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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 $\pi$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.

  • 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 rr is computable if there is a computer program which, when given as input a rational qq (in the form of a pair of integers aa,b b with bb nonzero - intuitively, q=aq = a/bb), will output either YES or NO according to whether qq is in the left Dedekind cut corresponding to rr. The computer program itself is the finite description of the real number rr. Every real number almost anyone has ever heard of is computable, including sqrt{2}√2 and piπ.

This is where model theory comes in. Given a structure M M - for example, the reals with addition and multiplication - we can talk about when an element of M is definable in M*definable in *M**. Note that MM 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 RR, with the "axes" corresponding to (1) what structure we put on R 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 sqrt{2}√2 are each definable by a first-order formula; however, piπ is not.

    • Piπ 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 sqrt{2}√2 and piπ are not.

  • In the reals with addition alone, no number other than 0 is definable - the point being that the map sending xx to -x-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, sqrt{2}√2, while definable, is not given by a term. However, sqrt{2}√2 is definable in a quantifier-free way - e.g. as "x\cdot x=1+1"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.

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 $\pi$ 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.

  • 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 sqrt{2} and pi.

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 sqrt{2} are each definable by a first-order formula; however, pi is not.

    • Pi 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 sqrt{2} and pi 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, sqrt{2}, while definable, is not given by a term. However, sqrt{2} is definable in a quantifier-free way - e.g. as "x\cdot 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.

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.

  • 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.

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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Let'sThere 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 moment grantimplicit 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?

Only numbers which have an exact representation can be said to exist.

ThisWhat it does show is something you've claimed withoutthat 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 $\pi$ 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 even ignoring thatsingle you still don't havenotion which is ultimately satisfying, but rather a real argument herespectrum of notions of exact representation. What(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 sqrt{2} and pi.

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 "representation" mean? You're privileging baseindicate 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 -ten decimal representations for example, but that's very arbitrarythe 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:

  • Why base 10?In the reals with addition and multiplication, the numbers 1/10, 1/3 has, and sqrt{2} are each definable by a terminating basefirst-order formula; however, pi is not.

    • Pi 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 representationare definable by a first-order formula but sqrt{2} and pi are not.

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

  • WhyWhat about when we go even finer than just first-order? In the structure of the real numbers with addition, multiplication, 0, 1, basesand division in(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 first place? "$\sqrtlogical apparatus of Boolean combinations and quantifiers is unnecessary. Meanwhile, sqrt{2}$", while definable, is itselfnot given by a representationterm. However, and it's completely exact and unambiguoussqrt{2} is definable in a quantifier-free way - e.g. as "x\cdot 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.

Even granting the restriction of existence to exact representability in some sense - which I don't really - you're ignoring the real issue of determining what counts as a representation (let alone an "exact" one). And until you address this, your objection against the square root of two on the grounds that it doesn't have a terminating base-ten representation carries no more weight than my objection against 10 on the grounds that it can't be written as a finite linear combination of powers of $\pi$.

Let's for the moment grant that

Only numbers which have an exact representation can be said to exist.

This is something you've claimed without justification, but even ignoring that you still don't have a real argument here. What does "representation" mean? You're privileging base-ten decimal representations, but that's very arbitrary:

  • Why base 10? 1/3 has a terminating base-3 representation - why isn't that good enough?

  • Why bases in the first place? "$\sqrt{2}$" is itself a representation, and it's completely exact and unambiguous.

Even granting the restriction of existence to exact representability in some sense - which I don't really - you're ignoring the real issue of determining what counts as a representation (let alone an "exact" one). And until you address this, your objection against the square root of two on the grounds that it doesn't have a terminating base-ten representation carries no more weight than my objection against 10 on the grounds that it can't be written as a finite linear combination of powers of $\pi$.

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 $\pi$ 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 sqrt{2} and pi.

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 sqrt{2} are each definable by a first-order formula; however, pi is not.

    • Pi 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 sqrt{2} and pi 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, sqrt{2}, while definable, is not given by a term. However, sqrt{2} is definable in a quantifier-free way - e.g. as "x\cdot 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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Let's for the moment grant that

Only numbers which have an exact representation can be said to exist.

This is something you've claimed without justification, but even ignoring that you still don't have a real argument here. What does "representation" mean? You're privileging base-ten decimal representations, but that's very arbitrary:

  • Why base 10? 1/3 has a terminating base-3 representation - why isn't that good enough?

  • Why bases in the first place? "$\sqrt{2}$" is itself a representation, and it's completely exact and unambiguous.

Even granting the restriction of existence to exact representability in some sense - which I don't really - you're ignoring the real issue of determining what counts as a representation (let alone an "exact" one). And until you address this, your objection against the square root of two on the grounds that it doesn't have a terminating base-ten representation carries no more weight than my objection against 10 on the grounds that it can't be written as a finite linear combination of powers of $\pi$.