statements using it could be reconverted (using mathematical operations) into ones that do pay rent.

If statement A can be converted into statement B and statement B pays rent, then statement A pays rent.

Only if the mathematical operation is performed by pure logically entailment, which - if a meaningless definition of infinity is used and that definition is scrapped in the final statement - it would not be. We will just go on about what constitutes a mathematical operation and such, but all I am saying is that if there is a formal manipulation rule that says something like, "You can change the infinity symbol to 'big enough'* here" (note: this is not logical entailment) then I have no objection to the use of the formal symbol "infinity."

*ETA: or just use the definition we agree on instead. This is a minor technical point, hard to explain, and I'm not doing a good job of it. I'll leave it in just in case you started a reply to it already, but I don't think it will help many people understand what I'm talking about, rather than just reading the parts below this.

I reject infinity as anything more than "a number that is big enough for it's smallness to be negligible for the purpose at hand."

Is a terrible one for most purposes, because for them, no matter how big you make a finite number, it won't serve the purpose.

For example? Although, if we agree on the definition below, there's maybe no point.

The immediate sense that a word means something is not, itself, the meaning, but only a reliable intuition that the word means something.

That's why I said "could potentially pay rent."

The epsilon-delta definition in my highschool textbook didn't use infinite sequences, except in the sense of "you could go on giving me epsilons and I could go on giving you deltas."

But then it did use infinite sequences.

That definition of infinity (if we'll call it that) directly means something to me: "this process of back and forth is not going to end."

Unlike your original definition, this is a good definition (at least, once it's been appropriately cleaned up and made precise).

Looks like we're in agreement, then, and I am not a finitist if that is what is meant by infinite sequences.

But then, to take it back to the original, I still agree with Eliezer that an "infinite set" is a dubious concept. Infinite as an adverb I can take (describes a process that isn't going to end (in the sense that expecting it to end never pays rent)); infinite as an adjective, and infinity the noun, seem like reification: Harmless in some contexts, but harmful in others.