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Set Theory and Topology

by Izzhov
Tags: theory, topology
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mathwonk
#19
Apr15-07, 11:06 AM
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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}.
MathematicalPhysicist
#20
Apr15-07, 11:24 AM
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Quote Quote by mathwonk View Post

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
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Apr15-07, 11:56 AM
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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.
Hurkyl
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Apr15-07, 01:19 PM
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Quote Quote by loop quantum gravity View Post
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
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Apr15-07, 02:33 PM
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Quote Quote by mathwonk View Post
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, ya 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
#24
Apr16-07, 01:21 AM
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Quote Quote by mathwonk View Post
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 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.
MathematicalPhysicist
#25
Apr16-07, 01:31 AM
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Quote Quote by Hurkyl View Post
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 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?
Hurkyl
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Apr16-07, 11:27 AM
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Quote Quote by loop quantum gravity View Post
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?
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
#27
Apr17-07, 01:08 AM
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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
#28
May29-07, 09:09 PM
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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
#29
May30-07, 12:34 PM
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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
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Feb10-08, 02:54 AM
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Free Cantor online text: http://www.openlibrary.org/details/c...nstot003626mbp
mistermath
#31
May31-08, 06:22 PM
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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.


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