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Set Theory and Topology

  1. Apr 12, 2007 #1
    I am interested in learning set theory. It is an independent study. I already have previous knowledge of logic and deduction. Does anyone know of any good resources for learning set theory?

    Also, the reason I plan on learning set theory is so I can learn topology afterward, so any learning resources for that as well would be much appreciated.
     
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  3. Apr 12, 2007 #2

    quasar987

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    Hi, I like chapter 0 of Munkres' book "Topology".
     
  4. Apr 12, 2007 #3
    Um... anything I can get online for free? that book costs more that $50.
     
  5. Apr 12, 2007 #4
    You can start here, i guess. (not my site) I'll be uploading the contents of this torrent soon, but if you see something you like on that list PM me I'll get it to you.
     
  6. Apr 12, 2007 #5

    quasar987

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  7. Apr 13, 2007 #6
    Halmos' Naive Set Theory is pretty much the standard text. It's pretty small and condensed, so it might be a hard read. Give it a try.
     
  8. Apr 13, 2007 #7
  9. Apr 14, 2007 #8
    if you want the most comprehensive textbook on set theory, you should try jech's book.
     
  10. Apr 15, 2007 #9

    mathwonk

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    when i was in high school i read halmos, and erich kamke, but a better text all round is hausdorff's set theory. and of course the original is georg cantor's contributions to the founding of the theory of transfinite numbers.
     
  11. Apr 15, 2007 #10

    mathwonk

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    the whole subject seems pretty trivial now, though, and about all I recall of interest was cantor's arguments that the rational numbers are no more numerous than the integers, but that the real numbers (thought of as infinite decimals) are uncountably infinite, i.e. more numerous than the integers.

    it is hard to recall that far back, but i do recall thinking it was fascinating at the time. the way of thinking about things in sets is so pervasive for the past hundred years, that it is hard to think otherwise now.

    of course category theory came in 50 years ago and tried to displace set theory as a model for thinking, where one goes up one level and thinks of maps between two things as more basic than the things themselves.

    I guess that never did really take hold in my being, but i am still trying to get it. if you read my intro to algebraic geometry posted here somewhere, you will see me mentioning Grothendieck's notice of K valued "point" in a space, as a map from spec(K) to your space.

    i enjoyed hausdorff's comment near the beginning of his book, dismissing attempts to define numbers, by saying that a mathematician does not care what numbers are, just how they behave.

    That always differentiated clearly for me the attitude of mathematicians from that of logicians, like Russel.

    I guess a non trivial aspect would be the indepoendence of the axiom of choice and the continuum hypothesis from the other axioms of set theory, as discussed in the book by the fields medalist paul cohen: set theory and the continuum hypothesis.
     
    Last edited: Apr 15, 2007
  12. Apr 15, 2007 #11

    mathwonk

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    heres another one i read as a kid, and the oprice shouldn't scare you:

    Introduction to the theory of sets (Prentice-Hall mathematics series) (Prentice-Hall mathematics series)
    Josef Breuer Bookseller: Prairie Hill Books
    (Brenham, TX, U.S.A.) Price: US$ 1.00
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    US$ 3.50
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    Book Description: Prentice-Hall, 1959. Unknown Binding. Book Condition: Very Good. Very nice copy 6th printing, firm binding, pages clean and tight, expected age related discoloration and tanning, minimum wear, name inside cover Ships Within 24 Hours - Satisfaction Guaranteed!. Bookseller Inventory # mon0000021922
     
  13. Apr 15, 2007 #12

    mathwonk

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    heres kamkes book:

    Theory of Sets
    E. Kamke Bookseller: WebBookStore
    (Pittsburgh, PA, U.S.A.) Price: US$ 2.87
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    US$ 3.95
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    Book Description: Dover Publications, 1950. Paperback. Book Condition: Brand New. Brand new, not a remainder, no marks, Paperback edition, Book Size: Length: 8.02 inches, Width 5.41 Height inches 0.32 Inches, Book weight: 0.36 pounds, This book will require no additional postage, Orders processed on AbeBooks Monday - Friday and ships 6 days a week and usually leave our warehouse in 3-5 business days, Synopsis: Requiring only some college algebra,contents include rudiments; arbitrary sets and their cardinal numbers; ordered sets and their ordered types; and well-ordered sets and their ordinal numbers. 1950 Dover translation of the second German edition., Barcode/UPC of the book/13 digit ISBN # 9780486601411. Brand New. Bookseller Inventory # 0486601412_N
     
  14. Apr 15, 2007 #13

    mathwonk

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    this is a really good book i taught out of to undergrad math majors in college back in the 70's:

    Sets, Logic, and Axiomatic Theories (ISBN: 0716704579)
    Stoll, Robert R. Bookseller: Whodunit?
    (Philadelphia, PA, U.S.A.) Price: US$ 3.50
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    US$ 3.99
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    Book Description: W.H. Freeman, San Francisco, 1961. Binding is Trade Paperback. Book Condition: Good. Good. Binding is Trade Paperback. Bookseller Inventory # 429332

