Re: Basic set question


agapito6...@xxxxxxx wrote:
> Say set A = {1,1,3,5,7}, and B = {1,3,5,7}
>
> By the definition of set equality A = B since each point of A belongs
> to B and vice-versa. By the definition of equivalence, A is not
> equivalent to B since they cannot be put in one-to-one
correspondence.
> So we have a case of a set not equivalent to itself, which is absurd.
> What is wrong with the argument?

"B = {1,3,5,7}" is just shorthand for the following statement:

For all x, (x is an element of B) if and only if (x=1 or x=3 or
x=5 or x=7).

{1,1,3,5,7} is not standard notation; if you want to interpret it as a
set, using the above convention, it would be the same thing as
{1,3,5,7}.

Remember the axiom of extentionality:
For all x and y, {if [for all z (z in x implies z in y)] and
[for all z (z in y implies z in x)], then x = y}.

.

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