Where Can I Find Resources for Learning Set Theory and Topology?

[Izzhov]

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.

 

[quasar987]

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

 

[Izzhov]

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

 

[Thrice]

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

 

[quasar987]

Here's a little something I found googling "Set Theory"

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.

 

[ZioX]

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.

 

[neutrino]

List of people working on set theory
http://www.ucl.ac.uk/~ucahcjm/stp.html

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

 

[MathematicalPhysicist]

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

 

[mathwonk]

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.

 

[mathwonk]

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.

 

[mathwonk]

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

 

[mathwonk]

heres kamkes book:

Theory of Sets
E. Kamke Bookseller: WebBookStore
(Pittsburgh, PA, U.S.A.) Price: US$ 2.87
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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

 

[mathwonk]

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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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] [

 

[mathwonk]

and here's 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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Book Description: New York Dover Publications circa 1960., 1960. Very Good condition,light shelfwear,wraps,ex-lib. Bookseller Inventory # 052412

 

[mathwonk]

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

 

[mathwonk]

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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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

 

[mathwonk]

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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Book Description: Prentice Hall, 1999. Hardcover. Book Condition: New. New. Excellent customer service. Bookseller Inventory # B0131816292Publisher Photo Topology (ISBN: 0131816292)
James Munkres Bookseller: viktree
(san jose, CA, U.S.A.) Price: US$ 10.76
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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

 

[mathwonk]

oh yes, here's halmos:

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

 

[mathwonk]

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}.

 

[MathematicalPhysicist]

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 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.

 

[mathwonk]

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.

 

[Hurkyl]

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.

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.

 

[Hurkyl]

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.

Formal logic is a form of mathematics, you know. And I don't really think you have a good picture of formal mathematics -- one of the major topics is studying theories and proof from a purely syntactic point of view. And logicians are concerned about what numbers "are", but only in a certain formal sense: the study of semantics formalizes the process by which we look at the finite ordinals (or a certain subset of R, or the collection of finitely generated free R-modules, or...) and say that they "are" natural numbers. (I.e. a model of the naturals)

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

 

[MathematicalPhysicist]

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.

im just saying that you can define the natural numbers set in set theory.
(and I am surprised of hausdorff's comment in a set theory book).
and to myself i don't think there's a big difference between logicians and mathematicians.
most logicians have an education in maths, their speciality is something else.

 

[MathematicalPhysicist]

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.

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 shouldn't 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?

 

[Hurkyl]

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 shouldn't 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?

Hausdorff was not saying that you should not make this construction -- he was not saying that you should not prove that the finite ordinals are a model of the natural numbers.

Hausdorff was saying that you shouldn't read too much into this construction -- you should not think of this construction as saying what the natural numbers "are".


When you prove that the finite ordinals under addition satisfy the monoid axioms, do you think "Aha! The monoid axioms were really just defining the finite ordinals!"? I assume not -- so when you prove that the finite ordinals under the successor operation satisfy Peano's axioms, why would you think "Aha! Peano's axioms were really just defining the finite ordinals!"?

 

[MathematicalPhysicist]

ofcourse not.
but from the comment quoted by mathwonk i inferred that he meant that you cannot define the natural numbers.
but here in set theory you obviously can do this.
if you may bring the full quote i will be convinced otherwise.

 

[mathwonk]

one big difference is that logicians seem lots smarter.

and i think anyone who knows what an isomorhism is, should understand that in math, what things "are' is unimportant compared to how things behave.

i.e. is this two: "2", or is this "II", or is this {. . } or this {0,{0}}, or all of them? it really doesn't matter, what matters is understanding bijections.

 

[MathematicalPhysicist]

in this case, just define cardinal numbers as russell and witehead did in their PM.
so there are more than one model to the number system, but it's still count as defining them in the particular model.

 

[EnumaElish]

Free Cantor online text: http://www.openlibrary.org/details/contributionstot003626mbp

 

[mistermath]

Here is my opinion on this. There are two paths you can take. One is, you can try and get an "ok" grasp on foundations of mathematics in order to prepare yourself for topology, or you can assume the foundations and go on.

If you are studying set theory, then why not study mathematical logic first (well formed formulas etc), then move up to set theory then move up to topology. This is imo a waste of time since most of the math you'll ever do assumes this stuff.

Or you can (this is the choice I recommend) just skip set theory and do topology.

All you need to know are the basics:
-What a set is.
-What a union and intersection is.
-De Morgan's Laws
-FACT: If I give you a set filled with an uncountable number of objects, you can pick an object from it. (called the axiom of choice).

I believe that if you pick up Munkrees (Like someone mentioned earlier) you can do the first 9 sections, skip 10/11 and learn all the topology you want, well worth the $50. Even feel free to skip the first 9 sections if you are decent with bijections, peano etc.