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21. 2. Continuum Hypothesis
8. MATHEMATICAL SCIENCES CLASSIFICATION BASED ON THE PERIODICAL SYSTEM OF HYPERREAL NUMBERS. COMPUTER MATHEMATICS (EXCLUDING BOTH w - Levels).
http://members.tripod.com/vismath1/zen/zen8.htm
##### 8. MATHEMATICAL SCIENCES CLASSIFICATION BASED ON THE "PERIODICAL SYSTEM" OF HYPER-REAL NUMBERS
COMPUTER MATHEMATICS
EXCLUDING BOTH - Levels)
CLASSICAL MATHEMATICAL ANALYSIS INCLUDING BOTH - Levels) (THE POINTS AT THE BOTH - Levels ARE IDEAL ELEMENTS,
IN LEIBNIZ'S AND HILBERT'S SENSE)
CANTOR'S THEORY OF TRANSFINITE INTEGERS
NON-STANDARD ANALYSIS AS A WHOLE
Fig. 4. Mathematical sciences classification based on
the "periodical system" of hyper-real numbers
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22. Mudd Math Fun Facts: Continuum Hypothesis
From the Fun Fact files, here is a Fun Fact at the Advanced level ContinuumHypothesis. This came to be known as the continuum hypothesis.
http://www.math.hmc.edu/funfacts/ffiles/30002.4-8.shtml
hosted by the Harvey Mudd College Math Department Francis Su
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From the Fun Fact files, here is a Fun Fact at the Advanced level:
##### Continuum Hypothesis
We have seen in the Fun Fact How Many Reals that the real numbers (the "continuum") cannot be placed in 1-1 correspondence with the rational numbers. So they form an infinite set of a different "size" than the rationals, which are countable. It is not hard to show that the set of all subsets (called the power set ) of the rationals has the same "size" as the reals. But is there a "size" of infinity between the rationals and the reals? Cantor conjectured that the answer is no. This came to be known as the Continuum Hypothesis Many people tried to answer this question in the early part of this century. But the question turns out to be PROVABLY undecidable ! In other words, the statement is indepedent of the usual axioms of set theory! It is possible to prove that adding the Continuum Hypothesis or its negation would not cause a contradiction.

23. Example 2.2.9: The Continuum Hypothesis
Example 2.2.9 The continuum hypothesis. Is card(P(N)) = card(R). In fact,this is a deep question called the continuum hypothesis. This
##### Example 2.2.9: The Continuum Hypothesis
Is there a cardinal number c with ? What is the most obvious candidate ? Back We need to find a set whose cardinality is bigger than N and less that that of R . The most obvious candidate would be the power set of N . However, one can show that
• card( P N )) = card( R
In fact, this is a deep question called the continuum hypothesis . This question results in serious problems:
• In the 1940's the German mathematician Goedel showed that if one denies the existence of an uncountable set whose cardinalities is less than the cardinality of the continuum, no logical contradictions to the axioms of set theory would arise.
• One the other hand, it was shown recently that the existence of an uncountable set with cardinality less than that of the continuum would also be consistent with the axioms of set theory.
Hence, it seems impossible to decide this question with our usual methods of proving theorems. Such undecidable questions do indeed exist for any reasonably complex logical system (such as set theory), and in fact one can even prove that such 'non-provable' statements must exist. To read more about this fascinating subject, look at the book Goedel's Proof or Goedel, Escher, Bach

24. Sci.math FAQ: The Continuum Hypothesis
http://isc.faqs.org/faqs/sci-math-faq/AC/ContinuumHyp/
##### sci.math FAQ: The Continuum Hypothesis
Newsgroups: sci.math sci.answers news.answers From: alopez-o@neumann.uwaterloo.ca (Alex Lopez-Ortiz) Subject: sci.math DI76Mo.8s1@undergrad.math.uwaterloo.ca alopez-o@neumann.uwaterloo.ca Organization: University of Waterloo Followup-To: sci.math alopez-o@barrow.uwaterloo.ca Tue Apr 04 17:26:57 EDT 1995 By Archive-name By Author By Category By Newsgroup ... Help
Send corrections/additions to the FAQ Maintainer:
alopez-o@neumann.uwaterloo.ca
Last Update March 05 2003 @ 01:20 AM

