Math was never my strong suit, but I think there’s an easier proof. There are an infinite number of real numbers between 0 and 1 with which to compare to the naturals. But then look at the reals between 1 and 2 and you now have another set of infinite numbers: infinity times 2. Then between 2 and 3, more, and so on. Math whizzes are probably chortling in their cereal as they read this, calling me infinitely stupid for forgetting the obvious effects of Blugglebots’ Conundrum or whatever. Oh, well… I’ll just have to live with the shame of not being nerdy enough for them.
Strange but True: Infinity Comes in Different Sizes
In the 1995 Pixar film Toy Story, the gung ho space action figure Buzz Lightyear tirelessly incants his catchphrase: “To infinity … and beyond!” The joke, of course, is rooted in the perfectly reasonable assumption that infinity is the unsurpassable absolute—that there is no beyond.
That assumption, however, is not entirely sound. As German mathematician Georg Cantor demonstrated in the late 19th century, there exists a variety of infinities—and some are simply larger than others.
Wouldn’t the amount of infinite number of reals between 1 and 2 be the same as between 0 and 1? I guess the infinite number of reals between 0 and 2 compared to between 0 and 1 would be infinity times 2……
dear god…….
my head hurts after reading that……..
It’s been 25 years or so, but I remember the short book “Asimov on Numbers” having a good discussion on the transfinite numbers.
Don’t forget the imaginary infinite numbers.
#1, it does not work that way. infinity + infinity = infinity and infinity x infinity = infinity but infinity ^ infinity = a completely different kind of infinity.
The number of real number = number of countable numbers ^ number of countable numbers
But there are even bigger infinities. Everyone knows that the shortest route from point a to point b is a straight line.
But it can be proven that the number of routes you can take between point a and point b exceeds the number of real numbers. In fact it is number of real numbers ^ number of countable numbers (“unreal” numbers?)
The next level of infinity exists mathematically, but to demonstrate it requires infinite dimensions.
Okay this is retarded. Infinite means infinite there is not less or more. The only time there is less is when the number or quantity really isn’t infinite and the word is being misused.
my head hurts too.
There can not be more infinities than one; for one of them would limit the other. –Sir W. Raleigh.
[1913 Webster]
Is math “real” or just a construct?
If its a construct, then infinity is just a definition, or how you think about it.
Two of the several types of “infinities” are very easy to understand:
1. Image an infinitely long series of natural numbers, such as (1,2,3,4,….). Pick any natural number and it will be in the series. So this countable infinite series catches everything.
2. Image an infinitely long series of 1’s and 0’s, such as 110101000101000110010…. Again you are allowed to pick an infinite amount of these series. Question: Have you got all possible series?
Answer no: Just write them all in lines. If you walk down the diagonal you will have a another infinitely long list if you just flip each entry in this list it won’t be in your infinite collection. Why? Well it differs from the first by the first entry (you flipped that, right?), it differs from the second by its second entry (again flipped) and so forth…
So infinity != everything. Pretty cool, eh?
But let’s not forget, NUMBER DO NOT EXIST !
Uncle Dave,
This is definitely headache material. I noticed most posts don’t know anything about math either.
Ya, infinity is kind-of a “self-identity” value for most operations (like “1 * x” or “0 + x”, or multiplying by a square matrix with 1’s from top-left to bottom-right , 0’s elsewhere). Not sure about ( inf ^ -1 ) – that might give you 1 like most everything else, or perhaps it’s still inf (Ask a mathematician).
(1) inf + inf = inf
:. (2) 2 x inf = inf
There are inf real numbers between 0 and 1. There are inf real numbers between 1 and 2. Therefore there are 2 x inf numbers between 0 and 2. (1) :- There are inf numbers between 0 and 2.
(3 – generalising) x * inf = inf (for all x where x > 0)
‘He assumes ‘ everything after that phrase is gobbdy gook.
Infinity is good for lines. Shoot a satellite into space and it will keep moving forever (infinity). Circles define finiteness. With circles, infinity is trapped. So when a satellite gets near a 3-dimensional circle like a planet it is pulled in by its gravitational force. Linearity is fine and infinite. But welcome to non-infinity brought to you by the Circle.
#15, MoL
but there’s still an infinite amount of locations that the Satellite can be at on the circle. Not to count the infinite amount of locations that it is not inside the circle.
However, the whole stuff about infinity x2 is bull.
Yes, there’s an infinite amount of real numbers between 0 and 1
There’s another infinite amount of real numbers between 1 and 2
and there’s an infinite amount of infinite amounts of real numbers between any given pair of real numbers. That’s just the nature of the beast.
However, there’s also an infinite amount of natural numbers.
