well, I doubt there’s going to be a clear answer on this, but…
which is bigger:
infinity with an infinite number of factorial signs after it,
or…
infinity to the infinitieth power an infinte mumber of times?
They are the same size.
Approximately the same size. But only approximately. I don’t have time right now to do the exact calculation
My vote goes to the first one! I just love something with factorials in it! I’ve always loved factorials!
But powers has POWER in it!
infinity can’t be multiplied. The question is meaningless.
is infinity + 2 greater than infinity + 1? Seems like it would be, but … if infinity is infinite… hmm
All infinities are equal. But some infinities are more equal than others.
Check a book called “1,2,3—Infinity” Can’t remember the author, but it was written in about 1949. An interesting point was some infinities have greater “strength” such as the number of points on a line vs: the distance between the points. The book was an interesting read even though I’m no mathemetician.
imagin you got a hotel with infinite number of rooms, and every room has a person in it. now imagin a new person shows up… they can give him a room by haveing everyone move down one room (so the guy in room 001 moves to 002 the guy in room 2333 moves to room 2334, ect) the building didn’t need to get any bigger in that case so adding must not make infinity bigger.
what if you multiplyed? well if you had a second hotel with an infinite number of people and it suddenly closed its buisness so that everyone has to go to the first hotel. well just haveing everyone move down an infinite number of rooms wouldn’t work, no one would ever be in a room again… so… it might seem impossible… but its not! you just have everyone move to a room thats two times the room number that they are in (001 moves to 002, 002 moves to 004, 003 moves to 006) and you get an infinite number of spaces. and once again the hotel didn’t get even a little bit bigger than it used to be, so multiplying can’t make an infinity bigger either!
It depends on what system of infinities you are talking about. If you are talking about cardinal numbers, then adding a finite number to an infinite number does not change it. For ordinal numbers, alpha + 2 is greater than alpha + 1 for all alpha.
The sum and the product of two cardinals is the larger of the two if at least one of them is infinite. For ordinals this is not necessarily the case.
Well, you can win an argument with “I got you infinity times plus one!” but if you try pushing it to “plus two”, you’ll be soundly and justly ridiculed by everyone else in the sandbox.
In this limited sense, the former quantity is definitely of lesser value than the latter.
“infinity” is a vague concept, and should be avoided. The factorial operator is only defined on finite cardinals (AFAIK), but if we extend it in the obvious way, aleph[sub]0[/sub] with an infinite number of factorial operators is still aleph[sub]0[/sub]. The latter is aleph[sub]0[/sub] raised to the aleph[sub]0[/sub] power raised to some power less than aleph[sub]1[/sub] (assuming GCH), so it’s equal to aleph[sub]1[/sub]. The latter is bigger.
For those who are curious, aleph[sub]0[/sub] is the number of integers, and aleph[sub]1[/sub] is the number of real numbers. aleph[sub]1[/sub] is larger than aleph[sub]0[/sub].
Not quite. c is the number of real numbers. It is indeed larger than aleph[sub]0[/sub]. This can be easily proved by contradiction. If the real numbers between 0 and 1 are countable, they can be listed in order. Each one has a decimal expansion, most of which are infinitely long.
Create a new real number as follows: Select its first digit different from the first digit of the first number on the list. Select the second digit as different from the second digit of the second number on the list, etc. As a result, you will have created another real number, which is unequal to each number on your list, which was assumed to be complete. That contradiction shows that the reals cannot be listed like the integers. In other words, it shows that c is larger than aleph[sub]0[/sub] (Also, there needs to be a small adjustment to this procedure to account for numbers that can be expressed two different ways, like .5 and .499999…)
aleph[sub]1[/sub] is smallest order of infinity larger than aleph[sub]0[/sub]. The continuum hypothesis is the hypothesis that aleph[sub]1[/sub] = c. However, IIRC the continuum hypothesis is unproved.
The continuum hypothesis is unprovable from ZFC. However, I earlier noted that I assumed GCH for this discussion, just to simplify things.
FWIW, these days most set theorists tend to think the continuum hypothesis should be false; c is quite possibly much much larger than aleph-1.
But wait, something can be both bounded and infinite, right? So, the amount numbers between 1 and 2 is the same as 1 and 1,000 because they both have an infinite amount.
Yes, the numbers between 1 and 2 and those between 1 and 1000 are both uncountably infinite.
Yeah, I know. But it’s a lot easier to explain things if you assume that it’s true.
For the record, the ways I can think of of defining inf. I’ll give some simple examples, but I don’t really understand these, and the second two are rather confusing. They are all well defined, but I won’t be able to justify this.
As a limit:
Define oo ! = lim { n! | n-> oo } = oo = lim { n | n->oo } In this case both expressions given are plainly equal.
As an ordinal:
This is similar to the intuitive way, in that you can say w<w+1, etc. But there’s some quirks to allow it to work:
w={1,2,3…}
w+1={1,2,3…,w}
1+w=w
I’ll think about this case later.
As a cardinal:
These are the sizes of sets. Eg. aleph_0 is |{1,2,3…}| = # naturals = # integers = # rationals. aleph_1 is the number of subsets of the naturals. Sets are the same size if they can be paired off 1-1 and not if not. Eg. size{1,2,3…}=size{2,4,6…} using the paring 1->2 , 2->4 etc. December shows that aleph_1>aleph_0.