25. Godel, K.: Consistency Of The Continuum Hypothesis. (AM-3).
of the book Consistency of the continuum hypothesis. (AM Consistencyof the continuum hypothesis. (AM3). Kurt Godel. Paper......
http://pup.princeton.edu/titles/1034.html
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##### Consistency of the Continuum Hypothesis. (AM-3)
Kurt Godel
69 pp.
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26. Cogprints - Generalized Continuum Hypothesis And The Axiom Of Combinatorial Sets
Generalized continuum hypothesis and the Axiom of Combinatorial Sets.Nambiar, Kannan (2002) Generalized continuum hypothesis and
http://cogprints.ecs.soton.ac.uk/archive/00002169/
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##### Generalized Continuum Hypothesis and the Axiom of Combinatorial Sets
Nambiar, Kannan (2002) Generalized Continuum Hypothesis and the Axiom of Combinatorial Sets. Full text available as:
PDF
##### Abstract
Axiom of Combinatorial Sets is defined and used to derive Generalized Continuum Hypothesis. Keywords: Generalized continuum htpothesis; Axiom of combinatorial sets. Subjects: Philosophy Logic ID Code: Deposited By: Nambiar, Kannan Deposited On: 07 April 2002 Alternative Locations: http://www.rci.rutgers.edu/~kannan/science/combinatorial_axiom_screen.pdf

27. The Continuum Hypothesis
The continuum hypothesis.
http://www.jboden.demon.co.uk/SetTheory/continuum-hypothesis.html
 The Continuum Hypothesis

28. Continuum Hypothesis
The continuum hypothesis Infinity has infinite ways to troubleour finite minds. This was proved by Georg Cantor in 1874. The
http://users.forthnet.gr/ath/kimon/Continuum.htm
 The Continuum Hypothesis Infinity has ... infinite ways to trouble our finite minds. This was proved by Georg Cantor in 1874. The "smallest level" of infinity has to do with countable things that can be put in some order. . This seems strange: one set is a proper subset of another and still they have the same number of elements. This is exactly the definition of infinite sets. What about rational numbers? These are a superset of the natural numbers but still of class aleph . It turns out that there is a way to put rational numbers in order: 1, 2, 1/2, 1/3, 3, 4, 3/2, 2/3, 1/4 ... (the pattern is based on a diagram so it is not obvious as shown here). Things change when we examine the real numbers. There is no way to create a complete list of reals and this was shown by Cantor with a beautiful argument, the "diagonal" one: Suppose we had such a complete list of real numbers between and 1 : r1=0.a a a r2=0.a a a r3=0.a a a Where a ij take values in 0,1,...,9 and all numbers are written with infinite number of digits (e.g. 0.2=0.20000..., 1/7=0.142857142857...). Then we can create another real r'=0.b

 29. Continuum Hypothesis From FOLDOC continuum hypothesis. Proving or disproving the continuum hypothesis wasthe first problem on Hilbert's famous list of problems in 1900.http://www.swif.uniba.it/lei/foldop/foldoc.cgi?continuum hypothesis

30. Woodin On The Continuum Hypothesis
a topic from sci.math.research Woodin on the continuum hypothesis.post a message on this topic post a message on a new topic 17
http://mathforum.org/epigone/sci.math.research/dwagruplix
a topic from sci.math.research
##### Woodin on the Continuum Hypothesis
post a message on this topic
post a message on a new topic

17 Feb 2003 Woodin on the Continuum Hypothesis , by David Madore
17 Feb 2003 Re: Woodin on the Continuum Hypothesis , by tchow@lsa.umich.edu
18 Feb 2003 Re: Woodin on the Continuum Hypothesis , by Carl Riehm
The Math Forum

31. Math Forum - Ask Dr. Math
The continuum hypothesis. Date Wed, 24 May 1995 090405 +0800 From SheparDSubject Math Problem Although I am not a K12 type person my daughter is.
http://mathforum.org/library/drmath/view/51437.html

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##### 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

32. Sci.math FAQ: The Continuum Hypothesis
Subject sci.math FAQ The continuum hypothesis. This See also NancyMcGough's *continuum hypothesis article* or its *mirror*. http
http://www.cs.uu.nl/wais/html/na-dir/sci-math-faq/continuum.html
Note from archivist@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 archivist.
##### 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

33. The Continuum Hypothesis
The continuum hypothesis. We showed in Proposition 1 that is the smallestinfinite cardinal. Cantor conjectured. continuum hypothesis .
http://mathcircle.berkeley.edu/BMC3/infinity/node12.html
Next: Problems Up: Infinity: cardinal numbers Previous: The cardinality of
##### The continuum hypothesis
We showed in Proposition that is the smallest infinite cardinal. It can be shown that there a next smallest cardinal called ; i.e., the only cardinals strictly smaller than are the finite ones and . Next come , .... Where does fit into this list, if anywhere? (A priori, it could be bigger than for every .) We know that , because we proved that . Cantor conjectured Continuum Hypothesis: In other words, he believed that there is no set whose cardinality is strictly between that of and that of The role of the continuum hypothesis in set theory is similar to the role of the parallel postulate in plane geometry. The parallel postulate (that given a line and a point not on , there exists a unique line through that does not intersect ) cannot be disproved from the other axioms of plane geometry, because it is actually true for the euclidean model of geometry. On the other hand, the parallel postulate cannot be proved either, since it is false in various noneuclidean models of geometry which do satisfy all the other axioms. Therefore the parallel postulate, or its negation, may be taken as a new axiom. Which one you choose will depend on your vision of what geometry is supposed to be. Similarly, whether you choose to accept the continuum hypothesis will depend on your idea of what a set is supposed to be.

