Georg Cantor
also famously claimed that the cardinality of the set of rational
numbers equals the cardinality of the set of integers because he
could devise a 1-1 correspondence or mapping between them as shown
here and here.

Relatedly,
the continuum hypothesis
is: There is no set whose cardinality is strictly between that of the
integers and the real numbers (rationals and irrationals).

Accepting
certain assumptions Paul Cohen showed that the continuum hypothesis
is neither true nor false. I think it’s false, the quantity of
rational numbers being greater than the integers, but less than the
reals (rationals and irrationals combined). I base this on two perspectives different from that of Cantor and Cohen. The first is the real number line. There is an unlimited count of rational numbers between 0 and 1, between 1 and 2, and so forth.

The second is part-whole logic. As
shown in the above links, a 1-1 mapping can be made. However, note
the first column in the array in either link, which is the integers.
A different 1-1 mapping or function f(x) = x can also be made.
Accordingly, the integers comprise a

*proper subset*of the set of all rational numbers, implying the set of rational numbers is larger than the set of integers.
“Cantor
was concerned to combat the Aristotelian view that there cannot be an
actual infinity, mainly because Cantor believed that God was
infinite.” – Michael Huemer,

*Approaching Infinity*, p. 71.
Regardless of Cantor's religious motivations, it is still the case that there are sets which can be put into 1-1 correspondence with proper subsets of the set. That is the difference between finite sets and infinite sets. A finite set cannot be put into 1-1 correspondence with a proper subset set.

ReplyDeleteWhy not? A = {1, 2, 3}. B = {1, 2, 3, 4, 5, 6}. Make a 1-1 correspondence between the members of A and the first three members of B.

ReplyDelete