> In article <3681B99D.66997C67@ashland.baysat.net>, Mike Deeth > <mad@ashland.baysat.net> wrote: > > > The table below shows a bijection between a Natural number and a Set with the > > same > > number of members. None can doubt this bijection continues into the infinite. > > Lets say that the last row of the table contains the infinite set of all > natural > > You call it a bijection, but there are sets, such as (2,4,6} for which you > have not provided a natural number.
Wrong! {2,4,6} maps to 101010 (binary) or 12 (decimal). I think you meant the set of all even numbers {2,4,6,...} which is mapped to the infinite binary string "...101010".
> It is only an injection, NOT a > bijection. THere is no such bijection of the integers ONTO the power set > of the integers. There will always be some sets of integers left out. >
I disagree. Do you realize that the odd natural numbers can form a bijection with the complete (odd & even combined) set of the natural numbers? Notice that all the even numbers are "left out" when the sets are compared non-bijectively. Does this example clear things up?
> > > You also speak of the last row of a list that has no last row. > If there were a last row, then there would be an integer so big that you > could not add one to it.
Yes!! My point exactly. This absurdity has its oragins in the "axiom of infinity".
