If cardinality is a measure of how large a set is, then why is ℤ = ℤ! = |ℚ|,when clearly ℚ > ℤ > |ℤ! | ?

The question seems to be about the concept of cardinality in set theory, a branch of mathematical logic.

First, let's clarify what these symbols mean:

• ℤ represents the set of all integers.
• ℤ! represents the set of all factorials of integers.
• ℚ represents the set of all rational numbers (numbers that can be expressed as a fraction of two integers).

The cardinality of a set is a measure of the "number of elements in the set". However, when dealing with infinite sets like ℤ, ℚ, and ℤ!, we need to use a different kind of "size" or cardinality, known as the "cardinality of the continuum".

The surprising result from set theory is that the cardinality of ℤ (the set of all integers), ℤ! (the set of all factorials of integers), and ℚ (the set of all rational numbers) are all the same. This is counter-intuitive because it seems like there should be more rational numbers than integers, and more integers than factorials of integers.

However, two sets have the same cardinality if there exists a one-to-one correspondence (also known as a bijection) between the elements of the two sets. In other words, if you can pair off elements from set A with elements from set B such that every element in each set has a unique partner in the other set, then the two sets have the same cardinality.

For the sets ℤ and ℚ, such a bijection does exist, even though it may not be immediately obvious. Therefore, they have the same cardinality. The same is true for the sets ℤ and ℤ!.

So, even though it seems like ℚ should be larger than ℤ, and ℤ should be larger than ℤ!, in terms of cardinality, they are all the same size. This is one of the surprising and counter-intuitive results in the study of infinite sets in set theory.