I would say a ring could never be an infinite set of symbols. Rings are created from a finite set of symbols, for example, the 0-9 symbols in an arithmetic ring. In fact, infinite symbolization would make operation based mathematics impossible.
BTW, to be exact, I meant UNIQUE infinite set of symbols. If we require the symbols to always be unique, there is where the whole system would break down. The precisian in me wouldn't let me not correct that comment.
Yes, precisely. the integers represent infinite augmentation with a finite set of symbols under the usual group operations. It is something that I've ruminated upon in a website article. If we could create an infinite alphabet (imprecise language here) of primitives, group operations would break down. So, think of having continually changing symbols to represent the integers and a group operation like + would be impossible. It was a minor criticism, all in all, your lectures are quite enjoyable, a little too slow in developing your concepts, but fun nonetheless.
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