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?
On
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.
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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.
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.