    [Bookseller & Payment Information] [
     
  15. Apr 15, 2007 #14

    mathwonk

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    and heres cantor:

    Contributions to the Founding of the Theory of Transfinite Numbers
    Cantor, Georg Bookseller: Abracadabra Bookshop & Book Search
    (Denver,, CO, U.S.A.) Price: US$ 5.50
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    US$ 4.95
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    Book Description: New York Dover Publications circa 1960., 1960. Very Good condition,light shelfwear,wraps,ex-lib. Bookseller Inventory # 052412
     
  16. Apr 15, 2007 #15

    mathwonk

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    and hausdorff:

    Bookseller Photo SET THEORY
    Hausdorff, Felix Bookseller: Ivan Luka
    (Kensington, MD, U.S.A.) Price: US$ 22.00
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    Book Description: New York: Chelsea Publishing, 1962, New York, 1962. Hard Cover. Book Condition: Very Good. No Jacket. 8vo - over 7¾" - 9¾" tall. Hard Cover. Very Good/No Jacket. 8vo - over 7¾" - 9¾" tall. Ex-library. Some wear on cover, mostly on corners and spine. Bookseller Inventory # 001841
     
    Last edited: Apr 15, 2007
  17. Apr 15, 2007 #16

    mathwonk

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    heres either a typo or evidence of the greed of some booksellers:

    THE CONSISTENCY OF THE AXIOM OF CHOICE AND OF THE GENERALIZED CONTINUUM-HYPOTHESIS WITH THE AXIOMS OF SET THEORY. Annals of Mathematics Studies, No. 3.
    Godel, Kurt / Goedel, Kurt Bookseller: SUNSET BOOKS
    (Newark, OH, U.S.A.) Price: US$ 599.00
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    US$ 3.50
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    Book Description: Princeton University Press, Princeton, NJ, 1953. Library Rebound. Book Condition: Very Good. No Jacket. Third Printing. 8vo - over 7¾" - 9¾" tall. 69pp, with full markings and pocket, minor wear and soil. Three pages of notes were added at the end of the by this third Printing to correct several misprints and slight errors in the first (1940) printing. The original was from "Notes by George W. Brown of lectures delivered at the Institute for Advanced Study, Princeton, New Jersey, during the autum term, 1938-1939." The original stiff wrappers were removed and the volume placed in a red cloth library binding, by a noted technical library. The internals are original. Ex-Library. Bookseller Inventory # 003877
     
  18. Apr 15, 2007 #17

    mathwonk

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    how interesting - these are the same book, but prices differ by $280:

    Publisher Photo Topology (ISBN: 0131816292)
    Munkres, James R. Bookseller: Limelight Bookshop
    (New York, NY, U.S.A.) Price: US$ 290.59
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    Quantity: 1 Shipping within U.S.A.:
    FREE
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    Book Description: Prentice Hall, 1999. Hardcover. Book Condition: New. New. Excellent customer service. Bookseller Inventory # B0131816292


    Publisher Photo Topology (ISBN: 0131816292)
    James Munkres Bookseller: viktree
    (san jose, CA, U.S.A.) Price: US$ 10.76
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    Quantity: > 20 Shipping within U.S.A.:
    US$ 6.50
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    International Edition

    Book Description: Prentice Hall, Lebanon, Indiana, U.S.A. Book Condition: New. BRAND NEW 2 edition textbook. This is an international edition and has no marks or writings anywhere and is 100% pristine. We give a 30-day full money back guarantee on all our books! SUPERFAST WORLDWIDE SHIPPING!. Bookseller Inventory # 0131816292
     
    Last edited: Apr 15, 2007
  19. Apr 15, 2007 #18

    mathwonk

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    oh yes, heres halmos:

    Naive Set Theory
    Halmos, Paul Bookseller: Kisselburg Military Books
    (Stillwater, MN, U.S.A.) Price: US$ 13.50
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    US$ 6.50
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    Book Description: N.Y. Springer 1974., 1974. VG+/ nice copy. Binding is Hardcover. Bookseller Inventory # 018014
     
  20. Apr 15, 2007 #19

    mathwonk

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    it seems hard to find free notes on set theory an an elementary level. the ones i found are at the research professional level. i am assuming you just want the basic language, since you want to study topology afterward.

    for that i suggest kamke.

    in the meantime here is a little exercise:

    prove the collection of all subsets of a given set S, is equivalent to the collection of all functions from S to the 2 element set {0,1}.
     
  21. Apr 15, 2007 #20
    i beg to differ, even if this is hausdorff's comment, i still think that if your'e learning set theory you should be aware that everything you are talking there is about sets, so you need to define the natural numbers from sets and not have numbers as given.
    im quite amazed of this comment in a set theory text.
     
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