Infinite breakthrough?
Ok, anyone who knows me, knows I like to think about infinities.
You may also recall that I've insisted from time to time that there
are many different sizes of infinities.
Take the set of even numbers out to infinity for example. That's one size.
Another one is the set of odd numbers to infinity.
Those two, the even and the odd, would be one size.
Now, an infinity that is twice the size of either of those would
be the set of all the integers, both even and odd, out to infinity.
An infinity that is only one fifth the size of that would be the set of integers
you get when you count by five all the way out to infinity.
Remember all that?
Well today I discovered a thing that collapses those different sizes and
makes all those infinities the same size.
Before I telly you what it is, I want to get a little feed back from people
on what's been said so far.
What do you think?
I know Bon always says that there's only one infinity, and I kind of think
that's right, but I never had a way to demonstrate it before.
Do any of you?
You may also recall that I've insisted from time to time that there
are many different sizes of infinities.
Take the set of even numbers out to infinity for example. That's one size.
Another one is the set of odd numbers to infinity.
Those two, the even and the odd, would be one size.
Now, an infinity that is twice the size of either of those would
be the set of all the integers, both even and odd, out to infinity.
An infinity that is only one fifth the size of that would be the set of integers
you get when you count by five all the way out to infinity.
Remember all that?
Well today I discovered a thing that collapses those different sizes and
makes all those infinities the same size.
Before I telly you what it is, I want to get a little feed back from people
on what's been said so far.
What do you think?
I know Bon always says that there's only one infinity, and I kind of think
that's right, but I never had a way to demonstrate it before.
Do any of you?
posted by Bill Fleming at 5:49 PM
18 Comments:
Well think about this Polyman.
Would an infinite number of elephants be heavier than an infinite number of fleas?
If so, then that would be an another example of two different sized infinities. I gave some more in the original post.
Now...how can you collapse all those different sized infinities into one size?
On a related point, Polyman, I'm not so sure about the circle either.
In fact, using some of this same new reasoning (new to me, but not to the ancients), I can show you how, regardless of what Euclid says, the diameter of a circle is the same as its circumference!
(Hint: You, in particular, Polyman, are gonna like this, I bet.)
BTW, don't you all just love that Polyman said:
"Infinity could be lots of things!"
Isn't that just the best?
Good one Polyman!
You are pretty much right (as usual) Polyman. The thing about those different sized infinities is precisely that they go on forever (if they are to qualify as "infinities").
Therefore, you can do a thing called "correllating" to them. In the case of the even numbers, you could assign a 1 to 2, a 2 to 4, a 3 to 6, a 4 to eight...etc.
Then you have both infinities running up the same track, just with different names.
Since neither sequences ever ends, they are thus both the same size. Same way with elephants and fleas. You would first have to find out how many "fleas" and elephant weighs, and do the same thing.
Let's say it takes 200 million fleas to balance the weight of an elephant (probably a teerible guess, but you get the idea).
Then, you assign one millionth of the elephants weight to each flea and you're on your way to infinity again, running up the same neverending (elephant) line.
Only problem with your "universe getting full" idea is that it sounds like you are assuming the universe is finite, not infinite.
Students of Einstein tend to say it is, for all practical purposes, because it is limited by the speed of light. But I, personally think the jury's still out on that one.
I'm kind of liking the idea that fundamentally, everything is the same as everything else.
Which leads to the circumference/diameter equality demonstration (coming soon to a blog near you!) But first, I'll let this stuff soak in.
Happy headscratching!
I'm sorry, Polyman, I didn't even use my own phoney math! I meant to write, "Assign one 200 millionth of an elephant to each flea." There, that makes a lot more sense now, doesn't it. (Sure, Owl, whatever..)
Ok, Poly, get your compass out.
I'm going to show you how the diameter of a circle equals the circumference.
First, draw a long, horizontal straight line. Next, somewhere towards the middle of that line, find a point and set a radius short enough not to go beyond the first line at either end, and draw a circle. Thus, a portion of the line you drew first will be the diameter.
Next, without changing the radius, take the point of your compass out to the sides of the circle and draw an arc on both sides so that they intersect with the main circle at the top left and bottom left and the top right and bottom right of the main circle.
Are you with me so far?
If so, draw a line from the intersection on the circle at the upper left intersection to the lower left intersection and do the same on the right side of the circle where the arcs meet the circle.
You will now have bisected the radius on both sides and established 2 new points, essentinally dividing the diameter into four equal segments, ok?
Let me know when you're ok with the directions so far.
Maybe the Blogoddesses can put up a picture so we can see where were at here.
This thing takes a whie to demonstrate.
Yeah, I'm sure you will PM.
As with art, it's easier to do
than to describe.
What I would have you draw are two circles, inside a circle on the diagonal of the main circle, tangent at the center of the main circle and touching the outside edges of it.
