buffordboy23
- 545
- 2
Hi, I am doing some self-studying on the topic of functional analysis, specifically set theory at the moment.
Suppose that we want to show that some set is denumerable. Is it required that we directly show the one-to-one correspondence between the elements of the set and the set of natural numbers? This seems to be the method of proof employed in the text for the given theorems, but the results of the theorems themselves seem to offer hand-waving. For example, the set that consists of the sum of a denumerable number of denumerable sets is itself denumerable.
Now suppose that the elements of some set that we are trying to prove is denumerable can be represented by a table of infinite elements. Must we use the "diagonal method" to prove that we can enumerate the elements of this set? Or is it sufficient to enumerate row by row, although the number of elements in each row of the table is infinite (we would never make it to the next row)?
Suppose that we want to show that some set is denumerable. Is it required that we directly show the one-to-one correspondence between the elements of the set and the set of natural numbers? This seems to be the method of proof employed in the text for the given theorems, but the results of the theorems themselves seem to offer hand-waving. For example, the set that consists of the sum of a denumerable number of denumerable sets is itself denumerable.
Now suppose that the elements of some set that we are trying to prove is denumerable can be represented by a table of infinite elements. Must we use the "diagonal method" to prove that we can enumerate the elements of this set? Or is it sufficient to enumerate row by row, although the number of elements in each row of the table is infinite (we would never make it to the next row)?