A Complex Numbers Not Necessary in QM: Explained

[Demystifier]

[Note from mentor: This was split off from another thread, which you can go to by clicking the arrow in the quote below]

2. To explain why complex numbers are necessary.

Actually they are not. See https://www.amazon.com/dp/3319658662/?tag=pfamazon01-20 Sec. 5.1.

 

[Peter Hearty]

Actually they are not. See https://www.amazon.com/dp/3319658662/?tag=pfamazon01-20 Sec. 5.1.

That book's been on my reading list for some time. One reason, although not a major one, is because of the complex numbers stuff. I presume something else must be used to accommodate the extra degrees of freedom that complex numbers gives, either via multiple simultaneous equations, or perhaps Clifford algebras.

Perhaps I should have said "why complex numbers are used", rather than "needed".

 

[atyy]

Actually they are not. See https://www.amazon.com/dp/3319658662/?tag=pfamazon01-20 Sec. 5.1.

Presumably one can correct it by saying they are needed up to isomorphism (or whatever the proper rigorous term is).

 

[Demystifier]

Presumably one can correct it by saying they are needed up to isomorphism.

Then classical Maxwell equations also need complex numbers up to isomorphism.

 

[atyy]

Then classical Maxwell equations also need complex numbers up to isomorphism.

In how many space + time dimensions?

 

[PeroK]

Actually they are not. See https://www.amazon.com/dp/3319658662/?tag=pfamazon01-20 Sec. 5.1.

In my view the question over whether to admit the use the complex numbers doesn't have the physical or mathematical significance that some people like to think it has. You could avoid calling them "numbers" or you could find a way to conduct the mathematics without anything that has the mathematical properties of complex numbers. But, really, why worry about complex numbers in particular?

Moreover, in a formal development of numbers it is the real numbers where a lot of the problems lie. And, especially, if you try to justify the real numbers as something physical. For example, a typical real number is indescribable, in the sense that it requires an infinite amount of information to quantify it. If you do something like take the position, $x$, of a particle at some time, $t$, where $x, t$ are real numbers, then you have something quite hard to physically justify. There is no way to write down a typical real number.

And yet, in all the physics I've some across measurement values are assumed to be real numbers. A formal justification of this would not be easy in my opinion.

Why, then, does no one try to do physics based on rational numbers and countable sets? I imagine that it is possible. But, I suspect that the sacrifice in losing calculus and much else besides is too much. Whereas, the sacrifice in losing the complex numbers is perhaps more manageable.

 

[atyy]

Why, then, does no one try to do physics based on rational numbers and countable sets? I imagine that it is possible. But, I suspect that the sacrifice in losing calculus and much else besides is too much. Whereas, the sacrifice in losing the complex numbers is perhaps more manageable.

The biggest obstacle is the chiral lattice fermion problem. Even if the chiral fermion problem is solved, one would still have to discretize the vector space, so it wouldn't be a complete solution. But for the present, we even fail at space discretization because of the chiral lattice fermion problem.

 

[vanhees71]

Actually they are not. See https://www.amazon.com/dp/3319658662/?tag=pfamazon01-20 Sec. 5.1.

Of course, you can express anything in real numbers what you can express with complex numbers, but why should you do so? It's more cumbersome at best. If you wish you can do all of analysis with natural numbers too; after all the real numbers are built step by step from the natural numbers via first algebraic then topological "completion"; but then it becomes a real nuisance ;-))).

 

[A. Neumaier]

in all the physics I've some across measurement values are assumed to be real numbers. A formal justification of this would not be easy in my opinion.

In electrical engineering, using complex observables is common. Of course, you can reduce everything complex to real by doubling the number of variables but often this is inefficient, and one loses all nice properties of complex numbers (analytic continuation, etc.) that are very important in the applications. Reducing to rationals is not possible, for you may measure the diagonal of a rectangle by measuring its sides and applying the theorem of Pythagoras, or the area of a circle by measuring its radius - and you have irrationals and even transcendentals...

 

[PeroK]

In electrical engineering, using complex observables is common. Of course, you can reduce everything complex to real by doubling the number of variables but often this is inefficient, and one loses all nice properties of complex numbers (analytic continuation, etc.) that are vvery important in the applications. Reducing to rationals is not possible, for you may measure the diagonal of a rectangle by measuring its sides and applying the theorem of Pythagoras, or the area of a circle by measuring its radius - and you have irrationals and enven transcendentals...

