Is the Cardinality of the Reals Equal to the Power Set of the Naturals?

[pjmunki]

C(reals) = C(P(naturals))??

hello,
Could someone help me please.
I am studying Cantor's set theory at present, but am a little confused as to why he concludes that the cardinality of the set of real numbers is equal to the cardinal number of the power set of naturals (2^aleph0).

Thanks.

 

[NateTG]

The elements of 2^N can be treated as sequences of 1's and 0's where a sequence corresponding to a subset A of N has a 1 in the n th position if n is in A.

You should have no trouble seeing that this mapping is bijective.

Now, if you look at the real numbers on [0,1] base 2, you get numbers like:
0.10101011101000110...
which are also sequences of ones and zeros. So there's a natural mapping.

Unfortunately there is a problem because
0.011111111111111111111111...=
0.100000000000000000000000...
in the reals.

But that only occurs a countable number of (N) times. So we can certainly construct a bijection to [0,1] + N.

So we have |2^N| = |[0,1] + N|

but you should already know that |[0,1] + N|=|[0,1]|=|R|
(Cantor certainly did)

So by substitution we get:
|2^N |=|R|

 

[tacman]

According to Cantor's theorem, the cardinality of the set of real numbers (C(reals)) is equal to the cardinality of the power set of naturals (C(P(naturals))). This means that for every real number, there exists a unique subset of naturals and vice versa.

To understand this, we need to first understand the concept of cardinality. Cardinality refers to the number of elements in a set. In other words, it is a measure of the size of a set. Cantor's theorem states that if two sets have the same cardinality, then they are considered to be of equal size.

Now, let's look at the set of real numbers. We know that real numbers are infinite and uncountable. This means that we cannot assign a natural number to each real number, as we can with the set of natural numbers. This is where the concept of power set comes into play. The power set of a set is the set of all possible subsets of that set.

In the case of the set of naturals, the power set (P(naturals)) includes all the possible subsets of naturals, including the empty set and the set of all naturals. This set is also infinite and uncountable, just like the set of real numbers.

So, if we think about it, for every real number, there exists a unique subset of naturals that corresponds to it. This means that the cardinality of the set of real numbers (C(reals)) is equal to the cardinality of the power set of naturals (C(P(naturals))).

In conclusion, C(reals) = C(P(naturals)) because there is a one-to-one correspondence between the elements of these two sets. I hope this helps to clarify the concept for you.