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Comment by pfortuny | Hacker News Reader

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”.

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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=...

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