Two points:<p>A) Euclid never assumes the existence of a finite set of all primes (doing so at the outset is a framing adopted only by later mathematicians); people just frequently erroneously think Euclid&#x27;s proof did this, never having actually read it.<p>Rather, Euclid constructively frames things, directly arguing that, for any finite set of primes, there is furthermore some prime outside that set. See <a href="http:&#x2F;&#x2F;aleph0.clarku.edu&#x2F;~djoyce&#x2F;elements&#x2F;bookIX&#x2F;propIX20.html" rel="nofollow">http:&#x2F;&#x2F;aleph0.clarku.edu&#x2F;~djoyce&#x2F;elements&#x2F;bookIX&#x2F;propIX20.ht...</a>.<p>B) Regardless, I don&#x27;t particularly care how Euclid phrased his proof; I never mentioned Euclid. Certainly, mathematicians have been aware that Euclid&#x27;s insight could be framed in the way I called an &quot;old proof&quot; for a long, long time. The distinctions between the various framings here are trivial.