- #1

Stephen Tashi

Science Advisor

- 7,112

- 1,298

## Main Question or Discussion Point

I think the cardinality of the set M of all 1-1 mappings of the integers to themselves should be the same as the cardinality of the real numbers, which I'll denote by [itex] \aleph_1 [/itex]. My naive reasoning is:

The cardinality of all subsets of the integers is [itex] \aleph_1 [/itex]. A subset of the integers can be identified with 1-1 mapping of the integers onto the set consisting of only two integers {0,1}. So the cardinality of M should be at least that large.

A 1-1 function is a set of ordered pairs of numbers that satisfy certain conditions. The set of all ordered pairs of integers has a cardinality [itex] \aleph_0 [/itex]. The set S of all possible subsets of that set has cardinality [itex] \aleph_1 [/itex]. The set M is proper subset of S so the cardinality M should be at most [itex] \aleph_1 [/itex].

Perhaps I need some famous "optional" assumption of mathematics, such as the continuum hypothesis, to justify some of those statements.

The cardinality of all subsets of the integers is [itex] \aleph_1 [/itex]. A subset of the integers can be identified with 1-1 mapping of the integers onto the set consisting of only two integers {0,1}. So the cardinality of M should be at least that large.

A 1-1 function is a set of ordered pairs of numbers that satisfy certain conditions. The set of all ordered pairs of integers has a cardinality [itex] \aleph_0 [/itex]. The set S of all possible subsets of that set has cardinality [itex] \aleph_1 [/itex]. The set M is proper subset of S so the cardinality M should be at most [itex] \aleph_1 [/itex].

Perhaps I need some famous "optional" assumption of mathematics, such as the continuum hypothesis, to justify some of those statements.