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Consider the fraction 10/3.

One way to interpret this is by stating the following: in the process of long division, which is a rule, take divisor as 3, and take dividend as 10, and initiate the process of division. Quotient generated will be the number that the above expression means. I believe there isn't anything more concrete than this definition of the expression '10/3'.

That is, this expression effectively indicates what to do with 10 & 3. Now if a computer is given input and is made to compute, then 3.33333333...……. will be generated, and it will go on infinitely. It can never represent 10/3 fully. That is, 10/3 doesn't exist on the number line, because it cannot be marked. And it cannot be marked not because of constraints of time or resources, but because of its very nature -what it means.

Same is the case with sqrt(2). It effectively means what inputs should go into the algorithm which computes square root of a number. For convenience we do not compute these 'incomputable' entities, and use these indicators to use them in our calculation.

This demonstrates that these numbers do not exist on number line, but we say number line contains all 'real' numbers. How can it 'contain' something if it doesn't exist on the line.

My question therefore is, in what sense non-terminating decimals exist, and in what sense are they different from the concept of infinity.

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    10/3 terminates in base 3. You're confusing numbers with their representation. – user4894 Jul 13 at 15:39
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    Aside: 3.33333333... is an exact representation of 10/3, and computers can be programmed to understand it. For example, wolfram alpha recognizes this notation. It also gives a similar answer if you ask for the value of 10/3. – Hurkyl Jul 13 at 18:35
  • it is possible to draw up a geometrical construct which has, as one of its lengths, a number whose decimal expansion either repeats forever or is irrational. If one of the other lines in the construct represents the number line, then you have geometrically mapped onto it a infinitely repeating decimal or an irrational number. – niels nielsen Jul 14 at 2:18
  • You claim that 10/3 does not exist on the number line, but every limiting approximation of it provides smaller and smaller neighbourhoods in which the point exists (e.g, 3, 3.3, 3.33, 3.333, etc). That is, even if we were not able to point to it 'exactly', it provably exists in these neighbourhoods. – Panda Jul 29 at 12:55
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There are two major issues here: the connection between existence and exact representability, and the notion of exact representability itself.

There is no justification given for the implicit claim that only numbers which have exact representations can exist. The OP merely makes the claim that numbers which can't be "marked" don't exist. This is not something that can be taken for granted. Even assuming we settle on a notion of "exact representation," why should that correspond to existence?

What it does show is that we understand different numbers to different degrees or in different ways. And while that's really interesting - see the section below - it's not really surprising.

In the absence of a justification for your claim there isn't really much to be said here. Instead, let me point out that in mathematics, existence claims are made and justified within precise contexts; e.g. ZFC proves, in a precise sense, that π exists, and this isn't questioned by anybody. Things get fraught when we step out of such contexts and try to talk about what mathematical existence really is. There is in my opinion a lot of interesting stuff to be said here (along with a truly gargantuan amount of pointless fluff), but one has to be very careful to recognize the non-universality of one's own underlying philosophical assumptions.

Now on to the other part of the question - what constitutes an "exact representation" of a number?

Mathematical logic, especially computability theory and model theory, provide a series of reasonable answers to the question - and more importantly, in my opinion, they make a convincing case for the idea that there is no single notion which is ultimately satisfying, but rather a spectrum of notions of exact representation. (All of this serving, unsurprisingly, serving to push back against the unjustified existence~representability connection.)

  • Let's start with a very permissive one: computable real numbers. According to this picture, we take as initially given the rational numbers, and will conflate a real number with its left Dedekind cut - the downwards-closed set of rational numbers whose supremum is the real number in question. (There are other equivalent approaches we can take here, e.g. via fast Cauchy sequences, but I like Dedekind cuts.) Identifying a real with a set of rationals, the question then becomes how complicated is that set, and computer programs provide a very satisfying answer to this question: a real r is computable if there is a computer program which, when given as input a rational q (in the form of a pair of integers a, b with b nonzero - intuitively, q = a/b), will output either YES or NO according to whether q is in the left Dedekind cut corresponding to r. The computer program itself is the finite description of the real number r. Every real number almost anyone has ever heard of is computable, including √2 and π.

Now this is the one that I favor, if I'm forced to choose, but it may seem too permissive; although the Church-Turing thesis does indicate that it is quite robust (as does the fact that we can replace the rationals as our "starting set" with many other things, like the algebraic numbers, without affecting the resulting definition), it may feel like it's talking about a "computational analogue" of the real line as opposed to the real line itself.

This is where model theory comes in. Given a structure M - for example, the reals with addition and multiplication - we can talk about when an element of M is *definable in *M**. Note that M depends on more than just its underlying set. We can also draw distinctions between formulas used in definitions, especially via syntactic considerations - e.g. first-order formulas, quantifier-free first-order formulas, monadic second-order formulas, ...

