“Infinity” and “number” mean different things in different contexts. In the context of set theory, its perfectly valid to talk about infinite numbers, e.g. https://en.wikipedia.org/wiki/Aleph_number
Those are very explicitly not referring to “the number infinity” though. They’re cardinalities. A number is used to represent the cardinality of the set, but that number is not the same as the set. It refers to the size of the set, not the value of the set. Many other sets could have the same cardinality.
I’m not quite sure what you mean by “value of the set”. Do you mean ordinal numbers? Yes, those are distinct from cardinal numbers, but they’re both referred to as “numbers” in set theory. My point is that the definition of “number” and “infinity” depends on which branch of math you’re working in.
“Infinity” and “number” mean different things in different contexts. In the context of set theory, its perfectly valid to talk about infinite numbers, e.g. https://en.wikipedia.org/wiki/Aleph_number
Those are very explicitly not referring to “the number infinity” though. They’re cardinalities. A number is used to represent the cardinality of the set, but that number is not the same as the set. It refers to the size of the set, not the value of the set. Many other sets could have the same cardinality.
I’m not quite sure what you mean by “value of the set”. Do you mean ordinal numbers? Yes, those are distinct from cardinal numbers, but they’re both referred to as “numbers” in set theory. My point is that the definition of “number” and “infinity” depends on which branch of math you’re working in.