What if I list the naturals like this: 0, 2, 1, 4, 6, 3, 8, 10, 5, 12, 14, 7, ... ? Is it anything more than a convention to take ordered intervals? All I noticed was that indexing the infinities, as in "aleph 0, aleph 1, aleph 2, ..." seemed to be implying that they were countable. I don't know anything about raising any of these aleph's to the power of one another...
It's not some arbitrary convention- the naturals come fully equiped with an order that's well, natural.
But what compels you to stick with this ordering when trying to find the ratio of odds/evens in the set of naturals? I agree that using ordered intervals seems like the most "natural" (and ultimately most useful) way of doing it, but does that invalidate all the other possibilities? My take is that reference to the ratio of odds/evens in the naturals is meaningless on its own.
Of course other possibilities aren't invalidated, an asymptotic density using the usual ordering is, as you admit, the most useful. There are different weights you can use, logarithmic density, zeta density, averages of these, that can lead to different interpretations, but all still reflect the usual ordering of the naturals. I would say that using some other ordering to the naturals is a bizzare way to interpret anything, the usual ordering is built in to the definition of the naturals, and I have yet to see anything fruitfull or otherwise come from this sort of consideration. While it's a colloquial phrase, if a number theorist says "half the primes are congruent to 1 mod 4" then I know they would put rigorous meaning to this in the normal asymptotic sense (though there is a context here). I wouldn't call it a meaningless statement, just an informal one. I would expect most people who say things like this can't put it on a rigorous ground, these are the people who want to select a random number from the naturals using a uniform distribution (this terminology gets abused by number theorists also, though 'random integer' will be made rigorous in an asymptotic density kind of sense as well).
I do not see how the right side is obviously c. What I want is a proof that c^aleph_0 = c (which is what the right side is, 2^alpeh_0 = c).
You can list them any way you want... just so long as it's a pattern that will tell me what the n'th number in your list is... not that we may know what this pattern is, just that it exists. I could do this: {2, 1, 3, 5, 4, 7, 6, 11, 10, 9, 8, 13, 12, 17, 16, 15, 14, ...} What's the pattern? Start with a prime number, then list every positive integer before it that has not been included in the list before it.
Well, if you have a countable number of reals you can map them to one real. I'll use numbers in (0, 1) in decimal to make the pattern clear, but obviously any base is equivalent. Suppose you have: 0.000000000... 0.111111111... 0.222222222... 0.333333333... ... You start in the top-left corner and take diagonals to create your new real. In this case: 0.0 01 012 0123 01234 ... Which gives a real 0.0010120123... and you know that the nth digit of the mth real will always occur at a countable position. If this looks familiar to proving the countability of pairs of naturals (or of the rationals, which is almost the same thing) it's because c = 2^<sup>aleph_0</sup> = 10^<sup>aleph_0</sup> (base doesn't matter) - so anything you can do to count naturals you can use to count the digits of a real. Since countability of pairs of naturals = aleph_0<sup>2</sup> = aleph_0, (2<sup>aleph_0</sup>)<sup>aleph_0</sup> = 2<sup>aleph_0^2</sup> = 2<sup>aleph_0</sup>
It's probably just me, but I think your explanation isn't telling me much. lol. All I can pull out is it looks like you are trying to show me the reals are not countable, which I already know. I know why 2^aleph_0 = c, but replace the base, 2, with c.
So you get what I'm saying about representing a real as a vector of binary digits, where the vector has countably infinite dimensions / 'slots'? So if you have a vector of things of cardinality m, and the number of dimensions (of the vector) has cardinality n, the cardinality of the set of all such vectors is m^n. Right? So to find c^aleph_0, you want a countably-infinite dimensioned vector of reals. That is, R<sup>aleph_0</sup>. Right?
Boy do a lot of you assume that the continuum hypothesis (and the generalized continuum hypothesis) are true. As I said before there is a proper class of cardinal numbers. This means they are too big to be a set, ie given any cardinal number, alpha, I can find more than alpha distinct cardinals. Thus, for those who seem to want to know more about this, not only are there infinitely many cardinals, there are uncountably many cardinals, indeed so many that the notion of many in the sense of Cantor cannot be applied to them: the class of all cardinals is sufficiently large that it does not have a cardinality.
Yes, the (G)CH is that 2^aleph_n = alpeh_{n+1}. I must have been imagining some things were being said that weren't.
