Teacher: “Well, there’s 100 million and one”
Child: “I was pretty close then!”
Twenty-four is the highest number! That's it. Let it go.
Or put another way, "what's ∞ + 1" basically invites the non-answer "that's not a well-formed question" whereas "what's ω + 1" gives you a whole intellectual thread to pull on.
It makes sense, since those are a lot less useful than the subjects that are taught, but something like number theory is incredibly approachable to a middle school student. And it can show students that math can be a lot less about memorization and a lot more about creative thinking w.r.t. proofs.
...and that the intellectual thread you are pulling on is a (more) artificial notion, constructed by set theorists for the sake of set theorists, not for the sake of counting or measuring in any real sense.
Or maybe put another way, taking the idea that infinity is just infinity makes a lot of sense when you’re primarily considering non-infinite numbers. When you’re primarily considering the concept of infinity and what you can do with it mathematically though, using systems that let you describe infinity with more nuance makes a lot of sense.
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.)
A number is represented in set theory as a set that contains all of the numbers before it. 0, 1, 2 is {}, {{}}, {{} {{}}}...
SO! If you start with a finite "a" and increment it infinite times, you still have infinity; you haven't broken out.
But if you start with Infinity, then adding anything to it gives you {Infinity}, {Infinity {Infinity}}, etc...
Transfinite addition is not commutative!
I am in no way a mathematician. My question about the definition of addition as it relates to set theory is just that; a question.
But your question actually hints at my most profound takeaway from that whole book. I think what you're saying is right, AND that foundations-of-mathematics folks spent a long intense period searching for different set theory axioms that did NOT lead to transfinite numbers. But anything anyone could come up with that included "the axiom of infinity" led to transfinites leaking in.
Which begs the question of how to think about these things. Are they "real"? Are they an oddball side effect that we shouldn't take seriously?
I think you've arrowed right to the philosophical heart of all of this.
I think we often end up at the end of logical thought processes back at the original question - how can we observe and describe a system that we are inherently a part of?
In that context the first quantity that you refer to above is nonsensical because you can't "increment infinitely many times".
Secondly, I'm not sure your construction is correct, since your Infinity+1 set cannot be a singleton (it must contain all the numbers less than Infinity).
For the first point, I went through the book long enough ago that I can't rebuild the proof here, but iirc the more rigorous idea is that you can construct a bijection between 1+ω and ω given the recipe I had above for how to represent numbers as sets, but you can't do it for ω+1, which is bijective with ω∪{ω}. The axiom of infinity declares that ω itself is a set, opening the door for transfinite numbers.
Better?
(Full disclosure: have three children and plenty of STEM in the family)
I'm not sure that's the _default_ answer, of course one might easily get that answer if at least one parent has a STEM background.
Schools don't teach about infinity to young children. A pity, really.
We don't live in the UK but our kids watch Numberblocks.
Our youngest started rattling off all kinds of number stuff which I know for sure she hasn't yet encountered in school.
Me: Wow ... how do you know that?
Her: Numberblocks!
Me: Umm ... OK!
> Numberwang theme tune
I do though. I still blow minds when I put my iPhone calculator in scientific mode. Math education is important for many reasons. But teaching it as a practical survival skill using no tools does a disservice to the student. Either it is useful as a problem solving exercise or it is a practical skill that should take advantage of tools. "Just memorize this stuff" isn't useful because it backfires into hating learning. Nothing about math makes it ideal for memorization and none of my math teachers spent any time on study skills.
Making a subject "fun" is alright, but making it entertaining (IME) makes for more productive engagement.
By age 6 to 7 they're expected to understand that addition and multiplication are commutative, while subtraction and division are not.
My experience, ultimately, was much less ... 'high-quality', let's say. When I left the Montessori school (by 3rd grade), I learned practically no math from then until after high school. First, in normal 'elementary' school (US), multiplication was still being covered in 6th grade. Then, suddenly (from my perspective), letters were being brought into the picture in 7th or 8th grade. So, in my arc, math started to not make sense, at all.
From my perspective, we had spent multiple years on multiplication and long division, which I already understood very well by the end of 2nd grade ... so, there was the period where I basically didn't learn anything, where it seemed like we'd reached the end of math or something. Or, perhaps, like there were some sort of subtleties remaining in multiplication and division. It just gave me a chance to be bored with all of it, boredom correlates heavily with mistakes with kids with attention issues (IMO), this fed into some sort of doubts about my understanding of everything etc., and then, suddenly, there was new material again starting in 7th grade. Material that was 'mechanical', and that didn't seem to have explanations I could understand.
Ultimately, I struggled along with that garbage through high school, then, after, took a course where we actually did PROOFS. Basic number theory stuff - modular arithmetic, etc. Bam, suddenly, the subject started to make sense.
Typing this out actually makes me slightly angry. I'm not sure I previously connected it all together - why I had so much trouble with math for some years ... how this 'arc' was pretty much perfectly engineered to make math a problem, for me. In any case, schooling through high school can be a really low quality experience at times - for some students, subjects, etc. The math curricula, methods of teaching, and progression I was exposed to, worked together, in some sense, to make the subject a problem for me. To do almost the opposite of what was intended - to pretty well impede learning. There's no one factor in that story I can point to and say 'here, fix this' ... no one involved in the story was actively attempting to do anything other than what they thought was best or what they were required to do, but, the net result was honestly worse - I now believe (and believed some years ago, even without quite this analysis) - than if I'd just been given some selection of math material to pick from and been allowed some sort of semi-self directed coursework.
Even better, though, if I'd simply had that course with proofs / basic number theory in, say, 8th grade ... guh, would have avoided so much pain, I'm pretty sure...