There are various ways to construct real numbers and complex numbers. One approach is given in Landau's Foundations of Analysis (a.k.a. Grundlagen der Analysis).

You might want to look at the Metamath Proof Explorer materials on constructing numbers, a good starting point is here: http://us.metamath.org/mpeuni/mmcomplex.html Metamath is a general tool that lets you specify axioms and proofs, and verifies that the proofs only depend on axioms and previously proved proofs. The Metamath Proof Explorer (MPE) is a particular application of it that uses classical logic and the ZFC set theory axioms. MPE shows how to construct numbers using these axioms, then proves a set of number axioms, and from then on uses only those axioms so that the details of any particular construction are not important. What's usually more important is showing that you can construct them.

Cool link. I know some construction of the real and complex numbers, I was asking about the construction of the hyperreals used in the textbook. I couldnt find info about that on the page.

The construction is fairly easy but requires some fairly hefty background knowledge to make formal. In brief, though: an ordinary real number can be implemented as a sequence of rational numbers, with two sequences considered to be "the same" if they eventually get, and stay, arbitrarily near to each other. A hyperreal can be implemented as a sequence of real numbers, with two sequences considered to be "the same" if they are equal at "very many places", for a certain strict and formal definition of "very many places". (Formally, fix a nonprincipal ultrafilter on the naturals; then the requirement is that the sequences be equal on a set of indices which is a member of that ultrafilter.)

A standard real z can be viewed as a hyperreal, by taking that sequence to be z, z, z, …. Another equivalent representation of that same hyperreal would be the sequence 0, z, z, z, … because that's equal to the first sequence in "very many places". There's an infinite hyperreal, implemented by the sequence 1, 2, 3, …. (In fact there are tons and tons of infinite hyperreals.)

You can prove that the space of hyperreals is a field, and moreover with some model theory you can show that actually a lot of structure (in fact, all first-order structure) transfers directly over to the hyperreals from the reals.

Very cool, thank you. Do you know what happens to measure theory in the hyperreals? Or integration in general?

I don't know much about it, I'm afraid. But there is the very cool fact that the usual Riemann integral can be written as basically just a sum in nonstandard analysis; it's covered very neatly in André Pétry, "Analyse Infinit́esimale: une présentation non standard" which is very readable. There's a lot more advanced stuff, including Brownian motion and (I think) some integration, in Hurd and Loeb, "An Introduction to Nonstandard Real Analysis".

For the foundations of nonstandard analysis, and a wide-ranging overview, I strongly recommend Robert Goldblatt, "Lectures on the Hyperreals: an Introduction to Nonstandard Analysis", one of the Graduate Texts in Mathematics.

I'm rusty, but did think about this a bit at one point. I recall you can't really do the usual constructions on hyperreals (problems with completeness?) but you can do something like approximate any real-valued measure closely.

Keisler wrote a monograph to accompany the textbook:

https://people.math.wisc.edu/~keisler/foundations.html

It's a bit more than an instructor's manual. It covers the construction of the hyperreals in two ways: One (in section 1E) is relatively short, and is like the compactness theorem approach Abraham Robinson originally used in his book ("Non-standard Analysis", Princeton U. Press).

The second construction uses ultraproducts (in Chapter 1*). Keisler gives a quick introduction to the logic needed to understand the ultraproduct construction, beginning with formal languages and sentential logic. So the prerequisites are there, and explained concisely but clearly. The ultraproduct construction comes up elsewhere; for applications to other areas of math, see Martin Davis's book "Applied Nonstandard Analysis" (Dover Publications). (There's a review of the books by Keisler, Davis, and Stroyan and Luxemburg: https://www.ams.org/journals/bull/1978-84-01/S0002-9904-1978...).

The rest of Keisler's monograph contains mostly nonstandard proofs of many of the results in the textbook. I like it a lot, and anyone who's interested in this stuff should grab the PDF.

Another book that is short and moderately elementary is "Infinitesimal Calculus" by James Henle and Eugene Kleinberg, also published by Dover [which also publishes a paper version of Keisler's text, and is generally a great publishing company].

When Keisler's book came out, someone decided that it would be reviewed for the Bulletin of the AMS by Errett Bishop, who (as I recall) was a noted constructivist. You can read the review here: https://www.ams.org/journals/bull/1977-83-02/S0002-9904-1977... Having a constructivist review a calc textbook that uses nonstandard analysis was probably not a great idea. The review ends with: "Now we have a calculus text that can be used to confirm their experience of mathematics as an esoteric and meaningless exercise in technique." I believe Keisler wrote a reply in a subsequent issue of BAMS, but I can't find it.

