a single fundamental relation, called membership . We say that A is a member of B (in symbols A B ), or that the set B contains A as its element. The understanding is that a set is determined by its elements; in other words, two sets are deemed equal if they have exactly the same elements. In practice, one considers sets of numbers, sets of points, sets of functions, sets of some other sets and so on. In theory, it is not necessary to distinguish between objects that are members and objects that contain members the only objects one needs for the theory are sets. See the supplement Basic Set Theory for further discussion.

 53. OUP: UK General Catalogue Iterated Boolean Extensions, Martin s axiom and Souslin s hypothesis. 7. BooleanValued Analysis. 8. Intuitionistic Set Theory and Heyting-Algebra-Valuedhttp://www.oup.com/uk/catalogue/?ci=9780198568520

54. Searching Cantor's Continuum Hypothesis
Results 1 to 3 for cantor s Continuum hypothesis (view as list tiles) teaching english, philosophical ethics, philosophy of math, axiom of choice,
http://ex.plode.us/search/cantor's continuum hypothesis
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55. Oxford Scholarship Online: Set Theory
Keywords lattice, Boolean algebra, Heyting algebra, Booleanvalued model, Continuum hypothesis, ultrailter, axiom of choice, forcing, generic, category
http://www.oxfordscholarship.com/oso/public/content/maths/9780198568520/toc.html
 About OSO What's New Subscriber Services Help ... Mathematics Subject: Mathematics Book Title: Set Theory Set Theory Boolean-Valued Models and Independence Proofs Bell, John L. , Professor of Philosophy, University of Western Ontario Third Edition Print publication date: 2005 Published to Oxford Scholarship Online: September 2007 Print ISBN-13: 978-0-19-856852-0 doi:10.1093/acprof:oso/9780198568520.001.0001 Abstract: This is the third edition of a well-known graduate textbook on Boolean-valued models of set theory. The aim of the first and second editions was to provide a systematic and adequately motivated exposition of the theory of Boolean-valued models as developed by Scott and Solovay in the 1960s, deriving along the way the central set theoretic independence proofs of Cohen and others in the particularly elegant form that the Boolean-valued approach enables them to assume. In this edition, the background material has been augmented to include an introduction to Heyting algebras. It includes chapters on Boolean-valued analysis and Heyting-algebra-valued models of intuitionistic set theory. Keywords: lattice Boolean algebra Heyting algebra Boolean-valued model ... category Table of Contents Preface BOOLEAN AND HEYTING ALGEBRAS: THE ESSENTIALS 1. BOOLEAN-VALUED MODELS OF SET THEORY: FIRST STEPS

56. Sci.math FAQ: The Continuum Hypothesis - Allanswers.org
sci.math FAQ The Continuum hypothesis . Article from category AC on site Various forcing axioms (eg Martin s axiom), which are maximality
 Home Science sci-math-faq AC sci.math FAQ: The Continuum Hypothesis Back to category AC Discuss "sci.math FAQ: The Continuum Hypothesis" Home Search ... allanswers.org

57. Continuum Hypothesis: True, False, Or Neither?
(See Ellentuck, GÃ¶del s Square Axioms for the Continuum , Mathematische Annalen 1975.) The strongest (Martin s maximum) implies that C = aleph_2.
http://consc.net/notes/continuum.html
##### Is the Continuum Hypothesis True, False, or Neither?
David J. Chalmers Newsgroups: sci.math
From: chalmers@bronze.ucs.indiana.edu (David Chalmers)
Subject: Continuum Hypothesis - Summary
Date: Wed, 13 Mar 91 21:29:47 GMT
Thanks to all the people who responded to my enquiry about the status of the Continuum Hypothesis. This is a really fascinating subject, which I could waste far too much time on. The following is a summary of some aspects of the feeling I got for the problems. This will be old-hat to set theorists, and no doubt there are a couple of embarrassing misunderstandings, but it might be of some interest to non-professionals. A basic reference is Gödel's "What is Cantor's Continuum Problem?", from 1947 with a 1963 supplement, reprinted in Benacerraf and Putnam's collection Philosophy of Mathematics . This outlines Gödel's generally anti-CH views, giving some "implausible" consequences of CH. "I believe that adding up all that has been said one has good reason to suspect that the role of the continuum problem in set theory will be to lead to the discovery of new axioms which will make it possible to disprove Cantor's conjecture." At one stage he believed he had a proof that C = aleph_2 from some new axioms, but this turned out to be fallacious. (See Ellentuck, "Gödel's Square Axioms for the Continuum", Mathematische Annalen 1975.)

 58. MBceo.com Continuum Hypothesis | Information About Continuum Hypothesis Believing the Axioms, I . Journal of Symbolic Logic 53 (2) 481Â511. Martin, D. (1976). Hilbert s first problem the Continuum hypothesis, inhttp://www.mbceo.com/more_information.php?c=Continuum_hypothesis

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