The power set of the continum

[Kent]

I read a popular science book, and in one chapter it talks about transfinite numbers.

i have some questions.

Can the power set of the continum be enumerated? I have no idea how you imagine a power set of a power set of a continum? Are their pictures?

what are the application for the power set of a set?

what are your thoughts on transfinit numbers?

any problems?

[maddog]

Originally Posted by Kent

I read a popular science book, and in one chapter it talks about transfinite numbers.

i have some questions.

Can the power set of the continum be enumerated? I have no idea how you imagine a power set of a power set of a continum? Are their pictures?

what are the application for the power set of a set?

what are your thoughts on transfinit numbers?

any problems?

I can help a little bit. You acknowledge that there are an infinite number of integers. In fact you can
enumerate them. The same for the rationals. There is a proof that I am unable to give there are as many
rationals as integers. However, the quantity of reals are more than the integers. Cantor came up with a
notion of transfinite numbers or "levels of infinity". The number of integers are assigned Aleph 0 (naught),
for the subscripted Hebrew character. Number of reals is Aleph 1. The number of continuous functions over
the reals Rn is Aleph 2. And so on. For more info, do a google search on Cantor and Transfinite numbers.

Maddog

[Kent]

Originally Posted by maddog

I can help a little bit. You acknowledge that there are an infinite number of integers. In fact you can
enumerate them. The same for the rationals. There is a proof that I am unable to give there are as many
rationals as integers. However, the quantity of reals are more than the integers. Cantor came up with a
notion of transfinite numbers or "levels of infinity". The number of integers are assigned Aleph 0 (naught),
for the subscripted Hebrew character. Number of reals is Aleph 1. The number of continuous functions over
the reals Rn is Aleph 2. And so on. For more info, do a google search on Cantor and Transfinite numbers.

Maddog

Well, i know that the set of natural , rational, intergers are denumerable so then they share the same cardinality, and i believe the book called it "aleph nunght"( called it N).

Then the book show that there is another set of number c(stand for continum) that are of greater cardiinary than aleph nunght.

Definition: c is the cardinaity of the set of real numbers

in other word

N < c

How to find a cardinary larger than c?

In order to do so, the book proof a general statement that is ---a set is always a proper subset of it `s power set.

in other word

Let `s denote suppose "K" as a set. then it`s power set, called it P[k].

then this relationship always hold:

A theorm: "cordinarity K < cordinarity P[K] "

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briefing on Power set:

suppose there are n element in a set K ( or the cardinity of K)

then the cardinairty of set K, P[K] is 2^n


example:

K={ a, b, c}

P[K] ={ {}, {a}, {b}, {c}, {a,b}, {a, c} , {b, c}, {a, b, c}}




---------------------------------------------------------------------------

In other word

By thinking of the cardinarity of c and it power set P[c]

then the relationship must also hold

c < P[c]


using the same type of reasoning

c< P[c]< P[P[c]]< P[P[P[c]]]<.............................

thus estable the order of the infinite


---------------

I Think

If say the set of P is {a, b }, then the number of element in the power set of p mush be 2^2

listing them: {{}, {a}, {b}, {a,b}}


If the cardinaity of the real is c, then the power set of c, P[c] is 2^c

continue the same reasoning

N < c < 2^c < 2^c^c< 2^c^c^c <....................................

----------------------------

How do you list the element of 2^c or 2^c^c or........?

To tell you the truth, it is soo counter intuitive that i have a hard time accepting it. The whole thing invoke the image of god to me. what are other people s thoughts? philosophies?...

perhaps it should be in the philosophy section.

It is such a amusement that it should be worth wild to discuss the philosophy or even God.

If human can tame the infinite, then can god be tame? or by reason alone, can we understand god itself??????????????[/B]

[maddog]

I don't want repeat your whole post by quoting it.

Instead I will add to what you have already done using your labels.

Your Cardinality of the Continum c which is the same size as the reals is Aleph 1 in my terminology.
What I was saying as an example of the next hierarchy was the set of continuous function of Rn being
the same size as your P(k) or Aleph 2 in Cardinality. I don't know of an example beyond that.

Maddog

[Kent]

Originally Posted by maddog

I don't want repeat your whole post by quoting it.

Instead I will add to what you have already done using your labels.

Your Cardinality of the Continum c which is the same size as the reals is Aleph 1 in my terminology.
What I was saying as an example of the next hierarchy was the set of continuous function of Rn being
the same size as your P(k) or Aleph 2 in Cardinality. I don't know of an example beyond that.

Maddog

You can always construct one infinite higher than another....
It is great for conversation and parties


Is 2^(aleph nunght) =aleph 1=c ? What do you think?

[maddog]

Originally Posted by Kent

You can always construct one infinite higher than another....
It is great for conversation and parties

Is 2^(aleph nunght) =aleph 1=c ? What do you think?

From what I remember from reading about Cantor's work, yes.

Maddog