I do not see how 1E shows existence. The definition of the natural extension needs the hyperreals in the first place. (reading on mobile, maybe I am missing something here)

Check the existence theorem on page 23 (I'm looking at the paper edition): "Let R be the ordered field of real numbers. There is an ordered field extension R* of R and a mapping * from real functions to hyperreal functions (i.e., functions on R*) such that Axioms A - D hold." Note that he just starts with the reals. The actual definition of R* is in the middle of the proof:

    R* = {tau[epsilon] : tau(x) is an element of T(M)}

Note that the proof uses Zorn's lemma, so it's definitely not "constructive" in the strict sense (in case that's what you meant).

Addendum to my post above - I didn't notice that mkl had linked to Keisler's "Foundations" earlier - my bad!

No worries, you provided useful details.

Well, no numbers exist in any real sense. Integers, real numbers, complex numbers, hyperreals, etc., they're all imagined, and built from axioms. Keisler does have a companion book that goes into more of the theory of hyperreals, and its bibliography (p209) points to deeper theoretical work too: https://people.math.wisc.edu/~keisler/foundations.html

Well, the real existence of abstractions is a pretty involved topic in philosophy of knowledge, and so you can’t simply say that, for example, integers do not ‘really’ exist. Because they indeed do, in a very important (and, in fact, obvious) sense: you need two cats to have a fight.

I flip between agreeing and disagreeing with you! I'm not well qualified to comment so I expect an informed rebuttal to this will be instructive for me. So here goes:

I still find myself in the "number systems are a cultural artifact" camp. We choose which details to keep and which to throw away in our abstractions. There are anumeric cultures and it's not clear to me at all that the existence of integers is obvious (beyond what humans can subitize) unless you're immersed in a culture that has impressed them upon you since early childhood.

Do you have "2 cats" or actually just "this cat and that cat"? This cat is a bit bigger but that one looks meaner.

Deep down I think that an insistence on the "reality" of integers versus reals (say) is purely aesthetic. (Ignoring for the moment that our conventional constructions build one from the other in a certain way).

Anumeric cultures might not really have existed (not conclusive evidence, but some evidence: https://www.jstor.org/stable/10.1086/667452?seq=1 )

The integers represent perhaps the simplest case. Another example of their actual existence is the fact that properties of atoms and their nuclei critically depend on the number of the nucleons of particular types in them; note that this fact does not in any way depend on whether a culture is anumeric or not. With some effort you can extend this idea to other abstractions (this process is usually greatly hindered by one's leaning towards solipsism).

The existence proof you describe is in terms of a yet more complex abstraction; we can imagine aliens might describe a more useful atomic theory than our own which does not involve nucleons or their discrete counts at all (ok, unlikely as I acknowledge that may be).

I don't want to descend into solopsism and apologise if I veer too far in this direction. Also thanks for taking the time to rebut what is probably a sophomoric argument.

I suppose my fundamental objection is that if integers are "real" then it seems to me that quaternions must be similarly "real" (since I can describe useful things with them) and so on, I can't see a boundary which would let me say "ok integers are the real deal but infinitesimals are just a thing we made up".

I think we have a zone of possible agreement if we decide either that all these abstractions are "real" or none of them are.

This may seem a bit extreme, but, in my opinion, while in mathematics (and especially in physics) there is a fair amount of scaffolding, what they do in mathematics is, actually, they discover things that do exist in reality in one form or another. (It is important to understand that few, if any, things in mathematics are pure fantasy; the objects and relations studied there are pretty much forced upon us - which, incidentally, is yet another evidence in support of their having real significance, i.e. as something lying outside our consciousness and acting there independently from it.)

Yes, but the reals (or the power set of the reals) are really a bit dodgy. The Nana h tarski paradox shows a big disconnect with reality. There is other axiom models without AOC, which allow for every subset to be measurable, but they seem to have their own problems.

The reals exist in nature in the same sense as, say, 𝜋 does (which occurs as the natural limit in a particular Monte Carlo experiment). The paradox that you mentioned is, rather, disconnected from our intuition, from which reality itself is, too, disconnected (if you think of quantum mechanics, for example).

> Deep down I think that an insistence on the "reality" of integers versus reals (say) is purely aesthetic.

I disagree! In fact, I think the difference between countability and uncountability is pretty huge!

