In mathematics, a real number is a value of a continuous quantity that can represent a distance along a line?! Any real number can be determined by a possibly infinite decimal representation?! You can't blame me if I'm confused as to whether mathematicians think of the Real numbers as integers plus a possibly non-zero decimal part, or if they think Reals are somehow all the quantities in a continuum. I would have thought you'd need to choose one or the other. And this being mathematicians, I would suggest they keep with the decimal numbers. Seems we know what we're talking about there. Less metaphysical, so to speak. How could we even prove that things like the decimal numbers map any continuum? EB
Speakpigeon: Don't rely too much on wikipedia for your mathematical definitions. You're probably better off consulting a textbook if you want precision. As far as I am aware, it's not proven. That's a conjecture known as the continuum hypothesis. There is, however, a rather ingenious proof that there is an uncountable infinity of real numbers (as compared to the countable infinity of the rationals, for instance).
This is consistent with other sources I could look at, including the work of professional mathematicians. Thanks. Yes, I was aware of that. However, I'm not clear why this should play a part. As I see it, proofs should concern constructed sets, like the Integers, the Rationals, the Decimals etc. And then, people would be free anyway to assume that what goes for those also applies to the Reals. Yes, I guess you must talking about Cantor's proof. I have to agree it's "ingenious". Apparently somewhat too ingenious for people at the time when he published it. Still, it seems to be one of the most well-known mathematical proofs. I assume that a whole branch of mathematics has come to grow essentially out of this one theorem. Let's hope it's not only ingenious but also correct. EB
Continuum hypothesis is NOT what Speakpigeon described. The basic idea behind it is there are different infinities. The "smallest" is called countable and describes the number of integers. The next important one is called the continuum and describes the number of reals. The continuum hypothesis states there are none in between these two.
The real numbers are the abstract thingies out there in Platonic land. Each real number has a decimal representation. But a real number is not its representation. There are many ways to represent the same real number, but there is only one abstract real number pointed to by all its representations. For example if I give you a list of left-right directions to traverse your way down the infinite binary tree, your path represents a real number. If I give you a decimal expression, it represents a real number. I could give you the decimal digits of pi, or I could give you the famous Leibniz series pi/4 = 1 - 1/3 + 1/5 - 1/7 + 1/9 - 1/11 + ... Each is a representation of pi, but neither are pi. Pi is the abstract thing they each represent. Like "snow" is not snow, it only represents snow. Same idea but for abstract objects. https://en.wikipedia.org/wiki/Leibniz_formula_for_π But a real number itself is the abstract thing pointed to by all its representations. Just as 2 and 1 + 1 are two different expressions that point to the same abstract thing, namely the number two. Which is yet another representation of the same number. In practice we don't worry too much about the philosophy, and we use whichever representation is useful for the problem at hand.
??? When you claim something about whatever I'm supposed to have said, please provide the relevant quote, or just shut up. Thank you. EB
Yes, I think I can recognise your position here. Clearly, that's one way to look at it which is perfectly legitimate. Still, this isn't my concern. I'm only interested here in whether there is a consensus view among mathematicians on the definition of the Real numbers, and if so, what it is. Indeed, and I'm not interested in the philosophy of it. I could make up my own whenever I would feel like it. My point is that it seems to me it would be a bad idea for mathematicians to be fuzzy about the definition of the mathematical objects there's working with. So, I would definitely expect them to have long decided what the definition of what they call the Real numbers should be. I understand Wiki isn't the reference in this respect and that's not the point either. The point is that if you know the one definition used by all mathematicians, I'm interested. EB
Oh, sure. The reals are generally defined either as Dedekind cuts, or else as equivalence classes of Cauchy sequences. The constructions are standard and taught to undergrad math majors in real analysis class. There are also some alternate constructions of interest. https://en.wikipedia.org/wiki/Dedekind_cut Although there are various constructions, everyone agrees on the axiomatic definition. The real numbers are uniquely characterized as a complete linearly ordered field. http://www-history.mcs.st-and.ac.uk/~john/analysis/Lectures/L5.html
Axiomatic definition is also in the Wiki page. https://en.wikipedia.org/wiki/Real_number#Axiomatic_approach
LOL. You don't get it. I didn't mean a Wiki quote! When I said "please provide the relevant quote", I meant that you ought to have provided the relevant quote of me saying what you are alleging I have said. Is that clear enough now? And wasn't that clear enough to begin with?! Further, I definitely haven't been insulting. You don't seem to know what an insulte is. EB
