> Unlike the case of even integers, one cannot go on to characterize even ordinals as ordinal numbers of the form β2 = β + β. Ordinal multiplication is not commutative, so in general 2β ≠ β2. In fact, the even ordinal ω + 4 cannot be expressed as β + β, and the ordinal number
For a six year old, I'd tell that infinite is not a number so it's not even or odd. If s/he even get's a Ph.D. in math, s/he will understand.
Moreover, I remember when I was a graduate T.A. that one day before lunch I went to a class to learn about the https://en.wikipedia.org/wiki/Alexandroff_extension in the morning. (The idea is that you add one ∞ to a set of numbers to get a compact set. And in the new set ∞ is (almost) a number as good as the other numbers.) After lunch, I went to teach limits to first years students, and with a total straight face I told them that ∞ is not a number.
> After lunch, I went to teach limits to first years students, and with a total straight face I told them that ∞ is not a number.
When you apply Alexandroff extension to add the point at infinity to, say, the real numbers, what you're left with is not a set of numbers (i.e. a field) anymore. So it makes sense to say that ∞ is not a number. Moreover, the way ∞ is used in analysis is different from Alexandroff compactification, in that you usually use two infinities (±∞) as a shorthand for quantification over increasing or decreasing sequences of real numbers (this can be formalized using extended real numbers [0] or other gadgets but doing so has no advantages in a first-year analysis class, and might in fact make matters worse).
It was a long time ago, something like an optional course in Advanced Functional Analysis. It was about the algebras of functions with and without unity, and how to complete the ones without unity using the compactification (i.e. including a ∞) and a few variants.
> two infinities (±∞)
It depends. In the real numbers it depends, but in most cases I agree that it's better to use two. In complex analysis it's much better to have only one infinity. And there are more weird case like the projective plane where you have one infinity in each direction.
> So it makes sense to say that ∞ is not a number.
I agree, it's not longer a field and the operation lose many properties if you try to extend them. So I said "(almost) a number". Anyway, the weird part is that in some cases you can write f(∞) in an advanced math course, but you can never write f(∞) in a fist year math course.
> The problem with transfinite is that you lose commutatively.
Depends on which transfinite algebra you're working with. If you restrict "number" to mean "element of an ordered field" (thus excluding things like the "complex numbers" but matching the usual intuition of how numbers should behave) then you can't include Cantor's ordinals but you can include the Surreal Numbers. Those include infinite ordinals and (due to being a field) have commutative addition and multiplication operations.
I'd say that the problem with transfinites is that you lose intuitive understanding of what's going on, and one of those intuitions is commutativity.
People seem to assume that they know a couple of tricks about infinity (adding, multiplying) and don't stop to think that there should be a much more rigorous definition. Which, they shouldn't -- the average person will never _actually_ care about transfinites.
Imagine the + in C++ when you have to add two complex numbers. They are just a struct with x and y, and some magic to make all operations work as intended.
The use of the + in this example is more like the concatenation of strings, like "Hello " + "World!" is "Hello World!". But in this case, the content of the string doesn't matter so "Hello " == "World!" and there are some magical strings that are infinite.
The idea is that anyone can overload the symbol + and sum whatever they want. It's not necessary to use + with numbers. Obviously, most silly overloads are ignored, and nobody use them. In this case it's a popular overload so it is teach in an advanced math course and has it's own Wikipedia article.
Flowing the standard notation, where the usual infinite in the integer or the real line is "ω = ∞ = 1,2,3,..."
ω+1 = ω+1 , i.e. "the next thing after infinity"
1+ω = ω , i.e. "the same infinity as before"
2ω = ω , i.e. "the same infinity as before", so it's even
1+2ω = ω , i.e. "the same infinity as before", so it looks odd, but don't fall in that trap
ω2 = ω2 , i.e. "two infinities chained together", that is weird
Two more weird example from https://en.wikipedia.org/wiki/Even_and_odd_ordinals
> Unlike the case of even integers, one cannot go on to characterize even ordinals as ordinal numbers of the form β2 = β + β. Ordinal multiplication is not commutative, so in general 2β ≠ β2. In fact, the even ordinal ω + 4 cannot be expressed as β + β, and the ordinal number
> (ω + 3)2 = (ω + 3) + (ω + 3) = ω + (3 + ω) + 3 = ω + ω + 3 = ω2 + 3
> is not even.
For a six year old, I'd tell that infinite is not a number so it's not even or odd. If s/he even get's a Ph.D. in math, s/he will understand.
Moreover, I remember when I was a graduate T.A. that one day before lunch I went to a class to learn about the https://en.wikipedia.org/wiki/Alexandroff_extension in the morning. (The idea is that you add one ∞ to a set of numbers to get a compact set. And in the new set ∞ is (almost) a number as good as the other numbers.) After lunch, I went to teach limits to first years students, and with a total straight face I told them that ∞ is not a number.