No. Aleph Null is the cardinality of the set of integers.
[/QUOTE]
I was under the impression that aleph[sub]0[/sub] was the cardinality of the set of rational numbers. Paging mathochist and other exotic-math experts…
[/QUOTE]
That’s the same number, but it’s more usual to define it as the cardinality of the natural numbers, i.e., the positive integers (including 0), since the natural numbers can be easily based on set theory. Every finite set has n elements, where n is a natural number.
[QUOTE=ianzin]
Specifically, it asserts that, given an infinitely long list of numbers, one can construct a new number by taking the first digit of the first number, the second digit of the second number and so on. This sounds seductive and plausible, which is why is tends to pass without remark or objection. However, this procedure is only legitimate and meaningful for a finite list of numbers. If you are starting with a list of numbers that you assert has an infinite number of members, each of which can be expressed as an infinitely long decimal (hence the use of an ellipsis at the right-hand end of each number in Anne Neville’s list), then the procedure described is meaningless.
[/QUOTE]
I think the problem is that such a procedure is dependent on the axiom of choice, and that you (ianzin) do not accept that axiom.
[QUOTE=Thudlow Boink]
I think the problem is that such a procedure is dependent on the axiom of choice, and that you (ianzin) do not accept that axiom.
[/QUOTE]
I don’t think it depends on the axiom of choice. If we can construct an explicit function to choose an element out of any given subset of a set, we don’t have to invoke the AoC. It’s only when you have to make arbitrary/undefineable choices that the axiom comes into play.
Somebody (I think Bertrand Russell) once noted that you don’t need the axiom to choose one shoe from each of a countably infinite set of pair of shoes, but you do need it to do the same with socks. I think we’re dealing with shoes rather than socks here.
[QUOTE=ultrafilter]
Somebody (I think Bertrand Russell) once noted that you don’t need the axiom to choose one shoe from each of a countably infinite set of pair of shoes, but you do need it to do the same with socks. I think we’re dealing with shoes rather than socks here.
[/QUOTE]
What if you’re like me, and keep your socks in pairs? (Not that I have a countably infinite number of socks in my sock drawer).
[QUOTE=Giles]
What if you’re like me, and keep your socks in pairs? (Not that I have a countably infinite number of socks in my sock drawer).
[/QUOTE]
You still need the axiom. With shoes, you can come up with a rule to choose (e.g., always pick the left one). At least with a brand new pair of socks, there are no distinguishing features, so you can’t have so succinct a rule.
This will be my last chance before someone who knows their stuff comes along and makes me look really bad.
The way I was taught to deal with infinity was not as a number, but as a limit. That is, there is no such thing as n/0, but there IS lim(x->0)n/x=infinity.
Apologies for being incompetent with the character map.
Infinity itself was on the border of “undefined.” Though certainly useful. There’s also bounded infinities, the easiest being the Koch Curve (we called it the Star of David Sequence), in which the boundary is infinity, but the area within is known.
Attempting to participate in this thread is gonna get me in a lot of trouble.
[QUOTE=ultrafilter]
I don’t think it depends on the axiom of choice. If we can construct an explicit function to choose an element out of any given subset of a set, we don’t have to invoke the AoC. It’s only when you have to make arbitrary/undefineable choices that the axiom comes into play.
[/QUOTE]
Yeah, I think you’re right. (But I still think ianzin sounds like the kind of person who wouldn’t accept the Axiom of Choice.)
[QUOTE=The Them]
This will be my last chance before someone who knows their stuff comes along and makes me look really bad.
The way I was taught to deal with infinity was not as a number, but as a limit. That is, there is no such thing as n/0, but there IS lim(x->0)n/x=infinity.
[/QUOTE]
Yes, thinking of infinity in terms of limits is right in analysis. However, when you talk about cardinal numbers, you are in set theory, where limits are not so important. (Though they are in the background: Cantor’s proof involves decimal fractions, and those decimal fractions are really limits of an inifinte series).
[QUOTE=Scuba_Ben]
I was under the impression that aleph[sub]0[/sub] was the cardinality of the set of rational numbers. Paging mathochist and other exotic-math experts…
[/QUOTE]
You and Rysto are both right. The cardinality of the rationals is equal to the cardinality of the integers.
Here’s a proof that the rational numbers in the open interval (0,1) are countable.
Recall that every nonzero rational number q in (0,1) can be written in a unique way as m[sub]q[/sub] / n[sub]q[/sub] with m[sub]q[/sub] and n[sub]q[/sub]coprime integers and 0 < m[sub]q[/sub] < n[sub]q[/sub]. Observe that for each positive integer N, there is a finite list of rational numbers q in (0,1) with n[sub]q[/sub] = N (ordered according to their position on the number line). Call this list L[sub]N[/sub].
By concatenating these finite lists
L[sub]1[/sub], L[sub]2[/sub], L[sub]3[/sub], . . .,
we get a countable list, which exhibits a bijection F between nonzero rational numbers and the positive integers.
