# Subsets of non countably infinite sets

• pyrole
In summary, the conversation discussed the concept of uncountably infinite sets and their subsets in relation to probability. It was mentioned that for sets with an uncountable number of elements, the number of subsets is not equal to 2^n. This concept may seem unintuitive, but it is related to the Continuum Hypothesis and the cardinality of different sets. The book may have been discussing the possibility of a transfinite cardinal larger than \aleph_0 but smaller than 2^{\aleph_0}.

#### pyrole

I was reading an introductory chapter on probability related to sample spaces. It had a mention that for uncountably infinite sets, ie. in sets in which 1 to 1 mapping of its elements with positive integers is not possible, the number of subsets is not 2^n.
I certainly find this very unintuitive, for eg. the set of all real numbers is an uncountably infinite set, I suppose.

Could someone throw some light on the topic with some examples. What happens when we look at probabilities of events in these sets?

Thanks

Hi pyrole!

What exactly does the book say. The number of subsets of every set A is

$$|2^A|=2^{|A|}$$

In particular, if A is finite, then the number of subsets is finite. And if A is infinite, then the number of subsets are uncountable.

pyrole said:
I was reading an introductory chapter on probability related to sample spaces. It had a mention that for uncountably infinite sets, ie. in sets in which 1 to 1 mapping of its elements with positive integers is not possible, the number of subsets is not 2^n.
I certainly find this very unintuitive, for eg. the set of all real numbers is an uncountably infinite set, I suppose.

Could someone throw some light on the topic with some examples. What happens when we look at probabilities of events in these sets?

Thanks

Can you copy the entire passage from the text? It sounds like you might be confusing this with the Continuum Hypothesis (CH).

First, the cardinality of the collection of subsets of any set $S$ is always $2^{|S|}$, where the absolute value bars $|S|$ denote the cardinality of $S$. So I don't believe that what you wrote is correct.

For example the cardinality of the set of natural numbers $\mathbb{N}$ is $\aleph_0$; and the cardinality of the real numbers is $2^{\aleph_0}$. It's easy to exhibit a bijection between the reals and the subsets of $\mathbb{N}$.

CH says that there is no other transfinite cardinal strictly between $\aleph_0$ and $2^{\aleph_0}$. CH is independent of the the usual axioms of set theory, known as ZFC. So there may or may not be some cardinal strictly larger than $\aleph_0$ and strictly smaller than $2^{\aleph_0}$. [Or the question may have no meaning, depending on one's philosophy.]

It seems likely (to me) that this is what your book was talking about; but in any event, if you post the relevant quote from the text we can have a better idea of what they are getting at.