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As a mathematical structure, I have no problem with the hyperreals. But I came across the following from Keisler's book "Elementary Calculus: An Infinitesimal Approach".

"We have no way of knowing what a line in physical space is really like. It might be like the hyperreal line, the real line, or neither. However, in applications of the calculus, it is helpful to imagine a line in physical space as a hyperreal line."

We get real number answers to physical problems in distance, from integers to transcendental numbers. However, we never get a nonzero number $w$ s.t. $w \lt 1/k, \forall k \in \mathbb N$ coming from a physical calculation, theoretical or applied. Isn't this what makes physical distances real and not hyperreal? Is it really possible that our space was never (locally) Euclidean all this time? Where are these infinitesmals and why are they hiding?

Secondly, we say the real line, by construction from completing the rationals, has "no holes". Yet, the reals are a proper subfield of the hyperreals. Where do these infinitesmals "fit" on the real line to make a hyperreal line when there is no room for them to fit? In other words, if we begin by assuming a physical line segment is a hyperreal line segment and then (mathematically) remove all the infinitesmals, we get a hyperreal line segment with "holes" in the form of missing hyperreal points, but this just gives a real line segment, which has no holes. There seems to be problems in assuming physical lines can be hyperreal lines.

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    $\begingroup$ Your first question seems to me to be based on an argument from incredulity. $\Bbb R$ offers a very useful mathematical model, but does it actually match physical space? You seem to be assuming that it does simply because you know of no evidence to the contrary. How would you even test? And in the other direction, how do you know that physical distance isn’t quantized? $\endgroup$ Nov 16, 2013 at 2:45
  • $\begingroup$ The set of all integers $\{\ldots,-3,-2,-1,0,1,2,3,\ldots\}$ also has no holes, although lots of real numbers are missing from it. My answer below explains. $\endgroup$ Nov 16, 2013 at 2:54
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    $\begingroup$ Many physicists, particularly in the old days, used intuition about "infinitesimals" (though not the Abraham Robinson version) as an aid to modelling. $\endgroup$ Nov 16, 2013 at 3:02
  • $\begingroup$ @AndréNicolas : Just in the old days and not now?? It would be unfortunate if that technique were forgotten. And I doubt it has been. $\endgroup$ Nov 16, 2013 at 17:46
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    $\begingroup$ I don't think it should be avoided in writing. Maybe it should be avoided in certain kinds of logically rigorous proofs, but not otherwise. $\endgroup$ Nov 17, 2013 at 16:23

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The assertion that $\Bbb R$ ‘has no holes’ is an informal paraphrase of the mathematically precise statement that $\langle\Bbb R,\le\rangle$ is a complete linear order. In other words, every non-empty $A\subseteq\Bbb R$ that is bounded above has a least upper bound in $\Bbb R$. This in no way prevents us from shoving new elements into $\Bbb R$. For example, I can take some object $p\notin\Bbb R$, let $X=\Bbb R\cup\{p\}$, and define a linear order $\preceq$ on $X$ by $x\preceq y$ iff

  • $x,y\in\Bbb R$ and $x\le y$, or
  • $x\in\Bbb R$, $y=p$ and $x\le 0$, or
  • $x=p$, $y\in\Bbb R$, and $y>0$, or
  • $x=y=p$.

This in effect inserts $p$ between $0$ and all of the positive reals. The various constructions of the hyperreals do something similar, but on a grand scale, surrounding each real number with a ‘cushion’ of infinitesimally different hyperreals, and moreover adding whole galaxies of infinite hyperreals at both ends of the line. This is very different from filling existing holes, which is what we do when we complete the rationals to form the reals.

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When it is said that the line has no holes, one must be precise about what is meant. Here's one way of looking at it, maybe the simplest: Suppose you divide the real line into two non-empty subsets $A$ and $B$, such that every real number is in one or the other of those, and every number in $A$ is strictly less than every number in $B$. Then there must be a boundary: a number $c$ such that every number less than $c$ is in $A$ and every number more than $c$ is in $B$. The number $c$ itself could be in either of those two sets.

