On Mon, Jun 20, 2016 at 09:35:17AM +0200, Edward Bartolo wrote:

[cut]

>
> 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?
>

Despite your question might be a bit off-topic in this list, I am
sorry but there is no paradox here. The two sets belong to the class
of numerable infinity and have the same cardinality (i.e., the same
number of elements), however strange this might seem at a first
sight.

The proof consists into showing that both have the same size of the
set of natural integer numbers N=0,1,2,... and proceeds by assigning
the "0" in each set to the number "0", positive elements to odd
integers and negative elements to even integers.

QED

KatolaZ

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