## Are all sets countable sets?

Can all sets be put into one-to-one correspondence with the counting numbers (size $\aleph_0,$ pronounced "aleph-null").

Consider the real numbers between 0 and 1.

If they are countable then there exists an (infinite) list that starts like this which will include all real numbers between 0 and 1.

 1 0.6735465 ... 2 0.1834892 ... 3 0.0001538 ... 4 0.9918771 ... 5 0.6881001 ... 6 0.2892353 ... 7 0.2596701 ...

But we can construct a number between 0 and 1 which is not in the list !!!

Let N = 0.7919162 ...

It differs from the 1st number in the list in the first decimal place (increased by 1)

It differs from the 2nd number in the second decimal place.

It differs from the 3rd number in the third decimal place.

• It is constructed to differ from the nth number in the nth decimal place so it is different from every number in the list.
So the assumption that the real numbers between 0 and 1 can be listed i.e. put in one-to-one correspondence with the counting numbers is wrong.

Therefore, the real numbers are uncountably infinite.

The size of the uncountably infinite real numbers is given the transfinite number c - called the size of the continuum.

Ummm interesting two different sizes of infinity !!!

# How are $\aleph_0$ and $c$ related?

Now that we know there is more than one infinity, mathematicians optimistically called "the next infinity" $\aleph_1.$ Now we know that

• $\aleph_0<2^{\aleph_0}$
• $\aleph_0<c$
The question is, how do $2^{\aleph_0}$ and c compare?

It turns out that $2^{\aleph_0}=c.$

Another question is this: is there an infinity between $\aleph_0$ and c ?

Surprisingly, it is formally undecidable as to whether there is an infinity between $\aleph_0$ and c. The Continuum Hypothesis was that they are equal, but Paul Cohen showed in 1963 that the question cannot be decided from the usual axioms of set theory (ZFC).

For more information see transfinite numbers, the continuum hypothesis and Georg Cantor.

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