cardinals and transfinite induction

[MandrakeRoot]

I would like to know the following:

 

Let kappa be a regular cardinal and B a kappa-club (kappa-closed and cofinal in kappa). And let lambda be another regular cardinal strictly smaller than kappa. Why is it possible to choose a strictly increasing sequence {alpha_nu : nu < lambda} in B, such that the suppremum of {alpha_nu : nu < lambda} has cofinality lambda ?

 

My first guess would be to use the fact that for every nu < lambda and every non empty A subseteq B intersection nu, sup A an element of B is.

And then do something with transfinite recursion, but why would B even contain any ordinals smaller then lambda ?

 

I hope someone could shed some light on this question.

Thanks in advance,

 

Mandrake

[bloodhound]

sorry dude, thats just too advanced. we are just first year maths students

[MandrakeRoot]

Let me include the definitions of the different things

 

a set B is kappa-closed if for every xi < kappa and every non empty A subseteq B intersection xi , we have sup A = union A element of B.

 

since a cardinal is in fact an ordinal, and for ordinals alpha, beta, alpha < beta if alpha is an element of beta. xi < kappa means xi element of kappa and union A is defined as the set of all elements x such that there is an y in A for which x in y.

Since each element of an ordinal is again an ordinal, it can be shown that sup A will be also an ordinal and thus the requirement sup A in B makes sense.

 

a set B is said to be cofinal in kappa if for every xi < kappa we can find an eta in B such that xi <= eta (<= meaning < or =)

 

The cofinality of a set is defined as the cardinality of the least set cofinal in it.

 

A regular cardinal kappa is a cardinal for which cofinality(kappa) = kappa, and thus the least subset of kappa being cofinal in it must have cofinality kappa.

 

(e.g. in N the set of even numbers is cofinal and omega = cardinality(N) is a regular cardinal).

By the way cf(kappa) <= kappa for all cardinals kappa.

 

Transfinite recursion is just some theorem saying that if you arrive to assign a set to each function f by some operation G, then there is an operation F

such that F(alpha) = G(F|alpha). So in fact if you can define something for ordinals/cardinals smaller then alpha, then you can somehow extend it to include all ordinals. Like recursively defining a sequence x_n+1 = x_n + 2 or something like so, you define x_0 = 0 and then you recipe and then you have a sequence.

 

I hope i could clarify the question ?

[Dave]

Sorry, it's still quite a bit too advanced for me. I don't know whether there's anyone else that has an answer to this question, but I'm sure it'll be answered (in time)

[alext87]

That's way too hard!!!!

[alext87]

However if somebody would like to tell me how to do it i would be interested!

[Dave]

I kindof get the jist of the question, but basically it's just abstract set theory. Doesn't really have many applications. I believe a cardinal describes the size of a set (i.e. the cardinality of a set, etc).

[MandrakeRoot]

I have to disagree about set theory not having applications. It answers questions like the place of the axiom of choice for instance, a vital theorem for Hahn-Banach and the existence of functionals, which is again very important for many questions in operator theory and thus for quantum mechanics. Also the axiom of choice is very often implicitely used in many proofs. Even though direct applications are maybe rare, the questions posed do have important implications in other fields of mathematics.

A cardinal is a set, but also the size of a set.

 

Two sets are said to have teh same cardinality if you can create a bijection in between them.

 

Mandrake,

[Dave]

To be perfectly honest, very little of modern mathematics has much direct implementation; I find myself sitting in lectures thinking 'what possible application could this actually have?' Don't get me wrong, I find it very interesting, and basically that's what modern mathematics is - investigating things we find interesting. More advanced concepts for the most part have little or no use in the real world.

[bloodhound]

dave, u maybe rite for now. But this mathematics will come very handy in future. Just like Differential and integral calculus was only used to describe motion of bodies relative to time when the tool was invented.

[MandrakeRoot]

Yeah i think i would agree with that. It might be very usefull in the future. The first physicist to occupy them with elementary particles and radioactivity, had no idea that would ever be used and in the time that was considered to be abstract physics without any use.

 

Mandrake