This gives a "two-dimensional array" of notions of representation of elements of R, with the "axes" corresponding to (1) what structure we put on R (addition and multiplication? just addition? addition and the number "1" labelled? addition, multiplication, and modular forms?) and (2) what sort of logic we use to formulate our definitions. For example:

  • In the reals with addition and multiplication, the numbers 1/10, 1/3, and √2 are each definable by a first-order formula; however, π is not.

    • π is, however, definable in this structure by a computable infinitary formula, as is every computable real (but the converse is not true). Finally, every real number is definable in this structure by an infinitary formula. But I think there are good reasons for considering non-first-order definitions to be not very satisfying, so from now on I'll just focus on first-order definitions.

  • In the reals with addition and 1 as a distinguished element, but without multiplication, the numbers 1/10 and 1/3 are definable by a first-order formula but √2 and π are not.

  • In the reals with addition alone, no number other than 0 is definable - the point being that the map sending x to -x is an automorphism of this structure, and definable elements cannot be moved by automorphisms.

  • What about when we go even finer than just first-order? In the structure of the real numbers with addition, multiplication, 0, 1, and division (either as a partial function or with the convention that division by zero results in zero, or something similar), 1/10 and 1/3 are each given by terms: the logical apparatus of Boolean combinations and quantifiers is unnecessary. Meanwhile, √2, while definable, is not given by a term. However, √2 is definable in a quantifier-free way - e.g. as "x · x = 1 + 1." Tarski showed that everything definable in this structure is definable by a quantifier-free formula, so in general this is something we'll always run into: we're never going to do much worse than term-definable without going fully undefinable.

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  • What kind of representation do you think would qualify as an objective representation of numbers? As for sqrt(2), I don't think it is something concrete. It is just a representation of a representation. What it really says is the following: In this system of representation, we define x * x = 2, and let's call x = sqrt(2) [symbol]. It is exact because we have defined it to be so -and if you say that denary system is itself a definition-a rule, I say yes, but now we are building more rules over it, and we are further away from exactness -unless you show that it collapses down to same level. – Ajax Jul 14 at 5:52
  • @Ajax I honestly don't understand your objection here, especially the last sentence - I can't follow what you're getting at there. First, how does the complexity of a definition affect its exactness? Second, what are these "levels" you're talking about? All of this seems very ad hoc - we grow up with positional representation systems so they must be right. This isn't to say that there aren't interesting distinctions to be made between representation systems - but you're trying to form a link (which you still haven't even tried to justify) between representability and existence. (continued) – Noah Schweber Jul 14 at 18:45
  • So you need to give some argument why your idea of representability is maximal - not just that other representations are less concrete, but that they are so much less concrete that they no longer entail existence. Incidentally, I think that removing the existential aspect from this question would result in something much more interesting, along the lines of "Are there meaningful ways to distinguish between (some of) ${1\over 10}$, ${1\over 3}$, $\sqrt{2}$, and $\pi$ (for example) in terms of 'concrete representability'?" That's already interesting, and doesn't lean on a strange assumption. – Noah Schweber Jul 14 at 18:48
  • On that note, and re: the first sentence of your comment, I'd say: the most obvious notion of representation here, in my opinion, comes from computability theory. We start by taking the rationals as given (which seems reasonable), and think of real numbers as Dedekind cuts. A real r is computable iff there is a (finite!) computer program which determines its Dedekind cut exactly: that is, such that when I input a code for a rational number q (that is, a pair of integers a,b with b nonzero - representing a/b), the program eventually halts and outputs either YES or NO, outputting YES iff q<r ... – Noah Schweber Jul 14 at 18:52
  • All the real numbers you've ever heard of are computable in this sense. Logic provides other notions too - for example, we can talk about definability in a structure. In the structure RF=(\mathbb{R}; +,\times), the reals 1/3, 1/10, and sqrt{2} are each definable; however, pi is not. So in that more restrictive sense, pi is meaningfully less definable than sqrt{2}. And RF has a nice logical property of decidability - roughly, definability in RF implies computability but not conversely. (cont'd) – Noah Schweber Jul 14 at 18:58
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The issue is with the exprsssion "number line".

Consider the cartesian plane with axis x and y; take the point P of coordinates (1,1) and with a compass draw a circle centered at (0,0) and through P.

This circle will intersect the x-axis exactly in one point : for sure on the line there is one point : what are its coordinates ?

All amounts to this : the assumption that to every point on the line we can associate a "suitable" number. This is the source of modern concept of real numbers.

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