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Set Theory and Topology 
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#19
Apr1507, 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}. 


#20
Apr1507, 11:24 AM

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


#21
Apr1507, 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. 


#22
Apr1507, 01:19 PM

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PF Gold
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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 settheoretic problems with natural number arithmetic. One rarely uses this model to solve natural number problems using set theory. 


#23
Apr1507, 02:33 PM

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


#24
Apr1607, 01:21 AM

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(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
Apr1607, 01:31 AM

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#26
Apr1607, 11:27 AM

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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!"? 


#27
Apr1707, 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. 


#28
May2907, 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. 


#29
May3007, 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. 


#30
Feb1008, 02:54 AM

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Free Cantor online text: http://www.openlibrary.org/details/c...nstot003626mbp



#31
May3108, 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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