# Are natural numbers always sets in set theory?

The case $0 \subseteq \emptyset$: If we define natural numbers as sets then we have $$0=\emptyset$$and $$\emptyset \subseteq \emptyset$$ and therefore $$0=\emptyset \subseteq \emptyset.$$

The case $2 \subseteq A$: In same manner, we can see $2$ as a set which is $$2=\{0,1\}=\{\emptyset, \{\emptyset\}\}.$$ As an example, given $$A=\{0,1\}$$ we have $$A=2$$ and therefore $$2=2\subseteq 2 =A.$$

Question: Do we always need to see a natural number as a set in set theory? Can we say that `$2$ is a number, not a set, therefore it cannot be a subset'?

• Well the only thing that set-theoretic language can talk about is sets. How else would you define what a number is in set-theory? – Stefan Jul 10 '17 at 12:18
• We can, in the version of set theory with urelemnts. – Mauro ALLEGRANZA Jul 10 '17 at 12:19
• But the "most common" version of set theory assume from the beginning that there are only sets in the (mathematical) universe. If so, the above is one way to define a proxy for numbers. – Mauro ALLEGRANZA Jul 10 '17 at 12:20
• "Do we always need to see a natural number as a set in set theory?" How do you define natural numbers in the first place? "Can we say that 2 is a number, not a set, therefore it cannot be a subset?" Again, what is your definition of the number $2$? – Jack Jul 10 '17 at 13:16
• There have been some very extensive discussions on this over on MathOverflow which are definitely worth the time reading them: mathoverflow.net/questions/90820/… (all the comments, the answers, and the comments on the answers). – Asaf Karagila Jul 12 '17 at 16:17

In formal mathematics you have to start somewhere with (essentially) undefined notions governed by axioms.

In the most common foundations, the undefined notion is a set, and everything is a set, pretty much by definition. There are other ways to develop contemporary mathematics that depend on other notions.

If you are interested mainly in arithmetic and number theory you can use a set of axioms (Peano postulates) where "number" is in fact the primitive undefined notion, whose behavior is governed by axioms.

In the usual development of axiomatic set theory we use sets to represent all of the mathematical objects we want to reason about, including numbers. This is a choice, not something that inherently has to be so.

An alternative would be to develop the axioms of set theory in a way that allows urelements -- things that can be elements of sets, but are not themselves sets -- and we could then extend the theory with axioms that assert that some of these urelements happen to behave like numbers; that there is a set of all numbers and sets that represent the addition and multiplication functions and so forth.

It is not usually done that way, because the main interest in writing down completely formal sets of axioms is to investigate the limits of what can be proved at all rather than conduct particular proofs within the system. For this purpose it is technically convenient to have as few axioms as we can get away with -- so given that the purely set-theoretic axioms turn out to be enough to build everything else on top of when we need it, it would just be dead weight to carry around additional axioms for numbers in particular.

In a common approach using superstructures one can take whatever one wants to be "atomic" elements, say the real numbers, and build the set-theoretic universe from there. If the collection one starts with is a base set then one can define a superstructure rank similar to the von Neumann rank in the approach starting with the empty set that you outlined.