... but, the point is that triangles, circles and all of Euclidean geometry are a mathematical abstraction. You could argue that these are just as much abstract mathematical tools as the complex numbers. In a way, it's not important whether physics works without real numbers or not. They are in many ways harder to justify physically then the complex numbers. My question is simply: why pick on the complex numbers for removal from QM?

 

[A. Neumaier]

... but, the point is that triangles, circles and all of Euclidean geometry are a mathematical abstraction. You could argue that these are just as much abstract mathematical tools as the complex numbers. In a way, it's not important whether physics works without real numbers or not. They are in many ways harder to justify physically then the complex numbers. My question is simply: why pick on the complex numbers for removal from QM?

Measurements are also mathematical abstractions.

In simple cases of everyday life, we see that a pointer is close to a particular bar of a scale, and translate that into a measurement value by an abstract process of estimating the distance to the next two bars, figuring out the label that these bars should have, from the few labels given, and then proceeding to linear interpolation. The result is our claimed measurement result.

To measure the (complex-valued) refractive index of an optical material, say, is a much more elaborate process. Scientific measurements are quite complex - some literally, most others even when they do not use complex numbers!

 

[A. Neumaier]

There is no way to write down a typical real number.

Here are some typical real numbers, written down in some unambiguous way: $0$, $1$, $-1$, $2/3$, $\sqrt{2}$, $\pi$, $(1+\sqrt{5})/2$, $1.05$, $3\cdot 10^5$, $10^{-12}$, etc.

 

[Demystifier]

In how many space + time dimensions?

3+1.

 

[vanhees71]

In how many space + time dimensions?

In (1+3) of course. The Riemann-Silberstein notation is indeed sometimes a very elegant and manifestly covariant description of Maxwell's equations, using the group isomorphism between the proper orthochronous Lorentz group and $\text{SO}(3,\mathbb{C})$.

 

[PeroK]

Here are some typical real numbers, written down in some unambiguous way: $0$, $1$, $-1$, $2/3$, $\sqrt{2}$, $\pi$, $(1+\sqrt{5})/2$, $1.05$, $3\cdot 10^5$, $10^{-12}$, etc.

Those are very atypical real numbers. Even the set of algebraic numbers is countable. To add to that we have a few special transcendental numbers and that's it.

 

[Demystifier]

But for the present, we even fail at space discretization because of the chiral lattice fermion problem.

I don't think that it is such a big problem. The Wilson method removes the doublers by a rather simple method. Essentially, one adds to the Lagrangian a discretized version of
$$a\partial^{\mu}\bar{\psi} \partial_{\mu}\psi$$
where $a$ is the lattice spacing. Sure, it violates the chiral symmetry, but so what? Lattice violates also the Lorentz, the rotational and the translational symmetry, and yet nobody gets too excited about it.

 

[Demystifier]

Here are some typical real numbers, written down in some unambiguous way: $0$, $1$, $-1$, $2/3$, $\sqrt{2}$, $\pi$, $(1+\sqrt{5})/2$, $1.05$, $3\cdot 10^5$, $10^{-12}$, etc.

The set of all "typical" real numbers is a set of measure zero.

 

[PeroK]

I just checked and even the set of computable numbers is countable. I guess the interesting question is whether you could do physics using the computable numbers instead of the real numbers?

 

[A. Neumaier]

Those are very atypical real numbers. Even the set of algebraic numbers is countable. To add to that we have a few special transcendental numbers and that's it.

I don't share your notion of typicality. The real numbers I gave are very typical when viewed in terms of the usage in the literature on mathematics, physics, and engineering.

I just checked and even the set of computable numbers is countable. I guess the interesting question is whether you could do physics using the computable numbers instead of the real numbers?

The set of definable real numbers (or any other mathematical objects) is countable and has all the properties of the reals (or the objects in question). It gives a countable model of them. This is called Skolem's paradox, but is not a real paradox. It has the consequence that one may restrict without loss of generality to the definable numbers. The ones I gave are typical examples, though one can easily give more complicated ones, e.g., ''the smallest prime with $10^{10}$ decimal digits''.

 

[PeroK]

I don't share your notion of typicality. The real numbers I gave are very typical when viewed in terms of the usage in the literature on mathematics, physics, and engineering.

The set of definable real numbers (or any other mathematical objects) is countable and has all the properties of the reals (or the objects in question). It gives a countable model of them. This is called Skolem's paradox, but is not a real paradox. It has the consequence that one may restrict without loss of generality to the definable numbers. The ones I gave are typical examples, though one can easily give more complicated ones, e.g., ''the smallest prime with $10^{10}$ decimal digits''.

