My answer to the child would be to talk about another number. The "number" larger-than-100. Is larger-than-100 even or odd? Hopefully the child would realize that neither of those answers are correct. Then the child will say that larger-than-100 isn't a real number. But it is just as real of a number as infinity (ie. it isn't a number either). However, infinity is one world, and there are no numbers that can be an example of infinity. Larger-than-100 doesn't have those characteristics, so that is why infinity feels like more of a number.
It's not a number. It's a process of generating an unbounded series of numbers. Then, what kind of infinity do are we talking about. Monotonic number series interleave odd and even numbers. The series of all positive odd numbers will by definition generate odd numbers. And so on.
It’s a cardinal number: the size of sets. Adding a single new element to an infinite set doesn’t change its size, so adding 1 couldn’t change an odd/even property of that cardinal number.
But there are also infinities as ordinal numbers, and for those there is a well-defined notion of odd/even, based on their total ordering.
Lets say you have a number x. x could be 2 or 3 or whatever. If you are only told it is x then you don't know if it is odd or even.
So x could be odd or even
If you add 1 to x you get x + 1
Now x + 1 could also be odd or even but it is the opposite odd/even to x
If x is 2 (even) then x + 1 is 3 (odd)
If x is 3 (odd) then x + 1 is 4 (even)
if x is infinite (all the numbers added up) then x + 1 is also infinite
but infinity is not equal to infinity + 1
Because there are different infinities, we could keep adding 1 to that number forever producing different infinities
we don't know that inifinity is odd or even but we do know that infinity + 1 is the opposite odd/even to inifinity.
Therefore there are infinities that are odd and infinties that are even but we cannot determine wether a specific inifinity is odd or even.
Consider the number 2. 2 is even. If we add 2 to iself we get 4 which is also even. Add 2 to 4 to get 6 which is also even.
We can continue adding 2 forever and all of the sums will always be even. Eventually this number will be infinite and also even. If we add 1 to that number it becomes odd.
I would say that by adding finite numbers you never reach infinity. You are not even approaching it, since your are always infinitely far away. Infinite sums need a special treatment, they can be tricky.
The set of all even numbers has infinite size. When you add an odd number to that set, you get a larger set (in the sense that the old set is a proper subset of the new set). That’s like adding 1 to its size. The size though remains the same infinity (one can define a bijection between the old and new set).
Shout out to my old math professor Fred Schaak who despised the word 'infinity' going so far as to ban its use from classrooms, preferring workarounds like "as n gets bigger" or "when n grows infinitely".
for my layman education, i came with the idea infinity is always divisible by one or by himself but it is "impossible" it can be divided by two, so it should be an odd number if the question has sense, right?
Wrong. For example, you could divide the set of all natural numbers into two sets of even and odd numbers. Of course, those two sets each have the same size (cardinality) as the original set.
