I understand clearly what you mean now. From the group theory perspective (what you call “structure”), infinity is less interesting than i. From the notational perspective, ∞ isn’t really a number and is therefore not relevant to this discussion.
I think the most interesting thing to come out of the study of infinity is the difference between countable and uncountable infinities. A countable infinity is the order of a set that can be put into one-to-one correspondence with the natural numbers. The rational numbers are an example of a countable infinity. An uncountable infinity is the order of a set too large to be put into one-to-one correspondence with the natural numbers. The irrational numbers are an example of a countable infinity.
Infinity is one of those concepts which seems fundamental but isn’t. Infinity isn’t fundamental because infinity isn’t really a single concept. It is an bundle of several related ideas. In mathematics, there are many ideas adjacent to infinity like “calculus” and “conditional convergence” which are important, but it is true they do not come from treating infinity as a number. Rather, they come from not treating infinity as a number.
I understand clearly what you mean now. From the group theory perspective (what you call “structure”), infinity is less interesting than i. From the notational perspective, ∞ isn’t really a number and is therefore not relevant to this discussion.
I think the most interesting thing to come out of the study of infinity is the difference between countable and uncountable infinities. A countable infinity is the order of a set that can be put into one-to-one correspondence with the natural numbers. The rational numbers are an example of a countable infinity. An uncountable infinity is the order of a set too large to be put into one-to-one correspondence with the natural numbers. The irrational numbers are an example of a countable infinity.
Infinity is one of those concepts which seems fundamental but isn’t. Infinity isn’t fundamental because infinity isn’t really a single concept. It is an bundle of several related ideas. In mathematics, there are many ideas adjacent to infinity like “calculus” and “conditional convergence” which are important, but it is true they do not come from treating infinity as a number. Rather, they come from not treating infinity as a number.