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Person A: The number of even and odd natural numbers is equal, because they alternate each time. Person B: There are far more even natural numbers. If you double an odd number, it's even. If you double an even number, it's still even.

How do I best explain to a person with only base mathematical knowledge the truth of the matter?

BmyGuest
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  • I'd explain that infinite is not a number, but a different, abstract, beast that sometimes can be be compared to a number, but most of times it has its own rules. – Ripi2 Apr 03 '21 at 19:53
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    Only 1/3 of all numbers are even. Take the asymptotic limit: 1,3,2,5,7,4,9,11,6, 13,15,8, ... – user4894 Apr 03 '21 at 19:53
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    Doubling introduces evenness and is therefore a failed attempt to classify this property. One way to approach this is to take an increasing interval from the natural numbers and count the even and odd members. – abiessu Apr 03 '21 at 19:54
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    Person C: There are far more odd numbers because every even number arises from adding 1 to an odd number. This "logic" as as false as that "doubles logic" – Ripi2 Apr 03 '21 at 19:57
  • Thanks to all who contributed in comments or answers. – BmyGuest Apr 04 '21 at 19:21

2 Answers2

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"There are more" is not well-defined for infinite sets. I think the real answer is that the question isn't meaningful. (The existence of an injective map is well-defined, but you can debate whether that means 'there are at most as many'.)

You may dispute the latter argument by pointing that "If you take a natural number and add its successor to it, the result is always odd". There are infinitely many functions $f : \mathbb N \times \mathbb N \rightarrow \mathbb N$ that always produce odd or always produce even numbers.

silver
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  • Would you have a suggestion how to best explain the issues of "infinity" to a person with mathemtaical knowledge of a ~9th grader? – BmyGuest Apr 04 '21 at 19:22
  • Good question. I think what I would do is emphasize 'infinite' as a property rather than a thing. The natural numbers are defined as "0 is a NN, and for each NN, the successor is another NN". This defines a set, and all we can do is study properties of this set, which turn out to often be unintuitive. If the person says that 'there are more' should be a meaningful property, just ask them how they would define it. Dk if that's helpful. – silver Apr 05 '21 at 13:25
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    Thanks. It is helpful for me. I did not convince the person I was speaking with - but it did end the discussion. – BmyGuest Apr 05 '21 at 17:56
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The second argument only shows that in the "double naturals", there are far more even numbers than odd ones. The argument says nothing about the naturals themselves.

Modern theory says that there are as many even as odd naturals, because you can associate every even number to an odd one and conversely (add or subtract $1$).

But it also says that there are as many naturals as there are even ones (multiply and divide by $2$) ! And there are as many primes as there are naturals.