There are really two kinds of infinity: Countable and uncountable. The set of integers is countable, the set of reals is uncountable. A set S is countable if there is a one-to one mapping with the set of integers. So if I define a set to be Z = (1/2, 1/4, 1/8,…,1/2^n,…) then Z is clearly countably infinite. However you cannot do such a thing with the set of reals. In fact you cannot even do such a mapping between any two rational numbers. These sets are uncountably infinite and in fact uncountably infinite sets can be divided into a countably infintie number of uncountable infinite sets. Say that five times fast.
Something just came to me -
It’s been a long time since I studie calculus and other higher forms of maths, but I don’t recall ever using numbers such as 0.4 or 0.888 repeating. Instead, I’d write 2/5 or 8/9. Aren’t decimal fractions (if that’s what they’re called in English) merely a shortcut, a simplification? How can you use them in complex equations?
In other words, 1/9=1/9
9/9=1
IIRC, every division is a fraction, so if you devide 18 by 4, you get 18/4, which equals 9/2 - at least for the pupose of solving the equation.
Or maybe they teach it differently in American schools.
In other words, when you said 8/9=0.888 repeating, you entered a 10 (or 2*5) into the equation, a 10 which has no right to be there.
<< There are really two kinds of infinity: Countable and uncountable. The set of integers is countable, the set of reals is uncountable. >>
Well, yes and no. An infinite set is deemed “countable” if it can be put in one-to-one correspondance with the positive integers, so that you can “count” the members of the set. Examples of countable sets include the positive integers (duh), all integers, all even integers, and all rational numbers.
An infinite set is called “uncountable” if it cannot be put in one-to-one correspondance with the positive integers. True enough. However, there are different levels of uncountable sets as well. For instance, the Real Numbers, and the Complex Numbers form uncountable sets that can be put in one-to-one correspondance with each other (commonly called Alef-1, IIRC). However, the set of all continuous functions (with Real domains and ranges) is LARGER than the set of all Real Numbers, so is a bigger uncountable set, commonly called Alef-2.
So while the classification of infinite sets as countable or uncountable is useful, I wanted to add the footnote that there are different sizes of uncountable sets.
The power set of a set is the set of all subsets of a given set. The power set of X is written 2[sup]X[/sup]. The power set is always strictly larger than the set itself. There is an elegant diagonal proof of this. It is known that the cardinality of the reals is equal to the power set of the integers, so it is at least aleph-one. The assumption that 2[sup]aleph-null[/sup] = aleph-one is Cantor’s continum hypotheses (CH). The generalized continum hypothesis (GCH) says that the power set of any aleph is the next aleph. CH and GCH are neither provable nor disprovable. I think it makes sense to assume GCH, but you could build a system where they are assumed to be false and you would not introduce any contradictions. You would have a non-Cantorian set theory much like non-Eucledian geometry.
Given any (finite or infinite) set you can always form a strictly large set by taking all the subsets of the given set.
Virtually yours,
DrMatrix - If I’ve told you once, I’ve told you 0.999… times.
Ah, yes, I remember Aleph-1 and Aleph-2, although only vaguely. Back in my more academic days I had passing experience with them. But, at this point I think I deserve a gold star for simply remembering countable and uncountable.
CK, I suppose I am agreeable with the spirit of this post, but I can’t help challenging a couple of things. We’re talking “meta-mathematics” now, but do you think numbers like one exist in the “real world”? Do circles exist “in the real world”? Then, if you really can come up with a circle of diameter one, then an arc of length Pi also must exist “in the real world”. It does by definition.
The thing I’m trying to counter is the notion that irrational numbers (in the case of pi) or infinite repeating decimals (like 0.999…) are somehow “fuzzy” and undefineable. This is really just an artifact of the choice of trying to represent them in the conventional decimal form, not anything to do with accuracy. For example the fraction 1/3 = 0.333… repeats endlessly, but I don’t think anyone has trouble thinking that a line divided into three equal parts is in any way as inexact or impossible to construct as you make it sound.
And, what the heck is a ‘definitional universe’ anyway? Is that outside of the “real world” too?
And in response to Alessian, decimal representations and rational fractions are just two different ways to represent numbers. Calculating with them requires different techniques, but you get the same numbers in the end. For some history on this look at http://members.aol.com/jeff570/fractions.html.
And, another note, people have been refering to Calculus in some of these posts. Actually, the things being covered here belong to the subject of “real analysis” which is foundational for calculus. In other words, you need to have a grasp of what things like converging sequences, limits, and infinite series are before you can go on to talk about derivatives and integrals (the stuff of calculus).
I had a circle with a diameter of 1 hotdog and I put a piece of string around it to measure its circumference. Then I straightened out the string and measured the length in hotdogs. And I ate it…
e will be a little harder to get though.
Oh, I dunno, I had -e^(i*pi) bagels for breakfast this morning. No joke.
Circles, according to the geometric definition, don’t exist in the real world, since the arc that comprises the circle would have to have zero width, which is impossible in reality. After all, let’s say you use a compass and make a circle with a diameter of one inch. Then the circle has a circumference of pi inches, right? Well, are you measuring the circumference around the outter edge of the pencil line or the inner edge?
Mathematical exactness does not exist in the real world. At all.
[QUOTE]
*Originally posted by kellymccauley *
**
Um, I think that was me again. I didn’t actually use Calculus in my post, though I said I was going to… Looking over what I wrote last night, I kind of wish I had waited until this morning to post. That’s two major inaccuracies pointed out so far.
Sorry about that. And thank you for not naming names.
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Some days, I feel like a one-bit processor with a parity error.
[troublemaker]I don’t know, Tzel, which edge did you use when you said the diameter was one inch? I’m sure we could go 'round and 'round on this…[/troublemaker]
smirk My point exactly.
I was trying to make the point that there is a universe of mathematics, that exists in pure thought, where lines have no thickness and irrational numbers are well-defined, within a logic system. There is also a universe of the senses, a “real” universe, where lines DO have thickness and no number needs more than a few hundred thousand decimal places to be pin-pointed pretty precisely [/alliteration] The mathematical universe is a theoretic model of the real universe that enables us to do marvellous things, but it is only a theoretical model.
So, kelly, when you ask me whether “one” or “circles” exist in the real world, I can answer, assuredly. I can certainly indicate “one” of an object, and I can draw a circle. These are not the same as “one” or a “circle” in the mathematical universe, of course.
Did that help?