That's fascinating. But, from what I understand Skolem's paradox does not mean that we can treat the reals as though they are countable. And assume that analysis and calculus can be done unimpaired by issues of uncountablilty.

 

[A. Neumaier]

That's fascinating. But, from what I understand Skolem's paradox does not mean that we can treat the reals as though they are countable. And assume that analysis and calculus can be done unimpaired by issues of uncountability.

Reals are intrinsically uncountable by Cantor's diagonal argument. But they are meta-countable (as objects talked about on the metalevel), and that is what counts on the level of usage.

 

[vanhees71]

The set of definable real numbers (or any other mathematical objects) is countable and has all the properties of the reals (or the objects in question). It gives a countable model of them. This is called Skolem's paradox, but is not a real paradox. It has the consequence that one may restrict without loss of generality to the definable numbers. The ones I gave are typical examples, though one can easily give more complicated ones, e.g., ''the smallest prime with $10^{10}$ decimal digits''.

The definable real numbers are not contable, as famously proven by Cantor. The real numbers are defined as the unique Archimedean ordered complete field of numbers (via Dedekind cuts or equivalence classes of Cauchy series with the standard topology).

 

[A. Neumaier]

The definable real numbers are not contable, as famously proven by Cantor. The real numbers are defined as the unique Archimedean ordered complete field of numbers (via Dedekind cuts or equivalence classes of Cauchy series with the standard topology).

The reals are not countable by Cantor's diagonalization argument. But most real numbers are not definable: Each definable real number is defined by a finite formula, and these formulas form a countable set. Thus the definable reals from a countable set only.

 

[vanhees71]

This is a very strange interpretation of "definable". I thought the modern definition of the real numbers is as I said above precisely for the reason not to be dependent on the necessarily only finite (and thus incomplete in the sense of the mathematical real numbers) "formulas". Of course, you cannot write a number like $\pi$ in terms of the usual decimal places, but it's still a well defined real number (e.g., by defining $\pi/2$ as the smallest positive solution of $\cos(\pi/2)=0$, which of course you cannot solve exactly with a finite algorithm or notation).

 

[A. Neumaier]

This is a very strange interpretation of "definable". I thought the modern definition of the real numbers is as I said above precisely for the reason not to be dependent on the necessarily only finite (and thus incomplete in the sense of the mathematical real numbers) "formulas". Of course, you cannot write a number like $\pi$ in terms of the usual decimal places, but it's still a well defined real number (e.g., by defining $\pi/2$ as the smallest positive solution of $\cos(\pi/2)=0$, which of course you cannot solve exactly with a finite algorithm or notation).

My usage of definable real numbers is standard; see the link!

There is a difference between defining the set of real numbers and defining individual real numbers. To use a number it must be defined as a particular number from the set of all possible numbers. $\pi$ is a definable real number, defined by a finite formula, e.g. $\pi=\int_{-1}^1 (1-x^2)^{-1/2}dx$. On the other hand, only countably many among the uncountably many real numbers are definable. This includes all algebraic numbers and many transcendentals, such as $e$ and $\pi$.

 

[Demystifier]

This is a very strange interpretation of "definable". I thought the modern definition of the real numbers is as I said above precisely for the reason not to be dependent on the necessarily only finite (and thus incomplete in the sense of the mathematical real numbers) "formulas". Of course, you cannot write a number like $\pi$ in terms of the usual decimal places, but it's still a well defined real number (e.g., by defining $\pi/2$ as the smallest positive solution of $\cos(\pi/2)=0$, which of course you cannot solve exactly with a finite algorithm or notation).

The $\pi$ of course is definable. But to define any conrete number, you must use some language (English, mathematical language, or whatever). A definition can be viewed as a sentence in the chosen language. The set of all possible sentences is countable, hence the set of all possible definitions is countable.

 

[PeroK]

My usage of definable real numbers is standard; see the link!

There is a difference between defining the set of real numbers and defining individual real numbers. To use a number it must be defined as a particular number from the set of all possible numbers. $\pi$ is a definable real number, defined by a finite formula, e.g. $\pi=\int_{-1}^1 (1-x^2)^{-1/2}dx$..

Is that integral with respect to all real numbers $x$ or only the ones that are definable?

 

[A. Neumaier]

Is that integral with respect to all real numbers $x$ or only the ones that are definable?

