The claim amounts to saying that the space of definable goals and algorithms for achieving is large, and it would be a strong claim to say some particular corner of that space is impossible. However, from "improbably thing X is not impossible" does not follow that "X is probable". The paper says nothing conclusive on how the mass of probable algorithms is distributed among that space; what it does is to present some arguments formalish-appearing manner which entice reader's intuitions to answer some questions in a way that they would not otherwise. It does not make their claims
sound.
When presented with abstract forms of argument, human intuitions are often way off. This is why freshmen in mathematics programs usually spend their first year or so proving calculus and many other theorems from scratch and quite laboriously considering how obvious they are when sounded out ( https://en.wikipedia.org/wiki/Intermediate_value_theorem ). The reason is that that with tools of mathematical analysis many unintuitive things ( https://en.wikipedia.org/wiki/Weierstrass_function , https://en.wikipedia.org/wiki/Banach%E2%80%93Tarski_paradox ) may also be said, and so the correct answer is not to trust intuition but proof until student has developed intuition about what really makes calculus to tick.
For example, to paraphrase the argument Armstrong presents on p. 16 "consider all superintelligences that we theoretically could build, is it likely that them having some particular goal would be impossible"? Frankly, I don't know: what is the typical goal, what is the typical path of superintelligence that could be built? Maybe the answer is yes, it would be impossible, and it is only wording that makes it sound unlikely because it invites us to think about large spaces ("all possible X") and small portions of them ("particular goal G") in a certain way. Thinking about space of all possible algorithms and goals, especially about the subset of algorithms that include all kind of intelligent behavior is bound to be unintuitive. The set to which they would "converge to" may not be small, but it still could exclude vast amount of goals, because why would a research team would want to (or even be capable of creating) a creature with blatantly orthogonal goals even if one can draw a hypothetical space of such goals, abstracting away all the important details?
(Secondly, rereading, I believe Armstrong misrepresents the counterarguments by presenting them in extremely strong-looking forms -- convergence thesis, incompleteness thesis -- and tearing them down by arguing that surely it is not totally impossible that something could happen.)
Translated to slightly more formal and mathematical-sounding argument, the author of the talk linked above claims that likely paths for complex minds will not involve them desiring orthogonal goals, because chances of a non-human artificial complex mind arising uniformly randomly from the space of potential algorithms with the vast space of potential goals is negligible: if such being will created, it will be precisely created (or evolved or whatever) following similar principles as the other complex minds on the planet (or by having them as a starting point). In other words, the space of potential algorithms we should concern ourselves with is severely restricted.
(All this assuming that agents and goals is even a sensible framework to model how creatures we call intelligent operate.)