What I get from this is that real numbers that are not rational numbers can only be identified as consequences of the way we specify operations on them, like addition, multiplication etc. A good example is that of the algebraic numbers. I'm not familiar enough with transcendental numbers except pi, and pi not being a rational seems a direct consequence of how we have to determine its value. OK, that's just a thought but there's something that's bothering me in there I can't articulate. EB
The real numbers are uniquely characterized as a complete, totally ordered field. That's the axiomatic definition. Anything and everything we ever wanted to do with the reals or know about the reals follows from this definition. However, what if someone says, "Well how do we know there even IS such a thing?" the answer is that we can construct a complete, totally ordered field in any of several ways within set theory. That's where the Dedekind cut or Cauchy sequence approach comes in. Having seen such a construction once, for the rest of our lives we can forget those details and just use the axiomatic definition. Now, what is a "complete, totally ordered field?" * Totally ordered means that it's a set that has a total order on it. There is a relation called '<=' such that for any two elements x and y, either x <= y or y <= x. That's a total order. * A field is just a collection of mathematical objects in which we can add, subtract, multiply, and divide. For example the rational numbers are a field. There are finite fields such as the integers mod 5. These are the symbols {0, 1, 2, 3, 4} with addition mod 5. Since for example 2 x 3 = 1 in the integers mod 5, we see that every number (except zero of course) has a multiplicative inverse. So the integers mod 5 are a finite field. They're not ordered though, because if you add 1 to 4 it "rolls over" to 0, so there's no sensible way to define an order relation that's compatible with the addition and multiplication. * Completeness is the defining property of the reals. The rationals are a totally ordered field, but there are Cauchy sequences such as 1, 1.4, 1.41, 1.414, ... that "should" converge to sqrt(2), but that don't converge in the rationals. That's because there's a "hole" in the rationals where sqrt(2) should be. The reals don't have any holes. Every Cauchy sequence (loosely speaking that's a sequence that "should" converge) does actually converge. So if you show me a mathematical collection of objects that's a field, that's totally ordered in a manner consistent with the addition and multiplication, and is complete, then that system is structurally identical to the real numbers. There are no other possibilities. With respect to your question, real numbers that are not rational are identified by the sequences of rationals that converge to them. In other words suppose someone says, "I don't believe pi exists." We can simply say: If you believe in the rationals, then consider the sequence of rationals 3, 3.1, 3.14, 3.141, ... We can if we like DEFINE pi to be this very sequence, or more precisely the equivalence class of all sequences that converge to the same "hole" in the rationals. So the real numbers are the completion of the rationals. If you take the rationals and you throw in all the limits of sequences of rationals, you get the reals.
Finite sums and products of rationals are rational. You can never escape the rationals via their arithmetic properties. The only way to escape the rationals is through infinite sequences.
Yes, I'm reasonably familiar with the notion of field ("Corps" in French, I think), things I remember from my two years at university as student in maths and physics, and topology was my preferred subject. I remember Cauchy very clearly but nothing about the Dedekind Cut. And I'm not clear what's better about Dedekind v. Cauchy. I think people would feel OK with transcendental numbers like pi if you could offer a finite and exact expression. There's no decimal value of the root of 2 that is finite and exact. Many people think it just doesn't make sense. You even have people uncomfortable with 0.333... even though they understand why we have to write it like this. So, broadly, algebraic numbers feel OK because their definition is exact and finite. I don't know enough about transcendental numbers but I suspect you can't say the same about them. Basically, they're all "infinite". Now, in Base 10, 1/3 = 0.333..., so it has an infinite decimal expansion,. But in Base 3, 3 is written 10 and 1/3 in Base 10 becomes 1/10 in Base 3, which is just 0.1, so now a decimal with suddenly a very simple expansion. So, 0.333... in Base 10 is equal to 0.1 in Base 3. Infinity suddenly disappears. And that goes for many other numbers. Essentially, the idea is that infinity may be just a "bad" consequence of the clumsy way we express mathematical relations. That's just an idea, though. And the Dedekind Cut, or any axiomatisation of the Reals I've seen, isn't there to remove this problem, if it is at all a problem. EB
