The one-to-one correspondence approach in set theory is based on Cantor-Hume principle, which postulates that two sets, whose elements can be set in one-to-one correspondence are equally big. On this princeple is based the Cantorian cardinality.

But there is an alterntive to this approach, and it is called Euclid's principle, which asserts that the whole is greater than the parts. It was Galileo who first pointed out the contradiction of Euclid's principle and the principle of one-to-one correspondence in his work about perfect squares. In this context people often talk about numerocity of a set. For instance, the numerocity even numbers is half of the numerocity of integers.

As you correctly noted, while cardinality can be defined on sets of any objects regardless of their position and order, to compare numerocities you need to introduce order or metric on the sets you are comparing (what you are calling "time").

For instance, a more densely distributed infinite subset of integers has greater numerocity than the less dense one. The numerocity of the odd positive integers has numerocity greater than the numerocity of even positive integers by 1/2 because one set is shifted towards zero, compared to the other.

The one-to-one correspondence approach in set theory is based on Cantor-Hume principle, which postulates that two sets, whose elements can be set in one-to-one correspondence are equally big. On this princeple is based the Cantorian cardinality.

But there is an alterntive to this approach, and it is called Euclid's principle, which asserts that the whole is greater than the parts. It was Galileo who first pointed out the contradiction of Euclid's principle and the principle of one-to-one correspondence in his work about perfect squares. In this context people often talk about numerocity of a set. For instance, the numerocity of even numbers is half of the numerocity of integers.

As you correctly noted, while cardinality can be defined on sets of any objects regardless of their position and order, to compare numerocities you need to introduce order or metric on the sets you are comparing (what you are calling "time").

For instance, a more densely distributed infinite subset of integers has greater numerocity than the less dense one. The numerocity of the odd positive integers has numerocity greater than the numerocity of even positive integers by 1/2 because one set is shifted towards zero, compared to the other.

The one-to-one correspondence approach in set theory is based on Cantor-Hume principle, which postulates that two sets, whose elements can be set in one-to-one correspondence are equally big. On this princeple is based the Cantorian cardinality.

But there is an alterntive to this approach, and it is called Euclid's principle, which asserts that the whole is greater than the parts. It was Galileo who first pointed out the contradiction of Euclid's principle and the principle of one-to-one correspondence in his work about perfect squares. In this context people often talk about numerocity of a set. For instance, the numerocity even numbers is half of the numerocity of integers.

As you correctly noted, while cardinality can be defined on sets of any objects regardless of their position and order, to compare numerocities you need to introduce order or metric on the sets you are comparing (what you are calling "time").

For instance, a more densely distributed infinite subset of integers has greater numerocity than the less dense one. The numerocity of the odd positive integers has numerocity greater than the numerocity of even positive integers by 1/2 because one set is shifted towards zero conpared to the other.

The one-to-one correspondence approach in set theory is based on Cantor-Hume principle, which postulates that two sets, whose elements can be set in one-to-one correspondence are equally big. On this princeple is based the Cantorian cardinality.

But there is an alterntive to this approach, and it is called Euclid's principle, which asserts that the whole is greater than the parts. It was Galileo who first pointed out the contradiction of Euclid's principle and the principle of one-to-one correspondence in his work about perfect squares. In this context people often talk about numerocity of a set. For instance, the numerocity even numbers is half of the numerocity of integers.

As you correctly noted, while cardinality can be defined on sets of any objects regardless of their position and order, to compare numerocities you need to introduce order or metric on the sets you are comparing (what you are calling "time").

For instance, a more densely distributed infinite subset of integers has greater numerocity than the less dense one. The numerocity of the odd positive integers has numerocity greater than the numerocity of even positive integers by 1/2 because one set is shifted towards zero, compared to the other.