It's the geometry you would need to make a yin/yang symbol. If you're still with me Polyman, notice that the circumference of the small circles are 1/2 the size of the main circle, so if you add the two small circumferences together you get the same distance as the circumference of the large circle. If you get it that far, you have the key to what I'm trying to show you.
I dont think I want to live in a world with that many fleas and elephants, and I know my dogs wouldn't.
Owl, I get what you are saying about how to do the drawing, but don't see how you're going to get to the diameter being equal to the circumference. Are we talking Euclidean geometry here - the circles are drawn in a plane?
Ok, if you follow that the two circles inside the main circle have a combined diameter equal to the outside diameter (of the large circle), then take the two small circles, and draw two smaller circles inside of them, the way you did with the large one.
(p.s. Hi Trick and Toad, nice job with Neo. That was pretty fun to watch!).
Anyway, you should now have four circles touching each other, running from the outside of the main circle to the other side, centered on the diameter. Again, if you add up the circumferences of all four of those circles, you'll get a measurement that is the same as the outside circumference of the main circle. The interesting thing is, more and more space is being created inside the circle.
Now keep doing that until you have 16 circles, then 32, etc. You'll get a row of smaller and smaller circles, all centered on the diameter, who's circumferences all keep adding up to be equal to the same as the outside diameter. Eventually, you will get to a point where the circles are so small, they are indistinguishable from the line (diameter). And their combined cirsumferences will be will be
(you guessed it) the same as the circumference of the circle. D=C (no pi tonight!)
Trick, yeah, it's straight Euclidian I guess, with a little Gödel and Zeno thrown in, huh? It's actually a kind of a sacred geometry meditation.
Yeah Toad, it's really nasty when your dog gets infested by elephants. Even worse when they get stepped on by a flea.
Ok, Now for a little Jonathan Swift, one of my faves:
On whales:
"Seamen have a custom, when they meet a whale, to fling him out an empty tub by way of amusement, to divert him from laying violent hands upon the ship. "
(Did JS know that whales have hands, like bears? Cool. They are bears, you know. We covered this on Toad's blog in a discussion about Creationism vs Evolution)
...and elephants (for Toad):
"So geographers, in Afric maps,
With savage pictures fill their gaps,
And o’er unhabitable downs
Place elephants for want of towns."
...and the all time best, for us all:
"So, naturalists observe, a flea
Has smaller fleas that on him prey;
And these have smaller still to bite ’em;
And so proceed ad infinitum."
That's a great quote about the fleas' fleas. And I really like the circle exercise as a meditation, but forgive me if I really don't think it works. It seems, looking at the diagram, as though the sum of the circumferences of the smaller circles approaches the diameter of the large circle - for each small circle, it's arc is barely distinguishable from the diameter of original. This small difference is actually compensated for by the large number of small circles. A very interesting exercise though. Much like the one about travelling half-way across a room, and then half again, and so on and never reaching the other side. It's the sort of ideas that led to the creation of calculus.
Actually Trick, it's the circumferences of the combined little circles that always exactly equal the circumference of the large (first) circle. The point is, there's no line. No real diameter, just the illusion of one when the circles get really small. It's like the fleas thing. The flaw in the reasoning is that if the diameter is drwn like that, so perhaps should the large circle's circumference be. BTW, in the days when people used to think like this there was no concept of zero (the null set) whatsoever. Whole different way of thinking.
It is indeed a meditation based on Zeno's paradoxes, which remained unsolvable until the invention of the calculus. I hear there is some dispute as to whether or not it was truly Isacc Newton who invented it (just as there is speculation that perhaps it was Einstein's
female friend who worked out relaticity theory).
I have a lot more to say about the circle thing, but it's all metaphysics. Those thoughts always sound like gobbldygook when you try to write them down. I guess that's why they call them "ineffable" huh?
Trick, it's key to get that the little circles' combined circumferences always add up to exactly the circumference of the main circle. That is the point! It looks like they equal the diameter, but they don't... they are bigger, (exactly by pi, actually.)
Do you see now?
Do you see, Polyman?
If not, come over to my house, and we'll draw, play music and talk.
Yeah, Polyman, that's the amazing part. The space goes away but the listance around the circles (combined) remains the same. It is really one of those mobuis strip thangs that makes you go "huh?"
Poly, et al. I made the art for this and sent it to her Spinnyness. Hopefully, she and the Blogoddesses will deem it worthy for presentation in the Spave. It's fun to look at.
(Hmmm... why do I think there are multiple Blogoddesses? It's a hunch... an intuition.. a channelling thing... a sign! That's it a sign! ...ok, I'm ready to write... the.uh....yes...that's it..... the Bible!!!!
Praise the Lordesses!)
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