Finite bugs have Aleph Naught bugs
Upon their backs to bite 'em
Aleph Naught bugs have Aleph One bugs
And so on, ad infinitum...
Aleph-naught bottles of beer on the wall, aleph-naught bottles of beer.
Take one down, pass it around.
Aleph naught bottles of beer on the wall.
I enjoyed watching this but it left me unenlightened (for which I blame only my own inadequacy.)
How is the set of all real numbers the smallest possible infinity? Isn’t the set of, say, all even real numbers also infinite, but half the size of that?
Something, something, -1/12?
My favorite explanation of this idea is from James Grime on Numberphile.
This may help to explain.
The natural or counting numbers make up the most limited set of numbers used in mathematics. It contains only the whole positive numbers (no zero).
The integers contain only whole positives and negatives with zero.
Rational numbers are all the fractions.
(There are others as well.)
So the idea being presented is this:
Even if all numbers go to the same forever place, the sets aren't subdivided equally. The different sets represent different opportunities to count the values. It makes no logical sense to say that something with only a positive set is the same size as something with both a positive and negative set, or divisions of the value. Even so, infinity requires that we accept this as true - or accept that there are "different sizes" of infinity which also makes no sense.
Also, we have to accept that infinity can never surpassed.
It exists only as concept, not as a number.
The smallest is really the set of natural numbers rather than real numbers. It's true the even numbers are a proper subset made by leaving out every second number. At the same time, though, halving the even numbers gives you all the natural numbers back:
0 → 0
2 → 1
4 → 2
6 → 3
That mapping means there can't actually be more natural numbers than there are even numbers. It may seem weird that the set didn't get any smaller by removing even half the elements, but that's how infinite sets work; in fact that's often used to define them.
If it would help to have more examples, looking up Hilbert's Grand Hotel might be a good place to start - this blog post is one explanation I found.
This on the other hand is going too far. Infinities are not real numbers, but they do for instance show up as cardinal numbers, which are no more or less conceptual than any of the other mathematical objects you might look at. They describe the sizes of sets.
And these can be surpassed; although there are an infinite number of rational numbers, there are more real numbers. The essence of the proof is shown in this video. This is vaguely related to a more general case, which is that for any infinite set, the set of its subsets - called the power set - will be a larger infinity.
(Also, whether the natural numbers include zero depends very much on who you ask. In set theory, zero is often thrown in so that they're all the finite cardinal numbers.)
Why is the cardinality of the set of curves Aleph 2, and not Aleph one, or aleph-null, or some other infinity? And what are Beth numbers?
Beth numbers are the infinities you get by taking power sets:
beth-0 is the number of natural numbers
beth-1 is the number of subsets you can get from something of size beth-0
beth-2 is the number of subsets you can get from something of size beth-1
In most cases when people talk about aleph-0, aleph-1, aleph-2, and so on they mean exactly the same thing. Formally, though, those are defined to be the first, second, and third infinity, etc. Are those really all the same as the beth numbers? It depends on what axioms you accept for your set theory.
How many curves are there? It depends by how you define them. There are obviously at least as many as there are points. There are just as many points on a plane or in space as there are real numbers. It's not a proof, but one way to sort of picture this is by interleaving the decimals from the coordinates:
(0.abcde..., 0.αβγδε...) → 0.aαbβcγdδeε...
So there are at least beth-1, and in fact if you look at continuous curves, that's exactly how many there are.
The best way I can think of to explain it is to imagine someone tracing each curve. The curve then becomes a function of time: at t=0 the pen was at this point, at t=1 it was at this point, at t=1/2 it was at this point, and so on.
For a continuous curve, all you have to worry about is where the pen was when t is a rational number, and then you can get all the others as limits. We saw each of these points can be expressed as a real number with beth-0 digits. The number of rational numbers is also beth-0.
If you look at the hotel example I linked, it shows how the hotel had room for infinitely many coaches with infinitely many guests. You can use basically the same approach to interleave these infinitely many real numbers, each with infinitely many digits, into a single string of infinitely many digits.
So we now we have a way to turn every tracing into a single real number, which shows there aren't more tracings than there are real numbers. And every continuous curve can be traced in at least one way - more, actually, since you could move the pen with different speeds - so there aren't more of them. The number of continuous curves is then beth-1.
On the other hand, you could mean curve without such strict requirements. I'm not sure it really counts, but the most general option would be to let people add or remove any points they want. That's basically talking about subsets of the plane, which make up its power set, whose size is called beth-2 by definition.
The proof that this is larger than the number of points in the plane works a lot like the one in the video; if somebody claims to have a way to match curves with points:
point 0.00000... → some subset they associated with 0.00000....
point 0.50000... → some subset they associated with 0.50000...
point 0.31415... → some subset they associated with 0.31415...