More precisely, we define the bijection F as follows: Given a rational number q = m[sub]q[/sub] / n[sub]q[/sub] in (0,1), let k[sub]q[/sub] be the position of q in the list L[sub]n[sub]q[/sub][/sub] containing it. Let
F(q) = |L[sub]1[/sub]| + |L[sub]2[/sub]| + . . . + |L[sub]n[sub]q[/sub] - 1[/sub]| + k[sub]q[/sub].
(Here, |L[sub]N[/sub]| denotes the number of terms in the list L[sub]N[/sub].) This defines our bijection in a way that hopefully even ianzin finds unobjectionable :).
[QUOTE=ultrafilter]
But by our construction, k[sub]n[sub]k[/sub][/sub] is not equal to f(n[sub]k[/sub])[sub]k[/sub], and it follows that f(n[sub]k[/sub]) is not equal to k. This implies that there is no bijection f between the positive natural numbers and decimal representations of real numbers in the interval [0, 1).
[/QUOTE]
Slight typo: what I believe you meant to write was that
He does. The reason why we get this unexpected result is that we cannot meaningfully do these sorts of operations on divergent series. a[sub]0[/sub] + 2 a[sub]1[/sub] + 2 a[sub]2[/sub] + … = a[sub]0[/sub] + 2 (a[sub]1[/sub] + a[sub]2[/sub] + …) makes sense only if the series a[sub]1[/sub] + a[sub]2[/sub] + … is convergent, which in this case it isn’t. At least, it isn’t convergent in the reals.
This is the standard high-school algebra example of why you have to be careful with divergent sums. Of course, once you get into complex analysis or p-adic integers and learn to be very careful, you can reasonably claim that 1 + 2 + 4 + 8 + … = -1. But that’s getting pretty far from the OP.
[QUOTE=ultrafilter]
** Of course, once you get into complex analysis or p-adic integers** and learn to be very careful, you can reasonably claim that 1 + 2 + 4 + 8 + … = -1.
[/QUOTE]
Hey, it could be worse. I’m not even convinced that you can always choose an element from a two element set (Pick one element of the set of square roots of minus one, and tell me which element you picked without resorting to circular arguments).
As for the infinite version, well, I’m quite happy to take the negation of the Banach-Tarski theorem as an axiom.
[QUOTE=ultrafilter]
This is the standard high-school algebra example of why you have to be careful with divergent sums. Of course, once you get into complex analysis or p-adic integers and learn to be very careful, you can reasonably claim that 1 + 2 + 4 + 8 + … = -1. But that’s getting pretty far from the OP.
[/QUOTE]
I see that Frylock has started [thread=458674]this thread[/thread] about this summation. I’ve worked with p-adic numbers before, but that was five years ago and I don’t remember all that much about them, so maybe you or someone else could explain in that thread how to make that series convergent in a suitable space.
[QUOTE=Hypnagogic Jerk]
Slight typo: what I believe you meant to write was that
[/QUOTE]
Good eye.
Are we using the standard construction of C from R? In that case, I choose (0, 1). If we’re using some other construction, I’ll still have a means for choosing.
[QUOTE=Frylock] ianzin, I’m interested in seeing a response to post 20 above (some of which I repeat in my last post).
[/quote]
I would be delighted to provide a response, but I am unable to do so. I don’t understand it. I’m sure the prevailing, majority opinion in this thread is that I don’t understand it because I lack the necessary education, or at least the necessary mathematical education. I don’t have a problem with this opinion, and I acknowledge that it may be correct. Nonetheless, it isn’t my opinion. Sometimes, a man doesn’t understand something because he lacks the ability to comprehend. Sometimes, he doesn’t understand something because he is being asked to understand something that doesn’t make sense. Here’s an example of a sentence that doesn’t make sense:
I cannot reconcile the different parts of this strange sentence. There is a shift from ‘is’ possible to ‘are’ possible that I find puzzling. I’m pretty sure that if there is a problem here, it’s not my inability to parse a sentence.
I cannot comment on the proof as expressed in post #17, as I do not pretend to even begin to understand it. I can only work with the material I can understand, including Anne Neville’s post and the description given in works such as Godel, Escher, Bach. I see that some posters are now asserting that these readable accounts may be sufficient for a non-specialist readership, but they are less rigorous than Cantor’s actual proof, and that the actual proof is robust. This may be so, but I have no way of knowing as the actual and rigorous proof, it seems, can only be expressed in terminology and symbols that I am not sufficiently educated to understand. Nonetheless, I’d like to thank those who have taken some time to at least try and shed some light on the subject for me.
Except that it seems to me that the standard construction still depends on the axiom of choice. If you produce that construction, and some other mathematician working completely independently produced a construction that looks just like it, how would we know that you’re both referring to the same number when you say (0,1)?
It may be that I’m acting under the assumption that the numbers themselves exist independently of our constructions of them, a view which I understand mathematicians try to discourage. It just doesn’t sit right with me, though.
Chronos, what you’re talking about doesn’t really have anything to do with the Axiom of Choice as it is understood in Set Theory. You’re talking about a more general issue with the philosophy of names and referents for abstract entities.
