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Cardinal numbers (size) and ordinal numbers (ordering) are both numbers. The numbers we're familiar with represent both concepts, sometimes simultaneously. I really don't think that block quoting ChatGPT is a good contribution.
It's the first time I've done it. I agree with you. But didn't know until I tried!
The way I would explain it to a 6 year old would be like this: Infinity isn't a number really, it's a concept, like the word many or the word few. If someone says they have many of something, you don't think is that odd or even you just know they have a lot of it. Infinity is kind of like that, it explains the idea of things going on forever, not an exact quantity of things like the number 10 or 11.
If I say someone had many of something, then I know for certain that they must have either an even amount or an odd amount. Same goes for few.
The way I would explain it to a 6 year old would be like this: Infinity isn't a number really, it's a concept, like the word many or the word few. If someone says they have many of something, you don't think is that odd or even you just know they have a lot of it. Infinity is kind of like that, it explains the idea of things going on forever, not an exact quantity of things like the number 10 or 11.
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OK, but then where do you go from there? There are infinity numbers. Then what?
Right. The problem with teaching infinity by starting with cardinal numbers is that it's either too trivial or too hard. You can establish that several other sets of numbers are identified by the same infinity but there's not much you can do. An old HN comment echoes the same sentiment: https://news.ycombinator.com/item?id=17677010 If we really are teaching kids, teach ordinals not cardinals.
Well, you can introduce them to the diagonal argument and the idea of a one-to-one correspondence. That's nothing to sneeze at.
But I think the real trick here is to teach them that numbers can stand for different kinds of ideas, and in particular, they can stand for "how many" or "what position", and that these are different. I would start, not with infinity, but with negative numbers. You can't have "one less than zero" because you can't take away anything from zero. That is the definition of zero. But you can have "the thing before zero", or, to be more precise, "the thing before the zeroth thing (where the zeroth thing is the thing before the first thing)", which we call -1.
Likewise you can't have "one more than infinity" because that's just infinity. That's the definition of infinity. But you can have "the thing after infinity" (or, to be more precise, "the thing after all the things that are the nth thing for all finite values of n", which we call ω.
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Never. Never in school? It's totally normal in the US, it's the reason as many people know that π starts "3.14" as do.
Yes never, not in school, not in analysis, and certainly not in numerical analysis. You've proved my point. It's either π or 3.14. Except that the latter is a rational number :)
Weird, I just assumed that was normal in most education systems. I don't know how you'd get a sense of the rough scale of various common irrationals, without having some idea what they look like when represented in decimal notation. Such representations are normal starting not later than when we start seriously working with circles, in US school, and never really stop coming after that. Estimation exercises lean heavily on having some idea of the decimal representation.
> You've proved my point. It's either π or 3.14. Except that the latter is a rational number :)
Never claimed π is 3.14, so no, I didn't at all prove your point. I wrote that it's very well known that it starts that way. When a normal person says "decimal number" they mean to include π, because any usefully-precise decimal representation of it's going to involve a decimal point. At least in the US, they saw it represented "3.14..." or "3.1459..." or whatever, many, many times in school. It's obviously, to a non-mathematician, a "decimal number". They mean "the real numbers" (or, perhaps, depending on context, exclusively the parts of the reals that aren't whole integers), except that name is harder to remember than the incorrect (but more common and intuitive) "decimal numbers".
The way I would explain it to a 6 year old would be like this: Infinity isn't a number really, it's a concept, like the word many or the word few. If someone says they have many of something, you don't think is that odd or even you just know they have a lot of it. Infinity is kind of like that, it explains the idea of things going on forever, not an exact quantity of things like the number 10 or 11.
This is true until you introduce transfinite numbers.
The problem with transfinite is that you lose commutatively. 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 =…
I find this concept perplexing. To me this implies that "infinity" has a value. How can you add 1 to a thing that by definition has no value?
What do I mean? 1/d retains the sign of d across all finite and infinitesimal values.
1/ε = +∞, 1/-ε = -∞
So what about 1/0? Neutral infinity.
I’ve always thought infinity is not a number but a concept that represents something which is ever increasing.