Neither is better than the other. They're just two different set-theoretic constructions of a complete, totally ordered field. They're not the only ones, I've seen others less well-known. Doesn't matter. As along as there's SOME construction within set theory of a complete, totally ordered field, that's all we need. But there are many such. For example the famous Leibniz series pi/4 = 1 - 1/3 + 1/5 - 1/7 + ... You can put that into a closed-form expression using summation notation, and a beginning programmer could code it up and run it. One of the coolest things about this particular series is that it converges incredibly slowly. I wrote a program to calculate it once and it only had 12 decimal places after four days. https://en.wikipedia.org/wiki/Leibniz_formula_for_π That's true, but it's a limitation of decimal representation. Sqrt(2) is the least upper bound of the set of rationals such that x^2 < 2. That's a finite description that uniquely characterizes sqrt(2). That's why there's education. Anyone who cared enough about the subject could read a book on real analysis or just work through the Wiki page on Dedekind cuts. Many people think the earth is flat. What of it? A course in calculus that include geometric series will educate them. The opinions of people uneducated in some particular technical discipline are not arguments against the basic facts of that discipline. "Many people" think someone with a brain tumor is possessed by the Devil. Brain surgeons simply cut out the tumor using their technical knowledge and skill. I agree with you that a lot of people take high school math and maybe a little calculus and are quite fuzzy about the nature of the real numbers. Perhaps your complaint is more about math education than about math, in which case I heartily agree with you. Yes, and in fact every computable number has a "exact and finite" characterization as a computer program. No, pi and e and every other well-known transcendental is computable and therefore has unique characterization as a finite-length computer program. Again, this is a limitation of decimal representation. And it's easily expressed finitely: "Apply the grade school division algorithm to 1 divided by 3." You're talking about decimal representations, not the real numbers themselves. Decimal radix representations in general have this property. But any computable real number has a finite expression. Decimal representation has that limitation. But so what? Well you've certainly identified some of the issues with decimal representation, but a representation of a real number is not the real number; just as the string "snow" is not snow. Correct. The Dedekind cut construction is there to show that we can indeed construct a complete, totally ordered field within set theory.
Computable isn't the same a computed. So, I agree it's a finite characterisation as you say but it's not a finite representation. There's a difference between saying there's somebody living at this address and providing the DNA of the person. Well-known transcendentals may be computable but I doubt most transcendentals are and we apparently know how to characterise more of them, like in the case of the Chaitin omega number, or the so-called Busy Beaver function, which seems to me, provides a blueprint for characterising more such transcendentals. These numbers may not be particularly useful, and it may even be difficult to decide whether they really represent something definite but they are Real numbers and non-computable, that much seems settled. So, for now, I think our notions about Reals are still remarkably muddled. If it's not the same thing then how can we be sure that the Decimals and the Reals are the same thing? As I see it, that's just the beginning. All the real work remains to be done. You have the adresse but you don't have the DNA. EB
I don't know what that means. Distinction without a difference. Of course I gave a finite representation of pi. I linked the Wiki page with the particular finite expression. I showed its expanded form. I can't fathom the point you're trying to make. But something more general is going on. In your most recent several posts you're simply asking about how mathematicians characterize and construct the reals. I've done my best to give you factually correct answers. But now you seem to want to dispute something or other, and I don't think you're making sensible points. Which has what to do with anything? Again, you're arguing about something but I'm not sure what. You asked how mathematicians regard the reals and I've told you. Correct, almost all (all but a set of measure zero) are not computable. There's a relatively small set of numbers that are definable but not computable. That's a subtle distinction. But almost all reals are neither computable nor definable so trying to characterize some of the definable reals won't support whatever point you're trying to make. Yes, there are numbers such as Chaitin's Omega that are definable but not computable. But most reals are neither. If you don't mind my honest observation, I think YOUR notions about the reals are remarkably muddled. Mine are perfectly clear. I've explained several times that they're not the same thing. Decimal expressions are representations of reals, in the sense that the string "snow" is a representation of snow. However, once we construct the reals (as Dedekind cuts or equivalence classes of Cauchy sequence or via any other such construction) we can then PROVE two things: * Every real has a decimal expansion (and some have two); and * Every decimal expansion corresponds to some real number (and sometimes two distinct decimal expressions correspond to the same real). The proofs are standard and not very difficult. So we are justified in freely conflating reals and decimal expressions; although when we put on our philosopher hats, we realize that one is an abstract thing and the other is a representation. I get that you have some philosophical unease, but I don't think you've expressed it with sufficient clarity for me to understand it.
