I don’t think that’s quite right, but I think you’re referring to inaccessible cardinals (or large cardinals in general). You can’t prove that such things exist in standard (ZFC) set theory, though; you need a new axiom.
Actually, aleph null to the power of aleph null equals the cardinality of the reals. The continuum hypothesis (CH) says that that is equal to aleph one, but CH is independent of ZFC; aleph null to the power of aleph null could be (much) bigger than aleph one.
Alephs are the trans-finite (infinite) cardinal numbers. Cardinal numbers refer to the number of elements in a set. The finite cardinals are the familiar whole numbers: 0, 1, 2, 3, . . . After 0, 1, 2, 3, . . . comes [symbol]À[/symbol][sub]0[/sub]. Then [symbol]À[/symbol][sub]1[/sub]. And it keeps going. I mean really keeps going. The subscript for the [symbol]À[/symbol] can be trans-finite.
There are [symbol]À[/symbol][sub]0[/sub] integers and [symbol]À[/symbol][sub]0[/sub] rational numbers. We know by Cantor’s diagonal proof, that there are more reals than integers. So we know there are at least [symbol]À[/symbol][sub]1[/sub] reals. Cantor knew that the cardinality of the reals was equal to the cardinality of the power set of the integers, and he thought that there were [symbol]À[/symbol][sub]1[/sub] reals, and tried to prove it. He couldn’t because, as it turns out, it can’t be proven (or disproven). The assumption that there are [symbol]À[/symbol][sub]1[/sub] reals is called the Continuum Hypothesis. I think it is convenient to assume the Continuum Hypothesis as an extra axiom.
So you’re right. What axiom is that?
Just for fun, this proof fails if you use binary numbers:
1 0.[1]000000…
2 0.0[1]00000…
3 0.11[0]0000…
4 0.001[0]000…
5 0.1010[0]00…
6 0.11000[0]0…
7 0.111000[0]…
…
where I’ve reversed the binary digits of 1, 2, 3, …, and placed them following the decimal (binomial?) point.
Since I’ve only got two digits, 1 must become 0 and 0 must become 1, giving
0.001111… which is, of course, equal to 0.010000…, which is already in my set.
Okay, I know that aleph-sub-two is the number of possible curves.
Can anybody explain relatively simply why the number of curves is demonstrably greater than the number of real numbers?
And do aleph-sub-three and higher have any relationship to “the real world” in the sense that alephs-sub-null through -two do to the cardinals, the reals, and the curves?
If aleph-2 is the cardinality of the powerset of the reals, then there’s a function corresponding to each subset, whose value is 1 if x is in the set, and 0 otherwise. So there are at least aleph-2 functions (or curves, as you call them–just be aware that not every function is a curve). In general, for any two sets A and B, there are |B|[sup]|A|[/sup] functions from A to B, and aleph-1[sup]aleph-1[/sup] = aleph-2 if you accept the generalized continuum hypothesis.
Actually, I’m not entirely satisfied with Cantor’s proof that the reals are uncountable. Is there a proof that doesn’t rely on representations in any particular radix?
P.S.: Zenbeam, the point you’re talking about is the binary point.
If I recall correctly, Martin Gardner illustrated the differences in infinity in either Gotcha or Aha thusly: a line has an infinite number of points between the endpoints (one could also imagine an infinite line), but the amount of points on a plane must be a greater infinity, as would then the points in 3d space. I guess that’s the same as multiplying infinity by infinity, and again by infinity. It’s still rather confusing for me.
Actually, a line, a plane, and a 3-D space have the same number of points. I agree; this isn’t necessarily the clearest subject.
I guess you’re right. Intuitively, one would think opposite- maybe that’s what he was trying to use to show the difference between [symbol]À[/symbol][sub]0[/sub] and [symbol]À[/symbol][sub]1[/sub] and so forth.
Actually that’s a problem in any base. The way to fix it is to include any number that ends in an infinite number of zeroes twice in the list, once in it’s usual form and once in it’s other form. In your example, if your list of real numbers included both 0.01000… and 0.001111… the proof would work.
The problem boils down to the fact that certain real numbers have two different infinite decimal (or binary, or whatever) expansions.
ultrafilter: try this proof. Let a[sub]i[/sub] be a sequence of real numbers between 0 and 1, indexed by the positive integers. Then for every i, let U[sub]i[/sub] be the interval of length 1/2[sup]i+1[/sup] centred on the point a[sub]i[/sub]. Then the union of all those intervals can’t cover the interval from 0 to 1, since 1/4+1/8+1/16+…<1, so there has to be a real number missing. Certain ugly details involving measure theory are left to the reader.
I like that. Thanks!
Actually you have the problem only in base two. For any other base, include each number only once.
For some values of n there will be two different representations:
F(n) = 0. d[sub]1n[/sub] d[sub]2n[/sub] d[sub]3n[/sub] . . .