I don't feel like going over the thread to see what I said, but I may have slipped up and said something like that. What I meant was 2^aleph_o = cardinality of the reals = c. asdf
Wow I really don't like the inconsistency. That bothers me. .9999.... should equal .9999... and 1 should equal 1. There is still a .0000...1 difference between .999... and 1 and yet they're both the same number. But there is no inconsistency. Your dealing with a number that doesn't exist. Such as you cannot divide two infinite numbers, you cannot do the series of steps you just did. By doing the series you just did, your assuming that at some point the .9999.... ends, which is evident right here: 10x - x = 9.999... - 0.999... Because, taking the same scenerio and instead using a number with ALOT of nines instead of infinite, 10 times that number is always going to be one decimal less than the original number in other words x = .99999999 10x - x = 9.99999999 - .999999999 9x = 8.999999991 x = .999999999 Therefore you performed an illegal mathetimal operation, you just can't deal with infinit decimal sequences like that, sorry.
Vortex... I got to say you are wrong. The fact that we can CREATE the number shows it exists in some sense. There is a theorem in calculus (or something) that says if (a<sub>1</sub> + a<sub>2</sub> + ... ) = L and (b<sub>1</sub> + b<sub>2</sub> + ... ) = K, then (a<sub>1</sub> + b<sub>1</sub>) + (a<sub>2</sub> + b<sub>2</sub>) + ... = L + K. If you let a<sub>0</sub> = 9, a<sub>1</sub> = .9, a<sub>2</sub> = .09... Or in general, a<sub>n</sub> = 9*10<sup>-k</sup> for k = 1, 2, 3, ... b<sub>k</sub> = -9*10<sup>-k</sup> for k = 1, 2, 3, ... We know the series a<sub>1</sub> + a<sub>2</sub> + ... converges. We also know that b<sub>1</sub> + b<sub>2</sub> + ... converges. To what, we do not know at this time (just assume). Well: (a<sub>1</sub> + b<sub>1</sub>) + (a<sub>2</sub> + b<sub>2</sub>) + ... = (0.9 - 0.9) + (0.09 - 0.09) + ... = 0. So it seems we are right to say 9.999... - 0.999... = 9 + (0.999... - 0.999...) = 9 + 0 = 9.
I still don't see it. To allow it to go to 9 like that the sequence would have to end, and when it does there's going to be a point where, according to what you said it'll still equal 9.0000....1. Ever done anything in computer programming? Such a sequence doesn't exist, its called an infinite loop, it never ends. Only when the sequence is interrupted by an outside occurance can an answer be viewed. Infinite cannot be used and applied this way, simply because its undefined.
One number having two different representations is not inconsistant in any way here. You're probably comfortable with 1/2=2/4. Go look up "Equivalence Relation". By .000...1 it looks like you mean an infinite number of 0's followed by a 1? This is total and absolute rubbish, there's no such real number. This isn't computer science, loops or reading variables is completely irrelevant here. Ever done any maths? Know what a limit is? Have you ever looked at one of the many (equivalent) ways to define the real numbers? Go do so, come back and explain what exactly the symbol "0.999...." means in mathematics. You will be enlightened when you have taken the time to study the real numbers beyond the vagueness seen in high school.
The rationals can be defined as the smallest field containing the integers. You can show such a field exists by explictly constructing one. The usual construction involves starting with the set of ordered pairs (a,b) where a and b are integers and b is non zero and defining an equivalence relation (you also need to define the usual algebraic operations) by (a,b)~(c,d) if ad=bc, i.e. (1,2)~(2,4). We end up with many representatives for the same equivalence class (in this case the thing that's supposed to be the multiplicative inverse of "2"). We get lazy and write (a,b) as a/b and use "=" instead of "~". Every rational has more than one representation as an ordered pair, choosing different representations for the same thing is whats going on behind 'reducing fractions'. (This construction works for integral domains in general). It would probably be worth it for you to look up the full details of this kind of construction to see how equivalence relations are used in mathematics in this more familiar context. If you are set on defining the reals in terms of decimals, they you necessarily have an equivalence relation for things like 0.99.. and 1, OR you explictly forbid things like 0.999... (this is generally not done) Why do you think they decimals are "set in stone one characteristic"? (what does this mean anyway?) I think you are hung up on "Real Number" somehow representing an object that exists in real life. They aren't. ? Why would logic tell you infinity-infinity is zero? If you stop to think for a second, you should ask why you think infinity-infinity is defined to be anything at all (though you do seem to get this). Decimals with an infinite number of non-zero digits are not in any way trying to treat 'infinity' as a number. They are very well defined objects. Alright, you know what a limit is. One way to define 0.999... (and likewise for all decimals), is as the sum of the infinite series 0.9+0.09+0.009+... The symbol "0.99..." is just shorthand to mean the sum of this infinite series. The sum of this infinite series is defined as the limit of the partial sums. The limit of these partial sums is 1, it's a geometric series. That's it. Note a big distinction here- 0.99... is defined as the limit of the sequence of partial sums, it does NOT equal any sum partial sum though.
When you do .999999 ^ 1000000 power, and do that for more nines and zeroes, the results converge on a number that is the multiplicative inverse of e. This demonstrates that both infinity and one over infinity can be manipulated as mathematical entities that produce precise numerical results.
