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**Some question about irrational numbers**

Our teacher showed us Cantor's second diagonal proof.

He said that by this proof we can show that there are more irrational numbers

than rational numbers.

He also said that the cardinality of natural numbers or rational numbers has a magnitude called aleph_0, where the cardinality of irrational numbers has a magnitude of 2^aleph_0.

He said that any set that its cardinal = 2^aleph_0 has the power of the continuum, so we can conclude that the set of all irrational numbers has the power of the continuum.

I thought about it and I have this question:

If we have two sets of irrational numbers (let us call them S1 and S2) such that:

S1={x: x < 1}

S2={x: x > 1}

then in this case the boundary point between S1 and S2 is the natural number 1.

In this case S1 and S1 are disjoint sets, because the boundary point is a natural number.

But then we can conclude that the irrational numbers do not have the power of the continuum, even if the magnitude of their cardinal is 2^aleph_0.

I think that from what I have shown, the irrational numbers are uncountable (by Cantor’s

Second diagonal method) but because S1 and S2 are disjoint sets, the set of all irrational numbers does not have the power of the continuum.

So, my question is:

What is the cardinality of the set of all irrational numbers, and how it is related to the continuum hypothesis?

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