Kinds of Numbers

[Bomb#20]

The reason why there is a notion "countable" is because there is an oppsite notion "uncountable" for sets where each pair of elements always has at least one element between them. (As for example real numbers)

Every pair of rationals has at least one element between them, and that's a countable set.

 

[Juma]

The reason why there is a notion "countable" is because there is an oppsite notion "uncountable" for sets where each pair of elements always has at least one element between them. (As for example real numbers)

Every pair of rationals has at least one element between them, and that's a countable set.

You're right... I shouldnt have tried to make an easy explanation of uncountable without checking it up...

Anyway, the set of all reals is a good example of an uncountable set.

A weird example is the cantor set...

 

[Speakpigeon]

Well, if you can't understand the process of counting on one's fingers, I will have to give up on you. Kids do it and humanity arguably started with it. What's wrong with you?
EB

People can do lots of things even when they don't understand the underlying process. I'm fairly certain you have very little understanding of the specifics of how your posts appear on my screen, but that keeps happening anyway.

My simple point is that what people call "counting" is closely related to the actual experience of counting on one's fingers or any similar setup.

The more difficult question is that the kind of counting people are familiar with is construed as a process that stops at some point in time, at which point you know the count.

There's a more subtle point point but these two are good enough for those who may be interested to think about it, which may not be you.

In case you weren't counting, this is the second time I'm asking to see if you understand the underlying process behind counting. You haven't given me much hope for your understanding thus far. Hint: It has very little to do with fingers.

I don't see how you could possibly prove that. But I'm listening. Give me a proof, or even just a compelling argument that counting as people understand it has "very little to do with fingers".

Take some time and think about it. And this time, see if you can try to actually be precise.

Sorry, I don't see what you are waiting for. I already said all that was necessary.
EB

 

[Speakpigeon]

Counting on one's fingers is a simple form of one-to-one correspondence: placing the entities that one is counting in 121C with one's fingers.

Exactly. I'm pleased one of you two isn't so ayatollah.

Also, in mathematics, it's irrelevant how long one needs to do something.

Only if you don't actually do it. But you therefore understand my point that any finite set could be counted at some point in time, at least in principle. An infinite set will never finish being counted in the ordinary sense of "counted", like "on one's fingers".

My point is also not about mathematics. I'm saying the mathematical sense of counted you use is not the same as the ordinary sense of counting on one's fingers. None of you have addressed these points so far.

Mathematicians had some vehement controversies over mathematical infinities in the late 19th and early 20th centuries, but those controversies seem to have died down. The controversies were, as far as I can tell, controversies about the legitimacy of various operations involving infinite sets. Georg Cantor was rather loudly criticized by his colleague Leopold Kronecker for violating what LK considered mathematical propriety. LK believed that the only real mathematical objects were those that could be constructed in a finite number of steps from the natural numbers, and GC's infinite sets violated that principle.

Yes.

Countably infinite sets are a mild violation of that principle, since each one of their elements can be constructed in a finite number of operations. That is evidently true for the natural numbers, and with 121C rules, that is also true for every other countably-infinite set.

This is what I dispute. To actually count a set, at least in the ordinary sense, is to provide the number of elements it contains. You can't do that with infinite sets.

Uncountably infinite sets are a stronger violation, since they contain some elements that cannot be constructed in a finite number of operations.

Good.

BTW, "countable" is often used as shorthand for "countably infinite".

And the question remains of whether this sense has anything to with the ordinary notion of counting.
EB

 

[Speakpigeon]

That's not the controversy here, though -- nobody's denying that finding a one-to-one correspondence is a legitimate method of counting a set. The point is that it isn't counting in the ordinary sense -- it isn't what nonmathematicians mean when they say "count". When we say we've counted the rationals and there are aleph₀ of them we're extending the concept, every bit as much as when we call "-1" a number.

