141–150 of 398 posts
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.
Sometimes I wonder if there's a better math language waiting to be invented that eschews the non-discrete.
Earlier quoted context omitted.
> Everybody experienced writing irrational numbers using decimal notation in school, To be pedantic, we experienced writing approximations of these numbers in decimal arithmetic.
Do you actually think I meant otherwise? Like, actually ? Do you think anyone on HN was confused about it? Did you really think that?
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.…
I would weaken the definition of even/odd to say that a set is even if /there exists/ a way to pair things off, and odd if /there is no way/ to pair things off (ie, not even). So the countable numbers would be even.
And you then you could retort with sets of size four, and I could use five, and then we can argue about whether we'll end up at the limit with more odd sets or even sets, and now we're arguing in circles. Reductio ad absurdum.
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For a child, I think the simplest kind of infinity to explain is the cardinality of integers—"how many numbers there are."
OK, but then where do you go from there? There are infinity numbers. Then what?
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.
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
In case anyone is curious, this person has experience teaching children mathematics. For example, on his blog, we have
http://jdh.hamkins.org/math-for-six-year-olds/ http://jdh.hamkins.org/math-for-seven-year-olds-graph-colori... http://jdh.hamkins.org/math-for-eight-year-olds/ http://jdh.hamkins.org/math-for-nine-year-olds-fold-punch-cu...
The most recent post in his category "Math for Kids" is in fact teaching how to count ordinals up to omega-squared: http://jdh.hamkins.org/counting-to-infinity-poster/
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Do you actually think I meant otherwise? Like, actually ? Do you think anyone on HN was confused about it? Did you really think that?
Yes, I thought you meant otherwise. Yes, I was confused about it. Yes, I really thought that. Truly.
[EDIT] Look, I don't mean to be a dick, performative misreading and plainly-unnecessary "correction" are just two of my least-favorite types of HN post. I probably should have just downvoted the original performative misreading (not yours, the one up-thread) and not Assumed Good Faith that the original poster genuinely doesn't understand what every non-math-nerd means when they say or write "decimal number" (it's the ones you write with a decimal. It's... so very simple, that's why non-math-nerds use that and not "real number", the definition of which they've long since forgotten. "Well but you can't actually represent irrationals them entirely in decimal notation" great, wonderful, has zero bearing on what people mean by it).
Earlier quoted context omitted.
I would weaken the definition of even/odd to say that a set is even if /there exists/ a way to pair things off, and odd if /there is no way/ to pair things off (ie, not even). So the countable numbers would be even.
But pairity is just a question of sorting the countable numbers into sets of size two, and the more general form even of that is sorting into sets of size N. It's just as easy to say that the countable numbers are odd if there exists a way to sort them into sets of size three. So I'd argue the countable numbers are odd. And you then you could retort with sets of size four, and I could use five, and then we can argue…
Earlier quoted context omitted.
For a child, I think the simplest kind of infinity to explain is the cardinality of integers—"how many numbers there are."
OK, but then where do you go from there? There are infinity numbers. Then what?