It is the standard Lebesgue integral, over all real numbers in $[-1,1]$. These real numbers are anonymous (the range of an integration variable), not defined ones.

 

[Demystifier]

Since we are now talking about continuum in physics, I think anyone interested in this topic should read https://arxiv.org/abs/1609.01421

 

[DarMM]

I think another version of this question is why does QM use amplitudes rather than dealing in probabilities directly?

A lot of work in SIC-POVMs and quantum information has it that this comes from QM being a probability theory with multiple sample spaces related via the uncertainty principle (rather than a single space like Kolmogorov Probability theory).

Within a single QM sample space you have the normal law of total probability relating the outcomes of two random variables:
$$P(B_j) = \sum^{m}_{i} P(B_j|A_i)P(A_i)$$
just as in Kolmogorov probability.

However between two of QM's sample spaces the law of total probability gets modified by an additional term:
$$P(B_j) = \sum^{m}_{i} P(B_j|A_i)P(A_i) + \sum_{k<m} 2cos\left(\theta_k\right)\sqrt{P(A_k)P(B_j|A_k)P(A_m)P(B_j|A_m)}$$
with $\theta_k$ measuring the angle between sample spaces (or "contexts" in Quantum Information language), i.e. a measure of how much Bayesian updating within one sample space updates the probability distributions in another. Complex number amplitudes are then just an alternate more compact way of encoding these Probabilities and the angles of interference between their contexts. However you could if you wanted use purely real numbers and deal with probabilities directly.

Another way of saying it is that QM's use of multiple sample spaces introduces the concept of the relation between these spaces. This is expressed as interference between their probability distributions as measured by the angle between the spaces. Thus complex numbers represent a geometric element to the probabilities in QM that isn't present in Kolmogorov probability.

 

[atyy]

I think another version of this question is why does QM use amplitudes rather than dealing in probabilities directly?

A lot of work in SIC-POVMs and quantum information has it that this comes from QM being a probability theory with multiple sample spaces related via the uncertainty principle (rather than a single space like Kolmogorov Probability theory).

Within a single QM sample space you have the normal law of total probability relating the outcomes of two random variables:
$$P(B_j) = \sum^{m}_{i} P(B_j|A_i)P(A_i)$$
just as in Kolmogorov probability.

However between two of QM's sample spaces the law of total probability gets modified by an additional term:
$$P(B_j) = \sum^{m}_{i} P(B_j|A_i)P(A_i) + \sum_{k<m} 2cos\left(\theta_k\right)\sqrt{P(A_k)P(B_j|A_k)P(A_m)P(B_j|A_m)}$$
with $\theta_k$ measuring the angle between sample spaces (or "contexts" in Quantum Information language), i.e. a measure of how much Bayesian updating within one sample space updates the probability distributions in another. Complex number amplitudes are then just an alternate more compact way of encoding these Probabilities and the angles of interference between their contexts. However you could if you wanted use purely real numbers and deal with probabilities directly.

Another way of saying it is that QM's use of multiple sample spaces introduces the concept of the relation between these spaces. This is expressed as interference between their probability distributions as measured by the angle between the spaces. Thus complex numbers represent a geometric element to the probabilities in QM that isn't present in Kolmogorov probability.

Is this the same reason, or a different one, from the reason that complex numbers are used in classical electrodynamics (see @Demystifier's post #13 and @vanhees71's post #14)?

 

[DarMM]

Is this the same reason, or a different one, from the reason that complex numbers are used in classical electrodynamics (see @Demystifier's post #13 and @vanhees71's post #14)?

Yes, I think. The physical reasons are different, but the basic reason of having a more compact algebraic expression for geometric relations holds I think. Ultimately in electromagnetism its a way of encoding a bivector and its Hodge dual into a single complex spinor (a special case of using Clifford algebras to simplify geometric quantities) with this spinor being in a rep of the double cover of the component of $SO(1,3)$ as @vanhees71 said.

EDIT: In general I think complex numbers in physics are often a way of "off loading" some of the geometry (e.g. overlap angles) into algebra.

 

[haushofer]

Historically, when did people realize the wavefunction needs to be complex/have 2 real degrees of freedom and one real degree of freedom (i.e. a real scalar function) does not suffice? Was it in the introduction of Heisenberg's commutation relations?

 

[A. Neumaier]

Historically, when did people realize the wavefunction needs to be complex and one real degree of freedom (i.e. a real scalar function) does not suffice?

The Schrödinger equation at its inception already contained a factor $i$. It is built into quantum mechanics quite independent of any interpretation. The canonical commutation relation also involves $i$.