Only finite amount of integers like 10^91 (or so, number of protons in the universe) exist in this sense.

On the other hand, it is interesting to think about the fact that, say, the Avogadro number is usually represented as a real number; the problem with the number of protons or other particles is that their number is unbounded: given enough energy, another particle can always jump into existence...

That's still arbitrary (designed by humans) because you decided to count each proton as 1. We could also count every proton as 2 and it would be just as arbitrary.

Unfortunately, the number of dollar bills in your wallet cannot be thought of as arbitrary.

The notion of existence of ideas has been discussed in philosophy since Platon and has not settled. Consider that to express an idea even a simple one like the numbers one-two-three one needs to use a language. And language has infinite number of interpretations that may be entirely different. So how do one know that his ideas are not misinterpreted by others? Moreover, memory is based on the language, so how does one know that things that he remembers from a day ago mean the same things as it was then?

They are not all build from axioms but from definition (apart from some models of natural numbers, or you go back to ZFC). Like C is build from R^2, R from the powerset of Q, Q from integer^2, integer from natural numbers.

Otherwise you do not easily know if your axioms are actually consistent.

Yes, but what I meant is that there are axioms at the base of this structure, so everything is ultimately built on them. Some people put axiom foundations higher up too, like axiomatic real numbers.

Yes, there are of course axioms, but if you have a rich enough set of axioms like zfc, then one should stop piling more axioms (if not absolutely necessary). So of course the question is existence in zfc

I agree with you (for some values of "exist") and furthermore think this is a good reason why "imaginary" is a terrible name for the number set they refer to.

(I prefer "pure complex", but that's got its issues too, of course.)

It is not complicated but it requires some work and more.

Define “f(x)” is an infinitesimal if lim f(x)=0 when x-> 0.

which is the same as saying “f(x)” is smaller in absolute value than any real number.

Nothing more.

The problem: you cannot divide by infinitesimals. Now you need to “create a field” from these numbers and the reals.: this requires the axiom of choice. There is also the sign problem.

But in the end, “that is all you neeed”. You think of infinitesimals as “numbers smaller than any true real number”.

Thanks! I know about axiom of choice (and I am fine with it), so how do you do the field construction?

I am also a bit worried, if you have a field it is also a division algebra. And I thought we know that the only divisions algebras are R, C, H? Like https://math.stackexchange.com/questions/2020399/division-al... (The link does not 100% fit). So how do hyperreals go around that 'no-go' theorem?

The Frobenius theorem characterizes finite-dimensional associative real division algebras as being isomorphic to the reals, the complex numbers or the quaternions. The hyperreals aren't finite-dimensional as a vector space over the reals (for example, if Ɛ is an infinitesimal hyperreal, then Ɛ, Ɛ², Ɛ³, Ɛ⁴ ... are linearly independent over the reals).

A simpler example of another real division algebra (and in fact, another field) is the field of rational functions with real coefficients. This field is also infinite dimensional over the reals (for example, 1, x, x², x³... are linearly independent).

Ah yes. But then if I think of infinitesimal als functions going to 0 at 0, I can have two infinitesimals (first one is constant zero for negative values other one constant zero for positive values), which multiplied give the constant zero function. How does a construction deal with that?

Although the original statement about “infinitesimals being functions that vanish at 0” was stated with confidence, it is wrong.

The usual construction of the hyperreals replaces real numbers with sequences of real numbers, and also introduces a nontrivial equivalence relation on the sequences, making two sequences equivalent if they agree on a “large” set of terms. The real numbers get represented by the constant sequences, infinitesimals get represented by sequences that approach 0, and infinite numbers are represented by sequences that grow without bound.

The magic is in how “large set of terms” is defined. You need a “large set” relation with the property that finite sets are not large, and for any set either the set or its complement is large. Then we can resolve your question: say you had two not-always-zero sequences that multiply to give the all-zero sequence. Then the set of zero positions is large for one of those two sequences. And that means one of your sequences is equivalent to the zero sequence. The field axioms are saved!

Thanks! Do you know of any source (textbook/paper) about this construction.

I learned it originally from Jim Henle, and iirc he had a textbook on the hyperreals (“Infinitessimal Analysis”, possibly?)

This honors project has what looks like an accurate write up of the construction along with proofs of some of the main theorems: https://ideaexchange.uakron.edu/cgi/viewcontent.cgi?article=...

A proof of existence for any number?

if you assume ZF, then 1={0} and existence follows from existence of empty set.