91–98 of 98 posts
Earlier quoted context omitted.
" a set of Aleph-0 points can completely cover a line " Aleph_0 is the cardinality of the integers. I don't think that'll cover a line. For that, you need the cardinality of the reals, C, which may or may not be Aleph_1.
OOPS. Thanks
The problem with "points on a number line" as a definition for real numbers is that it's not clear how you can tell if you have all of them. You can populate a number line as densely as you care to using just rational numbers, but that's not all of them, you're missing out on numbers like the square root of two. You can toss in the non-intergral powers of rational numbers, but you still won't have all of them, you're…
The set of numbers that can be uniquely defined in the English language in a finite number of letters is a countable set, because the set of finite sequences of English letters is countable.
Earlier quoted context omitted.
Sure. Chaitin's Omega is a good example. The question is whether such numbers occur in the real world.
The trick here is defining "occur" and "real world" precisely. Are you saying the act of me writing down those symbols and expressing the idea does not count as "occurring in the real world?" ;)
You're right though that a more precise definition of "real-world numbers" is needed, but I confess that my attempts to think of one in the past few minutes have been essentially circular (coming down to "the ones we know how to compute")!
Earlier quoted context omitted.
The trick here is defining "occur" and "real world" precisely. Are you saying the act of me writing down those symbols and expressing the idea does not count as "occurring in the real world?" ;)
This reminds me of the self-defeating property of an "uninteresting" number -- a reasonable definition might be "any number that does not have any property of human interest", but then of course there is a smallest such number, and so that has the interesting property of being the first uninteresting number, a contradiction! You're right though that a more precise definition of "real-world numbers" is needed, but I c…
It's not clear whether the universe is computable, however, in the sense that we only find computable numbers in nature. This is kind of an epistemological catch-22, though. How would we know whether this were the case or not?
Earlier quoted context omitted.
Infinity is a pretty strange concept. :) I'm not sure arguing over it in this format is meaningful, but for the fun of it: Consider that the integral of the ruler function from 0 to 1 is 0 (as is stated in your reference 1). In layman's terms you could express this as "there are infinitely more irrational than rational numbers between 0 and 1". At the same time, "for every two rational numbers there are infinitely ma…
I don't understand the point you're trying to make? The Cantor set shares the property that "for every two [points in the set] there are infinitely many [points in the set] in between", but no one would describe it as looking like a line. It's rather sparse.
The Cantor set is very different. It's even easy to give an example of two points in the set that can (sanely) be depicted with empty space in-between: 1/3 and 2/3. If I'm not mistaken that example also disproves your stated conjecture... ;)
Earlier quoted context omitted.
I don't understand the point you're trying to make? The Cantor set shares the property that "for every two [points in the set] there are infinitely many [points in the set] in between", but no one would describe it as looking like a line. It's rather sparse.
What I take issue with is an "image" of two rational numbers as two separate dots, with empty space in-between. That's a very deceiving image IMHO, since I cannot think of a sane way to produce it. The Cantor set is very different. It's even easy to give an example of two points in the set that can (sanely) be depicted with empty space in-between: 1/3 and 2/3. If I'm not mistaken that example also disproves your stat…
> It's even easy to give an example of two points in the [Cantor] set that can (sanely) be depicted with empty space in-between: 1/3 and 2/3. If I'm not mistaken that example also disproves your stated conjecture... [that between any two points in the set, there is a third one] ;)
Fair enough. Consider, then, the intersection of the Cantor set with the irrational numbers (you can think of this as the "open Cantor set"). It is, obviously, a subset of the Cantor set, and really does have the property described.
Since I'm feeling embarrassed about that last time, a proof follows:
-----
The Cantor set consists of all real numbers in the interval [0,1] which have a "decimal" expansion in trinary which does not contain the digit 1. That is to say, they can be expressed in terms of powers of (1/3) such that the coefficient of each power of 1/3 is either 0 or 2. (1/3 would usually be represented in trinary as 0.1, but is in the Cantor set because of its representation as 0.02222222...)
Let a,b be two irrational numbers in the Cantor set, a less than b. There is some decimal place at which they diverge, and since a is smaller, it has a 0 at that point, while b has a 2. Since a is irrational, it also has a 0 at some later point in its expansion (if every digit after that were 2, then a's expansion would be repeating and a would be rational). The number constructed by substituting a 2 for a 0 at that index is greater than a, less than b, and in the Cantor set.
Graphical representation of the proof:
a = 0.......0......
b = 0.......2......
a = 0.......0....0.....
c = 0.......0....2.....
b = 0.......2..........Earlier quoted context omitted.
The set of numbers that can be uniquely defined in the English language in a finite number of letters is a countable set, because the set of finite sequences of English letters is countable.
That's what I would think. But since the set of reals is clearly uncountable, it seems to me that the precise membership of the real numbers cannot be unambiguously defined. There must be uncountably many reals that are not the solution of any equation that can be made using a finite number of characters. But if a "real" number cannot be specified, in what sense does the number exist?
Personally, I'm not a constructivist; I think that these undefinable real numbers exist just as well as the ones that we can define. But that's a philosophical argument and I was never any good at those.
Earlier quoted context omitted.
I'm interested! I think Dedekind cuts are reasonably understandable, but infinitesimals are on the surface of much of our calculus syntax, so I'd be glad to understand where they become so tricky formally.
I wrote a short paper on the topic once upon a time[1] which you may find interesting. It's part history of math, part philosophy of math. It's not a great paper and most of the insights in it come from others but here is some of the arithmetic of nilpotent[1] infinitesimals as shown in the appendix. Imagine an entity which is not equal to zero but that when raised to the power of 2 or higher is equal to zero! Sounds…