inbetween

Yes. No. Both, and neither. Depends on what you mean by number, which is a fundamental question in mathematics. The foundations of mathematics are fraught with mystery, paradox, and unanswered questions. Take the Peano's Axioms for example.

$0$ is a number.

Everything has to start somewhere. Peano didn't explicitlly define what it meant to be a number, he chose something and made it a number

if $a$ is a number, so is its successor $a+1$

$0$ is a number, therefore $0+1=1$ is a number. Therefore $1+1=2$ is a number... The process repeats. Now we have a list of numbers. If you're given an object and asked if it is a number, you can work backward to see if it is a number.

If the successors of two numbers are equal, the the numbers are equal

When are two things equal? When are they different? This question is surprisingly difficult to answer. In Peano's system, the successor of $2$ is $1$, and the successor of $3$ is $2$, so $3 \ne 2$.

If a set $S$ contains $0$, and contains the successor of every it contains, then it contains every number.

This is a inductively defined set. Zero is in $S$, so $1$ is in $S$, and so is $2$...

In the Peano system, all equal numbers are equal: six is always equal to $6$. All of arithmetic can be built from these five rules, but there is a problem. There are statements about numbers which are true, but cannot be proven using the rules above. Mathematicians don't like that. As I mentioned before, in A Quick And Dirty History of Nothing, there are two kinds of numbers, ordinals and cardinals. There is the sixth day of the week, and the sixth month of the year. They are different things, but they have a commonality: there are five things before them. They are different things, but they have the same ordinal number (sixth). Most people have ten fingers and ten toes. Fingers aren't the same as toes, but the number of fingers is equal to the number of toes. There is only one cardinal number called ten. Any mention of ten is referring to the same concept, $10$. The cardinal number of fingers (or toes, or bowling pins) is ten.