Then you can make a new curve by picking:
point 0.00000... only if it isn't in the subset associated with 0.00000...
point 0.50000... only if it isn't in the subset associated with 0.50000...
point 0.31415... only if it isn't in the subset associated with 0.31415...
And we know it isn't covered by their matching. For instance if they said that was actually the subset associated with 0.42000..., you know they're mistaken, because your subset differs in whether it includes the point 0.42000... So you know there are more subsets than points.
That ended up really long. I hope it isn't too confusing; it's the best I could do to answer your questions.
As @chenille says, it's because you can create a direct one-to-one mapping between even integers and all integers. For every integer you give me, I can give you one unique even number that maps to your integer and no other integer.
Therefore, there are the same number of even numbers and integers.
The same is not true for real numbers (which includes never-ending decimals). There is no mapping you can come up with that gives you a one-to-one correspondence between arbitrary real numbers and integers. For any mapping you try to come up with, I can always give you a real number that won't fit your mapping.
That is why there are more real numbers than natural numbers, even though both are "infinite."
Eh? How can you quote me and then in the very sentence ignore what I said - that you quoted.
Infinity itself is not countable. Aleph-null, which is a cardinal number, is. Aleph-null describes the set of all the natural numbers up to infinity, and is the smallest countable measured set. Yes, it is an "infinite cardinal", but you cannot just blankly say "infinity comes in different sizes". What comes in different sizes (to our perception and reason) is different sets of numbers breaking down infinity.
Can you create a one to one mapping between the reals and the complex numbers?
Rap rap rappers
map map mappers
This is all just an infinite caper!
Yes! Complex numbers can be given as a pair of coordinates, their real and imaginary components, so it works the same as mapping between a plane and the real number line.
I mentioned interleaving decimals because it’s fairly intuitive, and can sort of be made to work. In truth it’s trickier than that, though, since 0.9999… = 1.0000… and so decimals aren’t always unique. To actually define a map in practice, it’s easier to start with space-filling curves.
Well, I did because to me saying infinities come in different sizes means the same thing as saying there are different infinite cardinalities. Especially given the context is a thread about their relative sizes and a question about the same, why one set wasn’t the “smallest infinity” when another seemed like it would be smaller still.
I now take it you meant something different by what you were saying, but it escaped me. When you say “different sets of numbers breaking down infinity”, for instance, it really isn’t clear to me what singular infinity you mean and how you are breaking it down.
In math, "infinity" is not the same as "aleph-null". They're totally different things. That's why Vi changed terms when she started talking about a countable set and explaining how aleph-null is the smallest aleph set, but since we can't count them (they're all infinite) we can't know what the next step would be (we can't just say "aleph-one" because we'd miss something).
You can't interchange the terms to make infinity suddenly be a cardinal number. Infinity is not aleph-null anymore than it is pi. It's a concept of something traveling on without end.
Here's one way to use "infinity" correctly:
In geometry, a "line" goes on forever - no end. It's infinite. You can't measure where it will end, or where it began. You don't know where it exists in space. We use points on lines to help us define them.
If you select an end in space for one end of the line, and then find a second point on the line that it can continue past - it becomes a "ray". You can only measure the origin and direction of a ray.
If instead, you define two end points, it's now a "line segment" and is fully-measurable all on its own.
That's one definition of the term infinity, yes, although I'm still not sure what you mean by sets of numbers breaking it down.
However, people do legitimately also use the term infinities to refer to infinite cardinals and ordinals, particularly in contexts like set theory. I missed that you weren't, and so we have been talking past one another.
Brilliant exposition, as usual.
But I have a longstanding question about the diagonal argument which I'd appreciate getting an answer to, if any math folk have the time to weigh in.
What troubles me about the diagonal argument is that first it generates a list of infinite decimal numbers, then it generates new numbers not on that list by performing a separate operation.
This seems like cheating. It's taking one set of numbers produced one way, then turning around and producing a new set of numbers with a different method. Isn't the fact that these new numbers are not on the original list a trivial result of the operation?
To put it figuratively: say I posit just one number, 4. Then I add 1 to it and get 5. Then I turn around and say, "See — nowhere was there a 5 associated with that original 4." Am I really demonstrating something about that original 4, or am I doing something to it -- adding 1?
Or again: just say I generate an infinite number list, and then, using the diagonal method, replace every "diagonal digit" with an emoticon, and say, "See — infinite list of emoticons not on the original list." Have I not done something to the original list that was not done to generate it?
The diagonal argument seems to take one list of numbers, then use that list as the input domain for a second operation. Is it really the case, then, that it demonstrates something about the properties of the original list — or has it simply done something to the original list?
It demonstrates something about the properties of the original list, namely that it's not comprehensive. We can look at it and find a real number that was not on it, so we know it's not a list of all the real numbers in the interval.
And since this applies to any possible list, that shows there's no way to match all the real numbers up with natural numbers. This is really a common way to prove something isn't possible: first you suppose there is a solution, and then prove it couldn't actually work.