Counting on one's fingers is much more than a simple form of one-to-one correspondence. Suppose you count these things on your fingers:

A B C D E

What you get when you place them into 121C with fingers is a collection of labels, with each letter having a different finger as its label. There are 10! / 5! ways to do that. Even assuming you always use up your fingers in the same order there are 5! ways to do it, 120, about 2⁷. When you finish creating your 121C you've accumulated 7 bits of information. When you report the result of counting the set you only deliver 3 bits: 101. That lossy information compression step is part of what ordinary people mean by "count" -- if somebody asks you to count them and you reply with "A-little D-ring B-middle E-index C-thumb", you're going to get a puzzled look.

In common-usage "counting", the compression step is to simply report the last correspondence in your 121C and throw away the rest of the information. What number had you counted up to when you ran out of elements? That's why having a last element is an essential part of counting in the ordinary sense. It's great that Cantor et al. figured out that we don't actually need that step and can open up a new world of interesting math using just the 121C; but let's not forget that when we made that conceptual advance we were also making a conceptual shift, and changing the language.

BTW, "countable" is often used as shorthand for "countably infinite".

Where "countably" is simply an assertion that the set is countable. It is in our sense; it isn't in ordinary people's sense. Consider that a set doesn't even need to be infinite for people to call it "countless".

All good to me. Excellent!
EB

 

[Speakpigeon]

So "countable" ought to mean "finite"?

Not necessarily. People use whatever language they want to use and good for you. My point is that it is ludicrous to use a different sense for a word in common use and then quip at people having difficulty understanding what you say! In fact, it's plain idiotic. If you wanted to be understood by non-specialists, you would start by using the same meaning as they do. It's much more simple that way. So, yes, you could avoid saying "countable" for infinite sets and use some more appropriate term. But, hey, it's not going to happen, right?
EB

 

[Speakpigeon]

That's right, the time it takes doesn't matter. But counting takes time and at some point you stop. How long would it take to count to infinity?
Waster.
EB

It would take you the same time as to count 1E100 apples: the rest of your life and you would still not be finished.

The point, that obviously was to fine to be noticed is that this has nothing to do with the timeconsuming human act of counting real objects.

The reason why there is a notion "countable" is because there is an oppsite notion "uncountable" for sets where each pair of elements always has at least one element between them. (As for example real numbers)

Well, I just ran out of time!
EB

 

[Juma]

It would take you the same time as to count 1E100 apples: the rest of your life and you would still not be finished.

The point, that obviously was to fine to be noticed is that this has nothing to do with the timeconsuming human act of counting real objects.

The reason why there is a notion "countable" is because there is an oppsite notion "uncountable" for sets where each pair of elements always has at least one element between them. (As for example real numbers)

Well, I just ran out of time!
EB

Why even respond when you have nothing to say? Why being such a jerk?

 

[lpetrich]

Now to algebraic numbers.

Let us consider some algebraic number defined as a root of some rational-coefficient polynomial. If it is a real one, then it can be found by using some approximation algorithm like  Newton's method. If one starts with a rational initial guess, and if it converges, it will make a rational-value Cauchy sequence.

Now consider one with algebraic-number coefficients, one which I'll call P. If those coefficients contain powers of some r that is not a rational number, a r given by some nth-degree rational-coefficient polynomial. Take powers P, P^2, ..., P^n, and divide by r's defining polynomial, leaving the remainder. These n quantities will then be polynomials in r with a degree of at most n-1. Treat the r powers as separate variables and solve for them. This will leave a polynomial with no powers of r in it.

Repeat this operation until one is left with all rational coefficients. That proves that the algebraic numbers are algebraically closed, that every algebraic-coefficient polynomial has an algebraic-number solution.

That is also true of complex (real) numbers: the  Fundamental theorem of algebra.

-

Now for defining arithmetic on algebraic numbers. One can do that without calculating the roots explicitly. One uses  Newton's identities, an interrelationship between "elementary symmetric polynomials" (sum of all products of a certain number of distinct elements of some set) and "power sums" (sum of a certain power of all elements of some set).

