121–130 of 398 posts
>To explain the idea to a child, I would focus on the principal idea: whether finite or infinite, a number is even when it can be divided into pairs. For finite sets, this is the same as the ability to divide the set into two sets of equal size, since one may consider the first element of each pair and the second element of each pair. The answer this quote came from is amazingly obtuse, but it does make me think that…
Infinities aren’t comparable for equality… are they?
Earlier quoted context omitted.
No it isn't. If you ask a child what comes after infinity, "Infinity + 1" is pretty much the default answer. Any kid who knows multiplication knows "Infinity + Infinity" is the same as "Infinity Times Two". The answer of "Infinity TIMES Infinity" is also popular for kids to say when they know a number bigger than their friend (who just proclaimed infinity is the largest number).
> Any kid who knows multiplication knows "Infinity + Infinity" is the same as "Infinity Times Two". Or is it "Two Times Infinity"? (Hint: It isn't, because "Two Times Infinity" = "Infinity", while "Infinity Times Two" = "Infinity + Infinity". Not sure every kid knows that.)
Earlier quoted context omitted.
My understanding is that this is true because there are infinite decimals between every decimal, infinitely. For example, there is infinity between 0.1 and 0.2, and infinity between 0.1 and 0.11, etc. i.e. infinite sets of infinity rather than one set of infinity. In the end it's all infinity, but their sets have higher cardinality described in Aleph terms ... (or something) https://en.m.wikipedia.org/wiki/Aleph_numb…
You can uniquely map all rationals onto the natural numbers, thus they are of the same quantity. That doesn't work for all real numbers thou.
Well, I asked ChatGPT: > Infinity is not a number, odd or even, but rather a concept or a mathematical idea that represents an unbounded or limitless quantity. Infinity is not a real number that can be used in ordinary arithmetic operations, but it is used to describe a quantity that is larger than any finite number. Therefore, the concept of odd or even does not apply to infinity.
In my experience with children, one of the easiest-to-grasp concepts of infinity is provided by the transfinite ordinals, since it can be viewed as a continuation of the usual counting manner of children, but proceeding into the transfinite: 1,2,3,⋯,ω,ω+1,ω+2,⋯,ω+ω=ω⋅2,ω⋅2+1,⋯,ω⋅3,⋯,ω2,ω2+1,⋯,ω2+ω,⋯⋯ Presumably this person has no experience with 6 year olds? This explanation is horrendous haha
Also my first thought. I assume he's writing this to other people who know what transfinite ordinals are (I don't understand the explanation) and would frame it differently with an actual kid. Even in context it's a hilarious quote though, I think it's possible this was on purpose
Explain everything like you're talking to a fifth grader. If you can't, you don't understand your problem fully.
He spend much of his professorship agonizing about how to fit all of physics into a freshman lecture. When he couldn't, he knew we needed to think more about that area.
It isn't clear to me infinity is a number in the first place. Reification and category mistakes are as much a danger in math as anywhere.
So there seems like a glaring hole in the answer, but maybe I'm missing something. Because: > It is easy to prove from this definition by transfinite recursion that the ordinals come in an alternating even/odd pattern, and that every limit ordinal (and hence every infinite cardinal) is even. Sure, if we use the natural numbers and start at 1, then we can group: [1, 2], [3, 4], [5, 6], ... and prove infinity is even.…
It's only even or odd with respect to other infinities which the cardinal numbers can count based on the presence of a bijection or not. It's a kind of relative parity.
I want to know if there are more decimal numbers between 0 and 1 than there are integers between 0 and infinity.
There are, and it turns out that this is a significant mathematical concept. The integers between 0 and infinity are defined as "countably infinite". Other infinities are considered countably infinite, or the "same" infinity, if and only if you can arrange it in a list such that each item in the list pairs to an integer in our 0 to infinity list. So the set of even numbers is countably infinite because for every i th…