What makes it clear that 1 precedes 2?

Question

In the construction of natural number system, I'm not sure how the ordering of elements of N is defined. It seems that almost every approach to that is quite abstract without mentioning an actual number except 1. Then, how do you determine whether 2 is bigger than 1? It would be great if you recommend a book that will be perfect for my question—traditional textbooks for set theory doesn't seem to cover this topic.

Answer 1

Standard construction is:

• $0:=\varnothing$

• $n+1:=n\cup\{n\}$

Order $<$ is actually the same as order $\in$.

Then $1=\{0\}$ and $2=\{0,1\}$ so that $1\in2$ or equivalently $1<2$.

Answer 2

As with almost every concept in mathematics there is not "the definition" of $2$. There are multiple ones. The most common ones are:

• $2$ is the successor of $1$
• $2 = 1 + 1$

But nothing stops you from using more complicated definitions, like:

• $2$ is the smallest prime (consider that primes have to be greater than $1$)

All these definitions are equivalent and the all imply that $1<2$. For example in the case $2 = 1 + 1$ we have $1<2$ because $a<b \Leftrightarrow \exists n>0 : a + n = b$ (take $a=n=1$ and $b=2$)