For a while, I thought I understood infinity. A while back, there was this character, Wayne, played by Mike Myers on Saturday Night Live. The name of the whole bit was “Wayne’s World”. And there was this running joke on it. Someone would say something and for emphasis say “…infinity!” after it, to which Wayne would say the same thing, but “…infinity plus one”. When I first heard this, I thought that was silly. Infinty plus one is just infinity, right?
Anyways, I don’t have a link. But I do remember hearing someone say on tv recently that if you go faster than the speed of light, you will have mass greater than infinity. (According to Einstein, at the speed of light, mass would be infinity, of course.) Did I hear that right? Something can be more than infinity?
Anyways, that is my first question: can anything be more than infinity? My second question is a thoughtful mental exercise I have thinking to myself for some time now: What is infinity taken to the infinite power an infinite amount of times? As I’ve said, for a while, I just assumed it was infinity. Right? (And if it isn’t, please clarify what it is. I’d like to know:).)
There are in fact an infinite number of infinities of different sizes. Boy, are you going to be sorry you ever asked this question once the real math nerds come in and start throwing alephs at you! :eek:
Assuming that by “infinity” you mean “the amount of integers”, and that by taking something to a power some number of times, you mean “take it to that power, then take the result to that power again, and so on”, then “infinity taken to the infinity power an infinite number of times” gives you a result that is equal to the quantity of the real numbers. As infinite quantities go, you can do a lot better than that.
It depends on what you’re doing. Do kings always move diagonally? Can you promote a piece to a king? The rules regarding kings in chess are different from the rules regarding kings in checkers, and similarly, the rules regarding infinities in some math-games are different from the rules regarding infinities in other math-games. If your application is to describe “How many items are in this collection?”, then you will likely be interested in games with rules like those Exapno Mapcase and Chronos are referring to (the rules of what are called “cardinalities”). If the use you wish to make of “infinity” is something else, then you may be interested in different games instead.
(And, of course, you may (and, indeed, often will) be interested in multiple games at the same time; you don’t have to pick just one and ignore the rest)
Also, to Chronos: I would interpret “((A^B)^B)^…^B, with C many Bs”, as A^(B^C). Accordingly, raising N to the Nth power N many times results in N^(N^N), which is beth_2, not beth_1 (i.e., not the cardinality of the reals, but rather the cardinality of the powerset of the reals).
I have always understood “infinity” to mean “without end”. I’ve always understood it as an abstract theoretical concept, that does not exist in any observable or measurable reality.
Even if you talk about “an infinite number of integers”, you’re just saying “there is no end to the number of integers”. But you could never “sum all integers” or “raise infinity to the power of infinity”. These are meaningless terms (as per my understanding).
Well, no, because things have meaning to the extent we define them. For example, ‘flooble’ is just a meaningless series of letters or sounds until I define it to mean ‘the feeling you get when you regift an ugly handmade item to someone who really likes it’.
Similarly, we can define ‘infinity’ and ‘infinite’ in rigorous ways that allow us to say, for example, there are more real numbers than integers, but the same number of complex numbers as reals, even though there are an infinite number of integers, reals, and complex numbers. This process of reasoning from rigorous definition is the very heart, soul, and foundation of mathematics, second only to strong coffee or tea in the elements necessary for mathematicians to function.
You did not hear that right; they did not say something can be more than infinity.
What you heard someone say on TV recently was a hypothetical situation (faster-than-light travel) that gives an impossible result (mass greater than infinity), suggesting the impossibility of FTL travel.
“‘A’ results in ‘B’. ‘B’ is impossible. Therefore ‘A’ must also be impossible.”
I’m no mathematician, but my point would be that as soon as you talk about is there “more” of this infinity or “more” of that infinity, you’ve already lost.
You can’t tell if there’s “more” of “less” of any given thing unless you can measure said things to a definable quantity.
There’s an endless number of integers, there’s an endless number of real numbers. To say there’s “more” of one compared to the other is a sentence without meaning, says this non-mathematician, because we’re not talking about quantities.
Okay, how 'bout this for a simpler approach to sizes of infinity:
There are an infinite number of integers 1, 2, 3, 4, 5…
There are an infinite number of even integers, 2,4, 6, 8… but this group must be roughly half as large as the first group, or if you prefer, this group must approach infinity roughly half as quickly as the first group.
You can prove that the infinity of real numbers is bigger than the infinity of integers, because for any mapping of integers vs reals, for example
1 to 0.1
2 to 0.2
3 to 0.3
4 to 0.4
etc
You can always find a new real number to fit in a gap in the sequence - so in this case, 0.15.
The point is that the infinity of integers is fixed in size - you can’t invent a new integer between 1 and 2, but you can do that with real numbers - therefore, the infinity of real numbers is larger than the the infinity of integers
No, you can compare infinities. You do it using set theory, by saying that if sets A and B can be put into a one-to-one correspondence, then they have the same number of elements. So, the set containing my parents has the same number of elements as the set of planets closer to the Sun that the Earth, with this correspondence:
my mother corresponds with Mercury
my father corresponds with Venus.
You can do that with infinite sets, only in this case an infinite set can be set into a one-to-one correspondence with a subset of itself. For example, there are exactly the same number of even numbers as there are counting numbers:
0 corresponds with 0
2 corresponds with 1
4 corresponds with 2, etc.
So, if you add one to infinity, you still have infinity, because you can still set up a one-to-one correspondence. (That’s true of the infinity of the counting numbers, or cardinal numbers, but it’s true of other infinities as well.)
However, you can’t set up a one-to-one correspondence between the infinity of the counting numbers and the infinity of the real numbers. How do you prove that? By showing that, kif you assume there is such a one-to-one correspondence, you can find at least one real number that isn’t in the correspondence. Therefore, the infinity of the real numbers must be bigger than the infinity of the counting numbers.
People on TV say all kinds of stuff, but the equations show that if an object is moving faster than the speed of light then its mass is imaginary. A lot of people don’t think that has physical application.
I would argue you can’t “approach” endlessness. If anyone ever tried to sit down and map integers against even numbers, they would not run out of numbers from either segment.
But I wouldn’t say that. You’re again trying to measure something that doesn’t have an end - it’s inherently meaningless. What is the “thing” you are measuring? What’s the unit of measure?
