Power set of integers

[Ruffgos]

I think I made a mistake somewhere in my reasoning but I cant find where:

For all the finite and infinite sequences of natural numbers we can construct a sum in a following way:
{1,2,3} -> (1/2)¹ + (1/2)² + (1/2)³
{10} -> (1/2)¹⁰
{1,3,5,6,7,8,9,12,...} -> (1/2)¹ + (1/2)³ + ...

For every sequence the sum will be equal to some real number in [0,1].
And for all real numbers there is some sequence corresponding to it in this way (because this is just representing real numbers in binary)

Only time when there are more than one sequence corresponding to one real numbers is when this number is rational (like 0.101 = 0.10011111...). And since rational numbers are countable, there are only countably many infinite sequences that correspond to rationals.

For every irrational real number, therefore, is a unique (infinite) sequence of increasing natural numbers. And for almost all sequences of increasing natural numbers there is a unique irrational number.

And since sequences of this nature can be bijected with all the subsets of integers, does that mean that the power set of integers is the same cardinality as the set of reals on [0,1]?
But continuum hypothesis is independent from ZFC, so where did I make a mistake?

 

[pka]

I think I made a mistake somewhere in my reasoning but I cant find where:
For all the finite and infinite sequences of natural numbers we can construct a sum in a following way:
{1,2,3} -> (1/2)¹ + (1/2)² + (1/2)³
{10} -> (1/2)¹⁰
{1,3,5,6,7,8,9,12,...} -> (1/2)¹ + (1/2)³ + ...

You need to study this link: the generalized continuum hypothesis

 

[lex]

And since sequences of this nature can be bijected with all the subsets of integers, does that mean that the power set of integers is the same cardinality as the set of reals on [0,1]?
But continuum hypothesis is independent from ZFC, so where did I make a mistake?

Yes the cardinality of the powerset of [imath]\mathbb{Z}[/imath] is the same as the cardinality of [0,1] (as the symbol [imath]2^{\aleph_0}[/imath] suggests). I don't see how this involves the continuum hypothesis; [imath]\mathbb{Z}[/imath] has cardinality [imath]\aleph_0[/imath] and [imath]\mathcal{P}(\mathbb{Z})[/imath] has cardinality [imath]2^{\aleph_0}[/imath] doesn't mean that another set cannot have a cardinality between these.