 

[A. Neumaier]

why does QM use amplitudes rather than dealing in probabilities directly?

You gave a complex reason :-)

The real reason is that amplitudes satisfy a simple differential equations, probabilities don't. Knowing all probabilities at a fixed time is not even enough to determine the future probabilities, since probabilities lack the phase information at each point in configuration space.

 

[DarMM]

You gave a complex reason :-)

The real reason is that amplitudes satisfy a simple differential equations, probabilities don't. Knowing all probabilities at a fixed time is not even enough to determine the future probabilities, since probabilities lack the phase information at each point in configuration space.

The expression in my post is actually equivalent to the Schrodinger equation in the case where $B$ is $A$ at a latter time. Certainly knowing the probabilities isn't enough, but the interference phases (angles $\theta$) are present in that relation.

What you're saying isn't incompatible with what I'm saying. The angles arise from the existence of relations between multiple sample spaces. However the evolution equations are cumbersome when dealing with the probabilities and angles directly, where as combining them into an complex amplitude is an alternate simpler expression of them that gives nicer differential equations, i.e. simplicity is the reason for using complex numbers as you said. I expressed this as "alternate more compact", one could say "simpler" as you have.

 

[A. Neumaier]

However the evolution equations are cumbersome when dealing with the probabilities and angles directly, where as combining them into an complex amplitude is an alternate simpler expression of them that gives nicer differential equations, i.e. simplicity is the reason for using complex numbers as you said.

Has anyone written down the evolution equations in terms of probability equations and angles? They must be ugly and impossible to motivate (without resorting to amplitudes), and they would have never been popular. Without complex numbers, the evolution equations probably would not even have been discovered...

 

[DarMM]

Has anyone written down the evolution equations in terms of probability equations and angles? They must be ugly and impossible to motivate (without resorting to amplitudes), and they would have never been popular. Without complex numbers, the evolution equations probably would not even have been discovered...

I've seen them, it's not too hard to derive, but I don't think anybody would use them. What I'm saying is that the existence of multiple sample spaces produces the multiple interference angles present in QM, a feature missing from Kolmogorov probability with its single sample space. However the resulting geometry of meshed sample spaces is cumbersome to deal with directly in terms of the probabilities and angles, hence the amplitude formalism.

Another thing is the constraints related to ensuring probabilities always sum to $1$ over exclusive outcomes imposes the structure of a Hilbert space on the amplitudes.

Adán Cabello's papers on the exclusivity principle focus on this where he shows requiring probabilities to be consistent across contexts implies they have relations between each other equivalent to them coming from (squares of) inner products on a Hilbert space. The Hilbert space being complex then occurs from requiring local tomography.

So I should say the specific form of the relation I posted is important as well, without the root you'd have a real Hilbert space.

Summing up, because I've blathered a bit:
The probabilities in QM involve multiple sample spaces that mesh together in a way that ensures consistency with probabilities summing to $1$ across contexts and consistency with local tomography*. This implies all probabilities and all relations between sample spaces can be encoded in a complex Hilbert space. Thus the use of complex numbers in QM.

*Local tomography is also basically imposed by special relativity, real Hilbert space QM has global degrees of freedom

 

[haushofer]

The Schrödinger equation at its inception already contained a factor $i$. It is built into quantum mechanics quite independent of any interpretation. The canonical commutation relation also involves $i$.

But what convinced Schrodinger then that the wavefunction can't be real? What's wrong with real wavefunctions and real operators?

I.e. what exactly becomes inconsistent if you try to build a quantum theory with a real wavefunction (1 real component)?

 

[DarMM]

But what convinced Schrodinger then that the wavefunction can't be real? What's wrong with real wavefunctions and real operators?

I.e. what exactly becomes inconsistent if you try to build a quantum theory with a real wavefunction (1 real component)?

It has global degrees of freedom inconsistent with special relativity.

 

[A. Neumaier]

But what convinced Schrodinger then that the wavefunction can't be real? What's wrong with real wavefunctions and real operators?

I.e. what exactly becomes inconsistent if you try to build a quantum theory with a real wavefunction (1 real component)?

The stationary states whose energies gave the connection to the older quantum theory from spectroscopy have complex phases. With real wave functions one can handle static issues only.

 

[Fra]

But, really, why worry about complex numbers in particular?
...
Moreover, in a formal development of numbers it is the real numbers where a lot of the problems lie. And, especially, if you try to justify the real numbers as something physical. For example, a typical real number is indescribable, in the sense that it requires an infinite amount of information to quantify it.

