You’re right that there is no greatest cardinal number. The number of ordinals is greater than any ordinal; I’m not sure whether that’s true for cardinal numbers.
You can sorta get around the arbitrarity by postulating the mathematical universe hypothesis, that all mathematical objects are real.
“Discrete Euclidean space” Z^n would be countably infinite, and the usual continuous Euclidean space R^n would be continuum infinite, but I’m not sure what a world whose space is more infinite than the continuum would look like.
It is also true that the number of cardinals is greater than any cardinal, leading to Cantor’s Paradox.
… Since every set is a subset of this latter class, and every cardinality is the cardinality of a set (by definition!) this intuitively means that the “cardinality” of the collection of cardinals is greater than the cardinality of any set: it is more infinite than any true infinity. This is the paradoxical nature of Cantor’s “paradox”.
You’re right that there is no greatest cardinal number. The number of ordinals is greater than any ordinal; I’m not sure whether that’s true for cardinal numbers.
You can sorta get around the arbitrarity by postulating the mathematical universe hypothesis, that all mathematical objects are real.
“Discrete Euclidean space” Z^n would be countably infinite, and the usual continuous Euclidean space R^n would be continuum infinite, but I’m not sure what a world whose space is more infinite than the continuum would look like.
It is also true that the number of cardinals is greater than any cardinal, leading to Cantor’s Paradox.