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Cantor showed that a multitude of infinities exist. $\aleph_0, \aleph_1, \aleph_2, \ldots$ and so on.

Are there a multitude of infinitesimals as well? My notion would be that the "normal" infinitesimal, $\epsilon$, would really be $\epsilon_0$ in a hierarchy of infinitesimals like Cantor's alephs, so we'd have $\epsilon_0,\epsilon_1,\epsilon_2, \ldots$ and so on.

So for $x + \epsilon_0$ there would be a $\epsilon_1$ such that $x < x + \epsilon_1 < x+ \epsilon_0$.

Does this make sense?

Michael Hardy
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MaxW
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    Cantor's theory of infinite sets has absolutely nothing to do with infinitessimals. – Mariano Suárez-Álvarez Aug 28 '16 at 19:05
  • Infinitesimals are developed in non-standard analysis, whereas Cantor's cardinal numbers are present in standard ZFC. To talk about infinitesimals rigorously, you can't work on the "usual" real line. – 57Jimmy Aug 28 '16 at 19:41
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    Interestingly, Cantor explicitly rejected infinitesimals. See http://philosophy.stackexchange.com/questions/33310/what-was-cantors-philosophical-reason-for-accepting-the-infinite-but-rejecting – user4894 Aug 28 '16 at 19:46
  • @57Jimmy, your comment reflects a typical misconception. The hyperreal framework as developed by Abraham Robinson operates within the standard foundational framework ZFC that you mentioned. See my answer below for more details. – Mikhail Katz Aug 29 '16 at 09:00
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    @Mikhail Katz Ok, then I must apologize for the above comment, thanks a lot for the explanation! – 57Jimmy Aug 30 '16 at 14:17
  • @57Jimmy, sure, no problem. It is not your fault actually. There is an atmosphere of a pogrom that was created around Robinson's discovery by the likes of professional mathematicians Errett Bishop, Alain Connes, and Paul Halmos who should have known better. We deal with these critics and others in published articles that can be found at http://u.cs.biu.ac.il/~katzmik/infinitesimals.html – Mikhail Katz Sep 01 '16 at 07:38

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The OP asked "Do Cantor's infinities imply a multitude of infinitesimals?" The question is a bit ambiguous because it is not clear what the term "Cantor's infinities" refers to exactly. So I will (1) define what this means in modern mathematics, (2) mention Cantor's own attitude, and (3) give an affirmative answer to the question.

(1) In modern mathematics it is customary to interpret "Cantor's infinities" in terms of a traditional set-theoretic axiomatisation called the Zermelo-Fraenkel set theory. This is usually taken to include the Axiom of Choice. The resulting axiomatisation is denoted ZFC.

(2) Cantor's own attitude was one of virulent hostility toward infinitesimals, as noted by his biographer Joseph Dauben. Not only did he not think that "his infinities" do not imply infinitesimals, but he was also convinced that infinitesimals were self-contradictory, and actually published an article allegedly "proving" this. Today we still live with widespread negative attitudes toward modern theories of infinitesimals that arguably stem from Cantor's hostility that was given currency by no less a heavyweight than Bertrand Russell, who was just as confused as Cantor on the issue.

(3) In the modern axiomatisation, ZFC, of Cantorian infinities outlined in item (1) above, it is very easy to construct suitable proper extensions of the real number system that contain infinitesimals and can serve a basis for calculus and analysis with true infinitesimals in the spirit of Leibniz, Euler, and Cauchy. For a freshman-level introduction see Elementary Calculus.

In more detail, the fact that such a number system, usually referred to as a hyperreal number system, is an elementary extension of the reals implies in particular that you have an entire hierarchy of infinitesimals. For example if $\alpha$ is a "base" infinitesimal (Cauchy used $\alpha$ for an infinitesimal) then the infinitesimal $\alpha^2$ will be "infinitely smaller" compared to $\alpha$, and so on: $\alpha^3, \ldots, \alpha^n, \ldots$

In more advanced hyperreal systems you can even have a hierarchy of infinitesimals $\alpha, \beta, \gamma, \ldots$ where $\beta$ is smaller than anything constructed out of $\alpha$, whereas $\gamma$ is smaller than anything constructed out of $\beta$, etc. Terry Tao has pointed out the usefulness of such hierarchies and exploited them in his own work.

Mikhail Katz
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    Wonderful. I was curious about whether the idea of a hierarchy of infinitesimals fit within the hyperreal system. – MaxW Aug 29 '16 at 17:39
  • By the phrase "Cantor's infinities" I meant "Cantor's hierarchy of infinities." The idea was exactly if a hierarchy of infinitesimals would also exist. – MaxW Aug 29 '16 at 18:28
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Cantor's original development of infinite ordinals and cardinals did NOT imply the existence of a corresponding set of infinitesimals. In fact, Cantor himself was stridently opposed to infinitesimals and repeatedly wrote against those who suggested that his work implied their legitimacy.

As for the suggestion that Cantor's ordinals are embedded in Conway's surreal numbers, that has to be taken with a rather large grain of salt. It is a theorem in surreal numbers that surreal addition and multiplication are always commutative, even for infinite surreals; this is decidedly NOT true of Cantor's ordinals. Further, in Cantor's ordinals $\omega$ is the smallest possible infinite ordinal, whereas for the corresponding surreal ordinal $\omega$ there are smaller infinite ordinals $\omega$-1, $\omega$/2, sqrt($\omega$), etc.

PMar
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    I don't want to make too much of a point of this--it's true that the surreal numbers aren't part of the standard development of transfinite numbers. But the ordinals are isomorphic to a subclass of the surreal numbers (in a canonical way). As for operations, addition of ordinals as surreals is the same as the so-called "natural addition" of ordinals from set theory (which is not the same as the standard definition of ordinal addition). And the surreals $\omega-1, \omega/2, \operatorname{sqrt}(\omega), etc.$ aren't ordinals, even within the surreals, although they are surreal numbers. – Mitchell Spector Aug 28 '16 at 21:48
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    The first paragraph answers the title question nicely. In retrospect I should have known the answer. Using one-to-one correspondence as a measure of sze (as Cantor does) is a different way to measure size than using the natural numbers. // The bit about surreal numbers is also a nice additional information. – MaxW Aug 29 '16 at 04:12
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Yes, this precise thing happens with Conway's surreal numbers; see https://en.wikipedia.org/wiki/Surreal_number.

Cantor's alephs (in fact, all ordinals) are embedded in the surreals, and the reciprocal operation is available. The reciprocal of an infinite surreal number is an infinitesimal surreal number, as one would expect.

Mitchell Spector
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    No, this is not precisely it. While in some sense cardinals and ordinals embed in the field of surreal number, the operations are quite different, so it is quite irrelevant in this context. – Mariano Suárez-Álvarez Aug 28 '16 at 20:49
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    See my reply on this to PMar's answer. In general terms, the surreal number system does precisely what the OP was asking about, even though it's not really part of the standard development of transfinite numbers. – Mitchell Spector Aug 28 '16 at 21:51
  • @Mariano The operations are different, but the surreal addition and multiplication can be derived from the usual operations with cantor normal form and writing things in the right order. – Mark S. Oct 13 '16 at 12:14