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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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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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quasar987

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Hi, I like chapter 0 of Munkres' book "Topology".

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Um... anything I can get online for free? that book costs more that $50.

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You can start http://cohomology.princeton.edu/books/Math/" [Broken] torrent soon, but if you see something you like on that list PM me I'll get it to you.

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quasar987

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http://www.ces.clemson.edu/~mjs/courses/misc/settheory.pdf

But it's pretty weak. Your best bet is to rent real books from the library.

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http://www.ucl.ac.uk/~ucahcjm/stp.html

You're bound to find notes on set theory at their homepages.

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MathematicalPhysicist

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if you want the most comprehensive textbook on set theory, you should try jech's book.

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mathwonk

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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.

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.

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mathwonk

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Introduction to the theory of sets (Prentice-Hall mathematics series) (Prentice-Hall mathematics series)

Josef Breuer Bookseller: Prairie Hill Books

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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

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mathwonk

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Theory of Sets

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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

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Sets, Logic, and Axiomatic Theories (ISBN: 0716704579)

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Contributions to the Founding of the Theory of Transfinite Numbers

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mathwonk

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and hausdorff:

Bookseller Photo SET THEORY

Hausdorff, Felix Bookseller: Ivan Luka

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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.

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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

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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)

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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

Publisher Photo Topology (ISBN: 0131816292)

Munkres, James R. Bookseller: Limelight Bookshop

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mathwonk

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Naive Set Theory

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mathwonk

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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}.

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MathematicalPhysicist

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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.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.

im quite amazed of this comment in a set theory text.

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mathwonk

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im sure you are right, after all, what does hausdorff know?

mores eriously i would suggest that your comment suggests to me that either you are a beginner, hence fascinated by these trivial questions, or in spirit a logician rather than a mathematician, hence fascinated by these esoteric questions.

mores eriously i would suggest that your comment suggests to me that either you are a beginner, hence fascinated by these trivial questions, or in spirit a logician rather than a mathematician, hence fascinated by these esoteric questions.

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Hurkyl

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Not all set theories study only sets. Allowing urelements is closer to how mathematics is actually practiced -- for someone whose goal is to use set theory in other areas of mathematics, it would be far more practical to learn a set arithmetic that includes urelements.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.

It is a nice metamathematical theorem that ZFC includes a model of Peano's axioms. But I really don't see any merit in intuiting that ZFC says what the natural numbers "really are." From a practical point of view, the reverse direction is far more important, the model allows one to solve set-theoretic problems with natural number arithmetic. One rarely uses this model to solve natural number problems using set theory.

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Hurkyl

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Formal logicmores eriously i would suggest that your comment suggests to me that either you are a beginner, hence fascinated by these trivial questions, or in spirit a logician rather than a mathematician, hence fascinated by these esoteric questions.

And, of course, categorical logic views all of these things in yet another (rather interesting, IMHO) way.

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- #24

MathematicalPhysicist

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im just saying that you can define the natural numbers set in set theory.im sure you are right, after all, what does hausdorff know?

mores eriously i would suggest that your comment suggests to me that either you are a beginner, hence fascinated by these trivial questions, or in spirit a logician rather than a mathematician, hence fascinated by these esoteric questions.

(and im surprised of hausdorff's comment in a set theory book).

and to myself i dont think there's a big difference between logicians and mathematicians.

most logicians have an education in maths, their speciality is something else.

- #25

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when i meant construction of natural numbers i meant you define 0=empty set 1={ES} etc, and you define what it means to be an inductive set. and then you need the axiom of infinity to deduce that there exists an inductive set (yes i can see hausdorff's point better to keep the numbers undefined (-:)... and so on.\now when i rethink it, it really depends in the context, if you were discussing it in any discpline besides logic and set theory you shouldnt have this construction but then agian hausdorff's comment is in a set theory textbook, you should expect this kind of constrcution would you not?Not all set theories study only sets. Allowing urelements is closer to how mathematics is actually practiced -- for someone whose goal is to use set theory in other areas of mathematics, it would be far more practical to learn a set arithmetic that includes urelements.

It is a nice metamathematical theorem that ZFC includes a model of Peano's axioms. But I really don't see any merit in intuiting that ZFC says what the natural numbers "really are." From a practical point of view, the reverse direction is far more important, the model allows one to solve set-theoretic problems with natural number arithmetic. One rarely uses this model to solve natural number problems using set theory.

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