I do enjoy how people think “If you don’t agree with the standard treatment of infinities, you must…
Steve Patterson
1

Russel’s Logicism didn’t pan out for a reason… not only is it a failed theory, it is so ungodly ugly to work with…

Anyway, the real point is that they aren’t conceptually incoherent. You just woefully misunderstand them. Infinite sequences, sums, continuity, and all of these arguments do not require infinities to make sense of them. That’s just a piece of notation used to show the intuitive idea. Sequences, sums, and all other infinite things converge if, for every finite number, you can get “close” to the thing you claim is the solution. Any introductory text in real analysis will explain to you very clearly that you do not “sum” infinite series. Sequences do not have “endpoints”, they have limits. You are simply saying that the collection of (finitely many!) partial sums gets close to some value, regardless of how large that finite number is.

In fact, sequences, series, continuity and all these other notions are workable in a world with no axiom of infinity. This is because the collection of such “accessible” numbers is countable, and finitism can handle the sizes of countable sets. If it works in the finitist world, I fail to see why extending the argument to the a set with a completeness axiom like the real numbers is an issue.

But the whole thing is nonsense anyway, because there has been an axiomitzation of analysis using “hyperreal” numbers, which is compatible with the basic premise of your view above, and has been shown to be consistent if and only if the standard model I’m describing is consistent. This should be enough for you to see that your inferences are wrong — they are glaring contradictions.

Like I said, you really need to revisit that text. I’m happy there are interested people out there in mathematics, but it does you no good to be on such poor footing.