This is opposed to a situation like this: consider the set of all non-zero real numbers. You can divide those into two subsets, namely the negative numbers $A$ and the positive numbers $B$, in such a way that every non-zero real number is in one or the other of those, and every non-zero number in $A$ is strictly less than every number in $B$. But if you take any negative number $x$, you can find a larger negative number, e.g. $x/2$, and for any positive number $x$, there is a smaller positive number $x/2$. Thus no negative or positive number $c$ can serve as a boundary, so that every number less than $c$ is negative and every number greater than $c$ is positive.

Even if we didn't know that irrational numbers exist, we could prove that for every rational number whose square is less than $2$, there is a larger rational number whose square is less than $2$, and for every rational number whose square is more than $2$, there is a smaller rational number whose square is more than $2$. Thus the set of all rational numbers has a hole, and in fact has many holes.

The hyperreals also have holes. For example let $A$ be the union of the set of all non-positive hyperreals (including $0$) and the set of all positive infinitesimal hyperreals, and let $B$ be the complement of that set. There's no hyperreal $c$ that can serve as a boundary. One can see that as follows. If $c\in B$, then $c/2$, being less than $c$, would be in $A$, but that would mean half of a non-infinitesimal positive number is infinitesimal, so that can't work. But if $c\in A$, then $2c$, being greater than $c$, would be in $B$, thus non-infinitesimal. But then again, half of the non-infinitesimal number $2c$ would be infinitesimal, and again that can't work.

If you think a missing hyperreal means a hole in the reals, consider this: the set of all integers has no holes. Lots of reals are missing, but that doesn't create holes in the sense defined above. To see this, consider the two sets $A=\{\ldots,-3,-2,-1,0,1,2,3\}$ and $B=\{4,5,6,\ldots\}$. Either $3$ or $4$ can serve as a boundary as defined above, since every number less than $3$ or less than $4$ is in $A$ and every number greater than $3$ or greater than $4$ is in $B$. And all other ways of dividing the set in two that satisfy the requirements above also yields a boundary. So there are no holes.

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The simplest intuition for "hyperreal line" is that it looks exactly like the standard real line.

Since you're coming from a physics background, this "functional" approach may appeal very much to you.

There is the "first-order language of real arithmetic". This language lets us talk about things like adding or multiplying real numbers. To talk about whether one number is bigger than another. To ask whether we can find solutions to an equation.

If we have a real line, there is an obvious way to interpret this language as asking talking about the real numbers. If we so interpret the question "given any two points, is there another point between them?" the answer is yes.

For comparison, there's an obvious way to interpret this language as talking about the integers. The answer to the above question would be "no".

It turns out that every question you can possibly ask in this language has exactly the same answer when you ask it about a real line versus when you ask it about a hyperreal line.

There is no "observable" difference between the real line and the hyperreal line, if we can only "observe" by doing first-order real arithmetic.


We can generalize this. We could use the more expressive "first-order language of real analysis", which is strong enough to express all of the mathematics we usually study. It lets us talk about things like intervals, Euclidean geometry, Taylor series, Hilbert spaces, Lagrangian mechanics, and so forth.

The same phenomenon exhibits itself in this language too: we have the "standard model" of real analysis, where we define everything in the 'usual' way, and there are also "non-standard models" of real analysis.

And once again, every single question we can ask using the first-order language of real analysis has exactly the same answer when asked of the standard model and when asked of any non-standard model.

e.g. if we use the language of real analysis to talk about Newtonian mechanics, and we design an experiment and ask "theoretically, what should the experimental result be?", the answer would be the same if we asked it about a standard model or a non-standard model. There is no physical experiment we can design to determine whether we live in the "real world" or the "hyperreal world"!


