Set Theory and Topology

[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 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]

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.