341–350 of 398 posts
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A circle is made up of an uncountably infinite set of points.
A line segment is also made up of an infinite set of points, but it's not endless.
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Not an expert but I thought it needed the 'power set' (set of all subsets) but maybe that's kinda the same as exponentiation in the end?
It is. The cardinality of the power set of a set S is 2^|S|.
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> 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. Exactly correct! This holds true of everything you can generate stepwise, even infinite sets. Cantor proved that you cannot "generate" (stepwise) all Reals between 0 and 1. Any infinite set you can generate stepwise is Countably Infinite. > I'll put Cantor and his proof in a box…
> And then infinitely later There can be no 'and then' after infinitely later. I don't see why stepwise is important but that must be the key to Cantor's proof. If he gives me 1.1 1.2 1.3 and I pair with 1 2 3, then he gives me 1.11 and I pair with 4, that seems fine as far as counting is concerned. The ordering could be entirely random, I don't see how it makes a difference. There will always be enough integers to m…
We can prove that no such 1-1 map exists between the integers (an infinite set) and the decimals in the interval [0,1] (another infinite set). The proof is by contradiction, meaning that we assume such a 1-1 map exists and prove it leads to a contradiction, therefore our assumption that the 1-1 map exists must be false.
So suppose we were able to construct a map from all decimals in [0,1], by enumerating them according to some clever rule. Let d_i be the ith digit of number i in your mapping. For each I pick another different digit d_i'. Let's construct the number with decimal representation D = . d_0' d_1' d_2' ...
Assuming we have our 1-1 map, it must be somewhere in our mapping. Let's say it's element k. By our labeling concention the kth decimal digit of D is actually d_k. However, this contradicts our method of construction of D. Therefore our assumption that there is a 1-1 map between decimals in [0,1] and the integers must be false.
It is in this sense that there are infinities of different sizes.
It’s an even number that fell on its side: ∞
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there is a difference between infinity and not a number, NaN isn't event equal to itself, in nodejs you get: > Infinity + Infinity Infinity > Infinity - Infinity NaN > Infinity == Infinity true > NaN == NaN false The fun thing: NaN to the power of zero (because technically the type of NaN is a ... number) > NaN ** 0 1 > typeof(NaN) 'number' By the same logic, NaN is also even because > NaN + NaN NaN but i think it's…
Huh. Most of those make sense to me, but infinity == infinity being true definitely feels like risky business. Algebraic limits is full of even some pretty trivial scenarios where infinity divided by a lesser-infinity turns out to be a real number— those cases where the two infinities are definitely not equal to each other.
It's been a while since I was doing this in a classroom, but I feel like those things you're thinking are nonuniform infinities could just as well be thought of as entities comprised of infinity and a (perhaps implied) coefficient. Divide out infinity to reveal the coefficient. (And the same for powers/logs, etc.)
In this model, infinities are indeed uniform (quite similar to a constant), though they are often augmented in any of infinitely many ways.
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A circle is made up of an uncountably infinite set of points.
A line segment is also made up of an infinite set of points, but it's not endless.
Earlier quoted context omitted.
Found the Brit! As an American I’d never heard the “lots of __” phrasing until I watched Numberblocks (a British show) with my kid…
I've never watched Numberblocks, but I do like to watch a good game of Numberwang!
> Numberwang theme tune