The utility of non-standard analysis comes from deciding to work with both the standard model and the non-standard model at the same time. Since they are 'physically indistinguishable', we can use whichever one is more convenient at the time.

e.g. it is often convenient to suppose a problem is about the standard model, but then take the standard objects and transfer them over to the non-standard model, so that we can make use of things like infinitesimals, or that the non-standard model has integers that it considers to be finite, but turn out to be larger than every standard integer.

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Answering your first question about physical intuition of the hyperreal line. In calculations using non-standard numbers you take the standard part of the result to get back to a real number. In terms of physics you can compare this to all the calculations with sub atomic particles such as quarks that can not be observed individually but we can see the effects of their existence at the atomic level.

Perhaps thinking of the infinitesimals as analogous to the folded extra dimensions that are compactified in string theory to get back to regular 4 D space time would help. So removing them doesn't leave holes in the real line because they are not part of the real line - they are a construction beyond it.

Just as removing the irrational numbers from the real line doesn't lead to any "missing" rational numbers in the rational line, so removing infinitesimals from the hyper-real line doesn't lead to missing real numbers.

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Imagine a small 'cloud' in place of each point of the usual real line, so that when you zoom on a point $x\in\Bbb R$, finally you would find a whole world of $x+\varepsilon$'s where $|\varepsilon|$ is 'infinitesimal'.
Then, also imagine the reciprocal of the infinitesimals in the cloud at $0$: these numbers have very big absolute value, bigger than all natural number, so should be placed on a schematic picture to the right (and to the left) of the original real line.

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If $\epsilon$ is infinitesimal and $r$ is a standard real, the real vector space spanned by $\epsilon$ and $r$ looks like the plane $\mathbb{R} \times \mathbb{R}$ with $xr + y\epsilon$ mapping to $(x, y)$. The ordering is the lexicographic ordering, i.e., you order the points first by $x$ and then by $y$ so the "clouds" mentioned in Berci's answer are vertical lines.The standard reals "fit in" as the points with $y=0$.

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To understand Keisler's comment:

"we have no way of knowing what a line in physical space is really like. It might be like the hyperreal line, the real line, or neither. However, in applications of the calculus, it is helpful to imagine a line in physical space as a hyperreal line,"

consider the fact that physical space, as we know from quantum mechanics, does that have the property of indefinite divisibility as does the real line or the hyperreal line. Both of these "lines" are idealisations that are useful in certain applications. If you were only interested in solving linear equations with integer coefficients, you would never need to leave the realm of rational numbers. Throwing in algebraic numbers allows you to express the diagonal of a square in a simple way, and throwing in transcendentals allows you to express the area of a unit circle in an elegant way. If you don't have problems that require the tools of the calculus, you will probably not need infinitesimals, either. An infinitesimal-enriched number system allows you to develop a formalism to express the calculus in a simpler way, and to express your solutions in a way that's closer to your original intuitions that led to the solution.

I would therefore agree with you that "our space was not (locally) Euclidean" all along.

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  • $\begingroup$ I would feel much more comfortable hearing your last line from a physicist! It does fall under physical inquiry, not mathematical. $\endgroup$ Jun 14, 2014 at 3:31
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    $\begingroup$ @JustSomeOldMan, A number of physicists have expressed themselves along these lines, including Brukner, Zeilinger, Durham, and Wheeler; see this article (section 8.4). $\endgroup$ Jul 28, 2014 at 14:17
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To respond specifically to the question: Secondly, we say the real line, by construction from completing the rationals, has "no holes". Yet, the reals are a proper subfield of the hyperreals. Where do these infinitesmals "fit" on the real line to make a hyperreal line when there is no room for them to fit?

It is worth keeping in mind that the claim of the absence of holes is correct when applied to an Archimedean continuum. The claim does not apply to more general continua, such as real closed fields extending the reals.

Some of these fields are quite easy to construct (e.g., the field of rational fractions).

This point is contained in other answers but does not stand out due to their length.

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