Link is good, but I guess direct explanation of this simple thing could be useful.
It is not hard to build explicit map between R and R² (more or less interleaving the binary notations for numbers).
So the claim of Continuum Hypothesis is:
For every property of real numbers P there exists such a property of pairs of real numbers, Q such that:
1) ∀x (P(x) → ∃! y Q(x,y))
2) ∀x (¬P(x) → ∀y¬Q(x,y))
(i.e. Q describes mapping from support of P to R)
3) ∀x1,x2,y: ((Q(x1,y)^Q(x2,y)) → x1=x2)
(i.e. the map is an injection)
4) (∀y ∃x Q(x,y)) ∨ (∀x∀y (Q(x,y)-> y∈N))
(i.e. map is either surjection to R or injection to a subset of N)
These conditions say that every subset of R is either the size of R or no bigger than N.
Link is good, but I guess direct explanation of this simple thing could be useful.
It is not hard to build explicit map between R and R² (more or less interleaving the binary notations for numbers).
So the claim of Continuum Hypothesis is:
For every property of real numbers P there exists such a property of pairs of real numbers, Q such that:
1) ∀x (P(x) → ∃! y Q(x,y))
2) ∀x (¬P(x) → ∀y¬Q(x,y))
(i.e. Q describes mapping from support of P to R)
3) ∀x1,x2,y: ((Q(x1,y)^Q(x2,y)) → x1=x2)
(i.e. the map is an injection)
4) (∀y ∃x Q(x,y)) ∨ (∀x∀y (Q(x,y)-> y∈N))
(i.e. map is either surjection to R or injection to a subset of N)
These conditions say that every subset of R is either the size of R or no bigger than N.