So really, there’s inf^inf real numbers just along the x-axis. That’s obviously bull, as infinity is not a number per se. It’s a concept. You can use it in some calculations, but very obviously, for most basic calculations, it doesn’t make a whole lot of sense.
inf+1 is still inf
inf+2 is still inf
inf/2 is still inf
the only interesting ones are those that include inf’s natural counterpart, zero.
inf*0=undefined
n/inf=0
so, inf is helpful in mathematical ways, but not really when you’re trying to look at countable things 🙂
An infinite amount of apples, transistors or pennies would be quite a problem.
pj
Once you push the boundaries of any science it starts sounding like religion anyway.
I believe Buzz Lightyear didn’t originate ‘To infinity and beyond’ as a phrase. It appeared sometime during the late ’50s or early ’60s in adverts for Alpla Cameras in National Geographic.
It’s years since I last saw the advert but given enough time I could probably find and scan one
There are provable sets and degrees of infinity, denoted by the symbol aleph from the Jewish alphabet. Aleph 0, Aleph 1, ….
Even Galileo (one paradox named for him) freaked out considering that the set of all even numbers was infinite, but didn’t include numbers like “17”. So yes, there are varying degrees of “infiniteness”.
An introduction to degrees of infinity is here:
http://www.mathacademy.com/pr/minitext/infinity/
“The article provides only a bare introduction to the topic of infinity, and there remain many beautiful ideas for the interested to discover: infinitesimal numbers, surreal and hyperreal numbers, and transfinite ordinals, to name just a few.” Just google “aleph infinity” and explore.
Everyone with an interest in this subject should check out “White Light”, Rudy Rucker. The protagonist is a mathematician whose thinking about Cantor during one of his many Out-of-Body (or maybe lucid dreaming) experiences. He’s end’s up traveling to heaven, which is built on the concept of multiple infinities. I must of read the book 20 years ago, and I still think about every once in a while.
Infinity is the reason I got disenchanted with Maths.
It’s intellectual dishonesty – you can’t treat something as a concept and a value. The problem of infinity shows that Math isn’t complete, it needs additional work and expression… but instead people go “on well, no, if we define this as a limit of this… etc.. it works!”
Yes, and if I define 2 as 2.5, 2 + 2 = 5, but that doesn’t make it good maths.
Call it a philosophical difference in what I wanted from maths – completeness, not cheating.
This kind of thing has always caused me brain pain.
Not all infinities are simply mathematical constructs though – one of my dogs has an infinite suppy of gas…
Everyone with an interest in this subject should check out “White Light”, Rudy Rucker.
A great book. It’s even greater if you’re doing a lot of drugs. I suggest some owsley.
He also wrote another non-fiction book on infinity which I have not read all the way through yet, despite multiple attempts. That single book made me seriously consider going back and getting another undergrad degree in mathematics.
My opinion is that humans are not truly capable of understanding the concept of infinity.
#16
However, the whole stuff about infinity x2 is bull.
…
the only interesting ones are those that include inf’s natural counterpart, zero.
inf*0=undefined
n/inf=0
Ah, but you don’t even have to go beyond calculus to see that all infinities are not equal. For example, why doesn’t the sum of 1/n converge as n approaches infinity, while the sum of 1/n^2 does? Infinity is infinity right?
As to the original idea, it seems quite intuitive that there are more real numbers than integers for the simple fact that for every single step you take on the number like counting integers, you are adding an infinite number of reals. In fact, no matter where you stop in your counting, you will always have counted infinitely more reals than integers up to that point.
Ah yes, the occasional flashback from the 60s comes calling. Here it is boys and girls and you won’t need to ingest 750 mics of Sandox first to go on this trip.
Take the concept of infinity. If it helps, by all means add the twist of greater or lesser infinity numbers or map space onto a Kline bottle… whatever gets you sufficiently bent about synaptically speaking.
OK, big deep breath, now, with I N F I N I T Y staring back from the void, slide the word ‘eternity’ into ye olde cerebellum. Yes, that’s it… juxtapose those two back and forth. Infinity – Eternity and Space – Time.
Infinity = Eternity, and Bob’s your uncle. 🙂
The thorazine is down the hall to the left.
@22, I think defining 2 as 2.5, then saying that 2 + 2 is 5 is about the most fundamental level of mathematics there is.
Quick tutorial. If you can write out the list, that is one type of infinity. For example, 1, -1, 2, -2, 3, -3, etc. If you can’t then you have something that’s larger than infinity.
And this is useful to who?
Infinity is not a number. Infinity is everything. Infinity goes in every direction. Since each infinity is infinite, one cannot be any larger than the other. An infinite amount of infinities is still infinite. They all go on forever. An asymptote approaches a point but never quite reaches it. That too is infinity. Infinity has no size. It’s stupid to say 2 x infinity. It’s the same thing. Everything becomes infinite in some way.
As an example, if you have all of a particular thing that exists in the universe, two times that is not more, because you have it all. Infinity is not a number, it’s a concept.
Now my hair hurts.