Schindler - Set theory: exploring independence and truth
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- Book:Set theory: exploring independence and truth
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- Publisher:Springer International Publishing
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- Year:2014
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onto the set of all prime numbers. Hence, it seemed, on the one hand the set of all primes is smaller than 
, whereas on the other hand it is as big as 
.
and 
be arbitrary sets. Define 
is smaller than or of the same size as 
(or, 
is not bigger than 
) as: there is an injection 
. Write this as 
. Define 
is of the same size as 
as: there is a bijection 
. Write this as 
. Obviously, 
implies 
. The theorem of Cantor Schrder Bernstein (cf. Theorem 1.4) will say that 
follows from 
and 
. We write 
if 
but not 
.
, i.e., if there is an injection 
, then there is a surjection 
. This is clear if 
is already bijective. If not, then pick 
(we may assume 
to be nonempty). Define 
by 
, if 
is in the range of 
and 
otherwise.
is surjective then there is an injection 
, i.e., 
. This is shown by choosing for each 
some 
with 
and setting 
. This argument is in need of the Axiom of Choice, AC , which we shall present in the next chapter and discuss in detail later on.
and 
as defined above. The proof of the following theorem may be regarded as the birth of set theory.
is trivial. We show that 
does not hold.
to 
, so that there is then also a surjection 
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