For r1, r2, ..., rn,
R(ESP):
1
r1 + r2 + ... + rn
r1*r2 + r1*r3 + ... + r(n-1)*n
...
R(PS):
n
r1 + r2 + ... + rn
r1^2 + r2^2 + ... + rn^2
...


To get the ESP's of the roots from a polynomial's coefficients, do
a(ESP,k) = (-1)^k * a(n-k)/a
with
a(ESP,k) = 0 for k > n

One then uses Newton's identities to calculate a(PS,k) to as high a value as one needs.
To get a polynomial back, reverse these operations.

I'll use A(k) for a(PS,k) in what follows.

For addition, x = a + b, the roots rx = ra + rb and their powers rx^k = sum over l from 0 to k of k!/l!/(k-l)!*ra^(k-l)*rb^l
For multiplication, x = a*b, the roots rx = ra*rb and their powers rx^k = ra^k * rb^k
Since the polynomials may have several roots, one sums over all selections of a root from each polynomial. This gives us

a + b: X(k) = sum over l from 0 to k of k!/l!/(k-l)! * A(k-l) * B(l)
a*b: X(k) = A(k)*B(k)

These operations are obviously commutative, and for associativity, one does addition or multiplication of A,B, and C:
a + b + c: X(k) = sum over l, m from 0 to their sum:k of k!/l!/m!/(k-l-m)! * A(k-l-m) * B(l) * C(m)
a*b*c: X(k) = A(k)*B(k)*C(k)

Multiplication being distributive over addition is easy.
a*(b+c): X(k) = sum over l from 0 to k of k!/l!/(k-l)! * A(k) * B(k-l) * C(l)
a*b+a*c: X(k) = sum over l from 0 to k of k!/l!/(k-l)! * A(k-l) * B(k-l) * A(l) * C(l)

Zero: A(0) = 1, A(k) = 0 for k > 0 -- additive identity, multiplicative zero
One: A(k) = 1 -- multiplicative identity

When one adds or multiplies the roots of polynomials a and b, the resulting number of roots is (number of roots of a) * (number of roots of b). That means that if one is looking for a specific root, one has to select it out. One can do that with a Cauchy sequence for approximating it.

However, this detail is shows that the algebraic numbers are closed under arithmetic operations -- one can construct a rational-coefficient polynomial that contains the sum or the product of two rational-coefficient polynomial roots.

 

[Speakpigeon]

I think it's better to say that natural numbers are only virtually (or potentially, notionally, theoretically etc.) countable. Because nobody ever actually counted them and we all accept we couldn't count them even if we had all the time in the world. Even if the universe was infinite and there was infinitely many people. So, basically, mathematicians are playing loose with the word "countable". Still, it's not so different from politicians and salesmen so it must be Ok and we all have to make a living.
EB.

 

[Juma]

I think it's better to say that natural numbers are only virtually (or potentially, notionally, theoretically etc.) countable. Because nobody ever actually counted them and we all accept we couldn't count them even if we had all the time in the world. Even if the universe was infinite and there was infinitely many people. So, basically, mathematicians are playing loose with the word "countable". Still, it's not so different from politicians and salesmen so it must be Ok and we all have to make a living.
EB.

Nah. Its you that are "playing loose". How the word "countable" is used in mathematics is perfectly clear.

 

[Speakpigeon]

I think it's better to say that natural numbers are only virtually (or potentially, notionally, theoretically etc.) countable. Because nobody ever actually counted them and we all accept we couldn't count them even if we had all the time in the world. Even if the universe was infinite and there was infinitely many people. So, basically, mathematicians are playing loose with the word "countable". Still, it's not so different from politicians and salesmen so it must be Ok and we all have to make a living.
EB.

Nah. Its you that are "playing loose". How the word "countable" is used in mathematics is perfectly clear.

Yeah, sure; it's creative countability.
EB