Part of the definition is each time we add 1, we get a number we haven't seen before; and so we have an infinite set by construction.
No. You have a rule that hypothetically would produce an infinite set if applied ad infinitum. This may seem like nitpicking but there is a difference between the concept of an infinite set and an actual infinite set, the latter can't be represented in a finite brain(I suppose).
I can write down the rules of a turing machine, but this doesn't produce a working computer to spring to life if you get my point.
No. You have a rule that hypothetically would produce an infinite set if applied ad infinitum.
Yep, exactly; no problem with that, that's how mathematics works. There is only a problem if someone wants to write down every element of an infinite set.
there is a difference between the concept of an infinite set and an actual infinite set
This is mathematics. The concept of a mathematical object is the object, because the "concept" version satisfies all the same rules (axioms) as any "actual" version, and these rules completely describ...
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