= 0. e[sub]1n[/sub] e[sub]2n[/sub] e[sub]3n[/sub] . . .
For other values of n there is one representation:
F(n) = 0. d[sub]1n[/sub] d[sub]2n[/sub] d[sub]3n[/sub] . . .
For these set e[sub]in[/sub] = d[sub]in[/sub]
Choose d[sub]n[/sub] to be different from d[sub]nn[/sub] and different from e[sub]nn[/sub]. You can do this because there are more than two digits in any base greater than two. The number 0.d[sub]1[/sub]d[sub]2[/sub]d[sub]3[/sub] . . . will not be equal to any number in the list.
Your proof is more elegant, but the diagonal proof can be patched if the base is greater than 2 without resorting to having some reals included twice.
I’m afraid I’m not that familiar with any of these particular axioms, I imagine some of them are simply of the form, “There exists a weakly inaccessible cardinal.” They’re all classified as “large cardinal axioms”, so searching on that may turn up something.
I just wanted to point out again that these relationships are not proven facts. “Aleph-one equals the cardinality of the reals” and “Aleph-two is the number of possible curves” are consequences of the continuum hypothesis and generalized continuum hypothesis, respectively; both hypotheses are independent of ZFC, so they can’t be proven or disproven within ZFC (but they can be taken as axioms, however).
Under the generalized continuum hypothesis, the power set of an infinite set always has the next higher cardinality. For example, the power set of aleph-one has cardinality aleph-two, the power set of aleph-two has cardinality aleph-three, and so on.
However, most set theorists these days think the continuum hypoothesis (and hence generalized CH) should be false, the reals are probably much greater than aleph-one. Here’s a quote from Paul Cohen (who showed that the negation of CH is consistent with ZFC, using his technique of forcing. C=cardinality of the reals=cardinality of the power set of natural numbers):
The quote comes from this (very interesting) link:
I’m afraid I’m not that familiar with any of these particular axioms, I imagine some of them are simply of the form, “There exists a weakly inaccessible cardinal.” They’re all classified as “large cardinal axioms”, so searching on that may turn up something.
I just wanted to point out again that these relationships are not proven facts. “Aleph-one equals the cardinality of the reals” and “Aleph-two is the number of possible curves” are consequences of the continuum hypothesis and generalized continuum hypothesis, respectively; both hypotheses are independent of ZFC, so they can’t be proven or disproven within ZFC (but they can be taken as axioms, however).
Under the generalized continuum hypothesis, the power set of an infinite set always has the next higher cardinality. For example, the power set of aleph-one has cardinality aleph-two, the power set of aleph-two has cardinality aleph-three, and so on.
However, most set theorists these days think the continuum hypoothesis (and hence generalized CH) should be false, the reals are probably much greater than aleph-one. Here’s a quote from Paul Cohen (who showed that the negation of CH is consistent with ZFC, using his technique of forcing. C=cardinality of the reals=cardinality of the power set of natural numbers):
The quote comes from this (very interesting) link:
Hey, thanks for the suggestion to read Rudy Ruckers’ Infinity and the Mind. I have just finished chapter 2, and it looks like my question was answered in that chapter by Absolute Infinity (a capital omega is used as its symbol)
Absolute Infinity seems to be defined such that nothing, even a transfinite, can be bigger than it. This is pretty much what I was looking for in my OP.
Of course, if I misunderstood what I read, please correct me.
I’ll grant you that, but you’ll also have to accept that there is no such thing as one, or two.
Uh yeah, but I can show you one apple, or two gorillas. Can you show me an infinity of anything? Is there anything that is infinite in nature? All the atoms in all the universe are still finite, and multiply them by the number of photons emitted since the dawn of time, and that is still finite. Multiply that number by all the positive integers ever generated by humankind, and that number is still finite. Infinity is only a concept useful for describing something that can never be, an adjective of the eternal in the religion of mathematics.
Show me precisely 348,384,488,764,386,049,103,776,498,861,551,051,643 apples. This “show me” rule of yours for whether something is only a concept doesn’t strike me as very useful. I know what it would mean to have 348,384,488,764,386,049,103,776,498,861,551,051,643 apples even though I can’t show them to you.
This isn’t true; there are just as many curves in the plane (or space if you like) as there are real numbers. This is because continuous functions f:R->R[sup]n[/sup] are determined by their values at rational points, so although there are 2[sup]2[sup][symbol]À[/symbol][sub]0[/sub][/sup][/sup] many functions g:R->R[sup]n[/sup], only 2[sup][symbol]À[/symbol][sub]0[/sub][/sup] of them are continuous.
QUOTE]*Originally posted by ultrafilter *
**There are also infinite cardinals that are bigger than any aleph, but that’s a technical matter. **
[/QUOTE]
If you don’t assume the Axiom of Choice you can have sets with “cardinality” not equal to any aleph, but it’s not because they’re too big, it’s because they can’t be well-ordered. Large cardinal axioms have nothing to do with this.