So if you've ever read Probability Theory, by E.T. Jaynes

I haven't; I probably should.

the position that, in order to make sense when applied to the real world, infinite things have to behave like limits of finite things.

Is this "limits" in the sense of analysis (epsi-delta limits), or is it "limit points" (like ω)? If the former, then that position involves not believing that arithmetic makes sense when applied to the real world. If the latter, then the position doesn't seem different from what most mathematicians believe, because allowing limit points gets you transfinite induction... but then, as I intend to show, that gets you the first uncountable ordinal, by the power of 'scary dots'. So... either EY doesn't believe arithmetic applies to the real world, or EY doesn't know the logical consequences of his beliefs, or EY doesn't believe the above position, or I've made an error. Of course, at this point the last of those is rather likely, which is exactly why I want to formalise my argument and lay it out as coherently as possible.

Also, if I can't say "exists", how come you can talk about "the real world"? Double standards if you ask me ;)