Earlier quoted context omitted.
Nobody ever finished writing an irrational number in school.
sqrt(2)
… Using decimal notation.
Goodness this is like talking to GPT at times.
Is infinity an odd or even number? (2011)
231–240 of 398 posts
Re: Is infinity an odd or even number? (2011)
#232Earlier quoted context omitted.
It follows from the way addition is defined on top of set theory. "a + b" is implemented as "increment a (the set that represents a) b times". 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, the…
Is addition defined _by_ set theory, or is set theory one way of defining addition? If it's the later, then there could be other ways of defining addition that don't have the same results for infinity (because our math system doesn't really "work" for infinity, or 0, depending on the circumstances). I am in no way a mathematician. My question about the definition of addition as it relates to set theory is just that;…
It's the latter; I'm also not a mathematician, just a guy who worked through Halmos's "Naive Set Theory" in intense detail...
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.
Re: Is infinity an odd or even number? (2011)
#233Earlier quoted context omitted.
I taught my kid that the way to think of infinity is that it's like hugs, there's always one more, unlike candy, which is limited and can be counted, infinity cannot be counted.
Hmm, that could potentially cause confusion later. There are 'countable' and 'uncountable' forms of infinity / infinite sets. A countably infinite set could be 'counted' (i.e., you could sit around labeling elements using the 'natural' or 'counting' numbers) in the sense that we might count candy. The issue for a human being is that you'd run out of time but not elements to count, at least, proceeding in the sense on…
> Hmm, that could potentially cause confusion later [...]
(Q: Do you have kids?)
Our experience is that pretty much everything parents tell young children could potentially cause confusion later.
In no particular order: Father Christmas aka Santa Claus, The Tooth Fairy, Where Babies Come From... it's a long list, our eldest is 13 and we're not done yet.
Re: Is infinity an odd or even number? (2011)
#234Definitely even, no chance half of infinity is a decimal.
Re: Is infinity an odd or even number? (2011)
#235I 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…
"Countably infinite" makes zero sense to me.
Whatever method you use to generate your decimals, you can just slap an integer on each step of the way. You'll never run out of integers.
I'll put Cantor and his proof in a box, tell him to give me his fancy decimals quick as he can, and I can match each one with an integer no problem.
And pairing one infinite list with another infinite list doesn't make either one any more countable, because however high you count, they keep on going.
Re: Is infinity an odd or even number? (2011)
#236In 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
Transfinite ordinals also known as hyperreals should really be taught in school as they make many parts of math easier: algebraic definition of derivatives (including algebraic derivative of step functions without dirac 'density') and yes: natural addition and multiplication. https://en.wikipedia.org/wiki/Hyperreal_number
> Transfinite ordinals also known as hyperreals should really be taught in school as they make many parts of math easier: algebraic definition of derivatives
Q: What proportion of children study maths long enough to understand derivatives?
Re: Is infinity an odd or even number? (2011)
#237Earlier quoted context omitted.
OK, but then where do you go from there? There are infinity numbers. Then what?
Then you explain the properties of that infinity, like how infinity + infinity = infinity, or (as per OP) that it's even.
But if infinity is even then infinity + 1 must be odd. But infinity +1 = infinity, so infinity must be odd as well as even.
Re: Is infinity an odd or even number? (2011)
#238I guess I would have thought the answer to this question would be "yes"?
Re: Is infinity an odd or even number? (2011)
#239-- the first comment here is the correct one, sorry for posting it without reading the comments. Infinity is not a number.
Re: Is infinity an odd or even number? (2011)
#240I want to know if there are more decimal numbers between 0 and 1 than there are integers between 0 and infinity.
If you thought of this question from no real math training then that's pretty interesting. You should have been a mathematician. Your question is one of the most important and concisely stated questions about infinity that you can ask!
I was thinking if Pi never repeats itself and infinity of integers can only go up then it seems to make sense that there are more decimals between any two numbers than infinite integers. I can't describe the thought process behind it just seems intuitive.