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This poses the problem of designating the cardinality of the natural numbers. The standard symbol for the cardinality of the natural numbers is ?o. The set of even natural numbers has the same number of members as the set of natural numbers. The both have the same cardinality ?o. By transfinite arithmetic we can see this exemplified.
1 2 3 4 5 6 7 8 ?
0 2 4 6 8 10 12 14 16 ?
When we add one number to the set of evens, in this case 0 it appears that the bottom set is larger, but when we shift the bottom set over our initial statement is true again.
1 2 3 4 5 6 7 8 9 ?
0 2 4 6 8 10 12 14 16 ?
We again have achieved a one-to-one correspondence with the top row, this proves that the cardinality of both is the same being ?o...
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