Cantor’s set theory was accepted and axiomatized, there has no longer been need to answer the questions above. The answer has ultimately been given in the form of an axiom: in the Zermelo-Fraenkel set theory, the axiom is called the *Axiom of Infinity:*[^1] $$(\exists z)((\emptyset \in z) \wedge (\forall x)(x \in z \Rightarrow x \cup \{x\} \in z)).$$

There is an important consequence of the Axiom of Infinity and of the *Axiom of the Power-set*: [^2] the existence of infinitely many infinite ordinal and cardinal numbers.

Measuring the actual infinity
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It is well-known that if we accept the existence of actual infinity and try to compare infinite collections, we have to choose one of the following two, mutually exclusive, principles:

1.  The *part-whole principle,* also known as the 5th Euclid Axiom: The whole is greater than its part.

2.  The *principle of one-to-one correspondence,* or Hume’s Principle: Two sets have the same size if and only if there is a one-to-one correspondence between them.

The two principles cannot be valid simultaneously for infinite collections. This fact is called the *Paradox of Reflexivity.* It was known already to Galileo Galilei who employed natural numbers and their squares to demonstrate it. 