I fullly agree with this stance.

Why, then, does no one try to do physics based on rational numbers and countable sets? I imagine that it is possible. But, I suspect that the sacrifice in losing calculus and much else besides is too much. Whereas, the sacrifice in losing the complex numbers is perhaps more manageable.

This is exactly what i am trying to do. Ie. to take representability and computability seriously. That means countable sets of distinguishable events are in the starting points. Real numbers can be thought of as an "approximation" that actually makes the math easier when you reach high complexity, but one must not forget that once you trace things back to LOW complexity (think big bang and primordal observers) the whole continuum mathematics are invalid as basis for physics IMO. This way of reconstructing measures will also automatically solve renormalization problems, that are really created simply because the limits are take and their orders are lost.

But there is not much published on this, and the mainstream paradigms also heavily rests of analysis and real numbers. It also requires a new understanding of symmetries, as the continuuum symmetries in this light may need to be reunderstood as approximations of complex systems rather than as fundamental mathematical truths. This complicates a lot of things in how we think of mathematics toolboxes of physics. I am personally convinced its the way to go though.

/Fredrik

 

[PeterDonis]

The set of definable real numbers (or any other mathematical objects) is countable and has all the properties of the reals (or the objects in question). It gives a countable model of them.

It has all the properties that can be described by a countable first-order axiomatization. But does that include all of the properties of the real numbers that are important for constructing physical models using, for example, calculus?

The particular property I'm thinking might be problematic is the least upper bound property: that every bounded set of real numbers has a least upper bound that is also a real number. But does every bounded set of definable real numbers have a least upper bound that is also a definable real number?

 

[A. Neumaier]

The particular property I'm thinking might be problematic is the least upper bound property: that every bounded set of real numbers has a least upper bound that is also a real number. But does every bounded set of definable real numbers have a least upper bound that is also a definable real number?

Yes. Defining $\sqrt{2}:=\sup\{x\in Q\mid x^2<2\}$ is a valid definition of a particular real number.

More generally, if $S$ is a set of definable real numbers then $\sup S$ gives a definition of its supremum, hence the latter is also definable.

 

[PeterDonis]

if SSS is a set of definable real numbers then $\sup S$ gives a definition of its supremum, hence the latter is also definable.

That's what I thought, but then I started reading about things like this...

https://en.wikipedia.org/wiki/Specker_sequence

...which is a sequence of computable real numbers whose supremum is not computable. But "computable" would seem to be a stricter notion than "definable", since the latter does not require that you actually know how to compute the supremum (for instance), only that you know how to define it using a countable set of axioms. So I guess that's the difference.

 

[A. Neumaier]

That's what I thought, but then I started reading about things like this...

https://en.wikipedia.org/wiki/Specker_sequence

...which is a sequence of computable real numbers whose supremum is not computable. But "computable" would seem to be a stricter notion than "definable", since the latter does not require that you actually know how to compute the supremum (for instance), only that you know how to define it using a countable set of axioms. So I guess that's the difference.

The computable numbers are meant in the sense of constructive mathematics, numbers defined in such a way that the definition implies an algorithm for computing arbitrarily close rational numbers. Unlike definable reals, computable reals do not form a countable model for the real numbers.

Being algorithmic is a much stronger condition than ''using a countable set of axioms''!

 

[DarMM]

This is very interesting! What can one not do with the definables? Can all of analysis be built atop them?

 

[PeroK]

Yes. Defining $\sqrt{2}:=\sup\{x\in Q\mid x^2<2\}$ is a valid definition of a particular real number.

More generally, if $S$ is a set of definable real numbers then $\sup S$ gives a definition of its supremum, hence the latter is also definable.

Let $S$ be a set of rational numbers. All rational numbers are definable. Therefore $\sup S$ is definable. But, every real number is the supremum of a set of rational numbers. Hence, every real number is definable.

In particular the set of all sets of rational numbers is uncountable. So you'll have to careful about how you define a set.

Something like the set of all rational sequences comes easily enough, from a finite set of axioms.

 

[PeterDonis]

Unlike definable reals, computable reals do not form a countable model for the real numbers.

This would seem to imply that there are more computable reals than definable reals. That doesn't seem right, since we can define numbers that are not computable.

 

[PeroK]

This is very interesting! What can one not do with the definables? Can all of analysis be built atop them?

If it can I'll eat my real analysis book.