On Mon, Jun 20, 2016 at 09:35:17AM +0200, Edward Bartolo wrote:
>
> You seem to work in a university's maths faculty. Can you explain to
> me this paradox?
>
> Consider Set I = {...., -3, -2, -1, 0, 1, 2 , 3, ....}, the set of
> Integers that is infinite in size having neither a lower bound nor an
> upper bound.
>
> Now, consider Set M = {...., -9, -6, -3, 0, 3, 6, 9, ....}, the set of
> multiples of 3 that also has neither a lower bound nor an upper bound.
>
> BOTH sets are infinite, yet, set I has 3 elements for EVERY element in
> set M! This gives the impression infinity is graded. But does it makes
> sense to claim a graded infinity? If it is graded, is it still
> infinite?

Yes, it is a paradox. It arises from thinking about infinite sets
with the same intuitions we get in the real worlds about finite sets.
It is resolved bu realising the differences.

Both sets have the same number of elements. They can be placed in
one-to-one correspondence:

...
-9 <-> -3
-6 <-> -2
-3 <-> -1
0 <-> 0
3 <-> 1
6 <-> 2
9 <-> 3
...

and one-to-oe correspondence is generally theway mathematicians define
the concept of two sets havein the same number of elements.

When I studied math in the 60's, we *defined* an infinite set to mean
a set that was in one-to-one correspondence with a proper subset
of itself.

For fun, google "Hilbert's Hotel", for example,
https://en.wikipedia.org/wiki/Hilbert%27s_paradox_of_the_Grand_Hotel

-- hendrik