If **natural numbers** refers to integers, removing all odds & all evens results in an empty set of numbers.

uhChem: I see no paradox, merely no more integers left in the set.

Your view that the final set is empty seems perfectly reasonable to me. I should have been more specific about what I was talking about here. I was addressing the idea that

**infinity - infinity = infinity**. Sorry that was not clear. Totally my fault. And here I will explain better. I hope.

1. The count of numbers in the set of natural numbers is infinity.

2. The count of numbers in the set of odd natural numbers is infinity.

3. So if we remove all of the odd numbers from the natural numbers, one would think that what is left is the set of even numbers, which has the count of elements at infinity.

4. So lets check, infinity - infinity = infinity. Looks OK.

5. Now remove the set of even natural numbers which also has the count infinity.

6. What is the count of the remaining set? Well, infinity - infinity = infinity. So the remaining set still has an infinite number of objects in it.

What are these objects? Well, we started by saying they were all natural numbers. And we removed all the odds and evens, so what is left are the natural numbers that are neither odd nor even. And there are mysteriously an infinite number of these things leftover because the count should be infinity, if you beleive that infinity - infinity is infinity. Sounds pretty weird to me. I found this proof that all natural numbers are either odd of even. I am going to assume it is correct. I don't have time to parse it but it seems reasonable from my scan.

http://math.colorado.edu/~jonathan.wise/teaching/math2001-spring-2016/proof09.pdf
My "proof" is about as far from rigorous as a person can get. I will state outright that I think it is complete nonsense. My original plan was start with this mess and begin to clean it up and point out the many errors in it. I would try and insert a little more rigor and remove the errors in a piecewise method. And at the end I was going to use my reasoning to show:

1. infinity is not a number.

2. infinity - infinity = infinity is nonsense.

3. Answer the OP's question about "computerised-random-number-generation-infinity".

Unfortunately, on rethinking the whole thing, I have vastly underestimated the amount of posts and time that would take. I am trying to think of a better way to proceed. I think I can do it. The question is if I have the time and writing skill to get my points across.

SUMMARY: I think you are right. What is left should be the empty set.