# Infinite sets don’t exist!?

This accomplished mathematician gives his opinion on why he doesn’t think infinite sets exist, and claims that axioms are nonsense. I don’t disagree with his arguments, but with my limited knowledge of axiomatic set theory and logic, I am unable to take sides. Would someone be so kind as to enlighten me on why his arguments are/aren’t correct? Thanks

I stopped reading the article at this point:

(6. Axiom of Inﬁnity: There exists an inﬁnite set.

….

And Axiom 6: There is an inﬁnite set!? How in heavens did this one sneak in
here? One of the whole points of Russell’s critique is that one must be extremely careful about what the words ‘inﬁnite set’ denote. One might as well declare
that: There is an all-seeing Leprechaun! or There is an unstoppable mouse!

Quite frankly, he is using an layperson’s interpretation of the axiom and then critiquing this interpretation for being imprecise, when the entire point having these interpretations is to give the gist without being too technical. The common form of the Axiom of Infinity used today is the following (put into words instead of logical symbols):

There is a set $X$ having the property that $\varnothing$ is an element of $X$, and whenever $x$ is an element of $X$, then $x \cup \{ x \}$ is also an element of $X$.

This is a very precise formulation which one can show yields a set which is not finite (hence infinite):

• As $\varnothing$ is in $X$, then $\varnothing \cup \{ \varnothing \} = \{ \varnothing \}$ is an element of $X$.
• As $\{ \varnothing \}$ is in $X$, then $\{ \varnothing \} \cup \{ \{ \varnothing \} \}= \{ \varnothing , \{ \varnothing \} \}$ is in $X$.
• As $\{ \varnothing , \{ \varnothing \} \}$ is in $X$, then $\{ \varnothing , \{ \varnothing \} \} \cup \{ \{ \varnothing , \{ \varnothing \} \} \} = \{ \varnothing , \{ \varnothing \} , \{ \varnothing , \{ \varnothing \} \} \}$ is in $X$.

You see that these elements of $X$ get larger and larger without (finite) bound, and so it stands to reason that such an $X$ must be infinite.