34. What Is CH?
What is the continuum hypothesis? The continuum hypothesis is the first problemon David Hilbert's famous list of 23 unsolved mathematics problems.
http://www.wall.org/~aron/whatisch.html
##### What is the continuum hypothesis?
The continuum hypothesis is the first problem on David Hilbert's famous list of 23 unsolved mathematics problems. The history of this problem stretches back to the brilliant man who created (or discovered, whichever you prefer) set theory.
Georg Cantor
1, b 2). So far this is quite intuitive. But Cantor applied this principle to infinite sets as well. So the set of all even natural numbers has the same cardinality as the set of all natural numbers, even though the latter has all the elements of the former and more! The mapping looks like this:
In the same way you can map any infinite set of natural numbers to any other infinite set of natural numbers. They are the same size. Cantor gave this infinity the name aleph_0, after the first letter of the Hebrew alphabet. Now, infinity would be kind of boring if aleph_0 was all there was. But Cantor proved that some sets are larger than aleph_0. The set in question was the set of real numbers, or the set of points on a line. Cantor proved that the cardinality of the reals cannot be mapped on to the natural numbers. This is the proof: Take the real numbers between and 1. Now suppose there was a mapping of the real numbers between and 1 to the natural numbers. It would look something like this:

35. CH Directory
Enter at your own risk. A cool page on the continuum hypothesis Toward any dissenters,I quote the first Peano Postualate. What is the continuum hypothesis?
http://www.wall.org/~aron/chdir.html
##### Warning:
Severely complicated mathematics within. Enter at your own risk. A cool page on the continuum hypothesis
Note: The reader should be warned that throughout these pages I include as a natural number. Toward any dissenters, I quote the first Peano Postualate.
What is the continuum hypothesis?

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36. The Continuum Hypothesis
The continuum hypothesis. Follow Ups Post Followup Geometry II FAQ continuum hypothesis. What do we mean when we say continuum ?
http://superstringtheory.com/forum/geomboard/messages2/117.html
String Theory Discussion Forum String Theory Home Forum Index
##### The Continuum Hypothesis
Follow Ups Post Followup Geometry II FAQ Posted by sol on September 13, 2002 at 17:35:56: Continuum Hypothesis
What do we mean when we say "continuum"? Here's a description Albert Einstein gave on p. 83 of his Relativity: The Special and the General Theory:

The surface of a marble table is spread out in front of me. I can get from any one point on this table to any other point by passing continuously from one point to a "neighboring" one, and repeating this process a (large) number of times, or, in other words, by going from point to point without executing "jumps." I am sure the reader will appreciate with sufficient clearness what I mean here by "neighbouring" and by "jumps" (if he is not too pedantic). We express this property of the surface by describing the latter as a continuum.
So here we are describing something that is inherent in Superstring theory, and who is going to pave the way for me to understand this? By coming to the realization of the continuum, spoken by Einstein, it has made me realize, the value we could have assigned energy, and what did Kaluza do for Einstein that Einstein did for us?

37. Re: The Continuum Hypothesis
Re The continuum hypothesis. In Reply to Re The continuum hypothesisposted by mathman on September 13, 2002 at 203957 Mathman,.
http://superstringtheory.com/forum/geomboard/messages2/120.html
String Theory Discussion Forum String Theory Home Forum Index
##### Re: The Continuum Hypothesis
Follow Ups Post Followup Geometry II FAQ Posted by sol on September 13, 2002 at 20:42:22: In Reply to: Re: The Continuum Hypothesis posted by mathman on September 13, 2002 at 20:39:57: Mathman, I believe, this is where you are wrong:)Thanks anyway:)
Sol
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38. Remarks Concerning The Fascinating Continuum Hypothesis
http://www.rpi.edu/~bestlj/COURSES/PARA/continuum/continuum.html
##### Remarks Concerning the Fascinating Continuum Hypothesis
Selmer Bringsjord
Spring 1999 Two sets A and B are said to be of the same cardinality iff there is a bijection from A to B . A set is finite iff it is of the same cardinality as some set . A set is countable iff it's either finite or of the same cardinality of N , the natural numbers. As you now know from seeing earlier proofs, both N ) (the power set of the set of natural numbers) and R (the reals) are uncountable. Now, the conintuum hypothesis is:
(CH)
Every infinite subset of R is either countable or of the same cardinality as R
• If ZFC is consistent, then ZFC CH.
In 1963 Cohen proved that
• If ZFC is consistent, then ZFC CH.
So, if we assume that ZFC is consistent, then neither CH nor of axioms for set theory, there exists an assertion about sets that is such that

• 39. No Title
The continuum hypothesis. The logician K. Gödel (19061978) establishedthat the continuum hypothesis is consistent with set theory.