Zermelo–Fraenkel set theory the natural numbers defines $1$ as $1 = \{\{\}\}$ but this does not seem right If 1 can be defined as the set that contains only the empty set then what of sets which contain one thing such as the set of people who are me.
number 1 does not just mean $1$ nothing, it means $1$ something. The definition does not seem to capture what we mean by numbers when we use them in our everyday lives.
I think I may be missing something in the explanations I have read.
Can anyone put me right on this.
 A: The set theoretical definitions of basic mathematical objects should absolutely not be thought of as mathematical definitions in the traditional sense of simply being a more precise version of an intuitive idea. The formal definitions of the integers are not somehow tapping into the true nature of "what integers are".
I think the definitions really come from the Peano axioms. We simply want a set theoretic construction that satisfies the Peano axioms. Essentially any such construction will do.
The idea that the set theoretical foundations are somehow "true" or give you insight into the fundamental nature of mathematical objects is a common misconception among beginners. This may disappoint you (it did me at first), but the intuitive definitions of numbers that you've been using since kindergarten are still the only real definitions available, and there is no real mathematical proof that $1+1=2$. Foundations are here to answer philosophical questions like "can we reduce math to the pure mechanical application of logical rules", and to clear up paradoxes involving things like infinite sets, they do not constitute an investigation of the true underlying nature of mathematical objects.
A: $\{\{\}\}$ does contain one thing.  The thing that it contains is $\{\}$, which is something.
A: This definition has almost nothing to do with the "meaning" of $1$.
This is a familiar concept, just in an unfamiliar context. What is "sun"? Is it a gigantic ball of burning gas? Nope; it is simply three curvy lines drawn next to each other. It is a word that refers to a thing, but is not the thing itself.
The definition you cite is almost completely analogous: it's just that the purpose is to write one as a set in a set-theoretic universe so that we can treat it set-theoretically, rather than writing it as a word in the English language that we can treat linguistically.
But it does turn out that in this writing scheme, the set chosen does have some relation to the meaning: we have encoded one as a set whose cardinality is one (i.e. as a set with one element).
This is sort of like the word "sesquipedalian" which happens to be an instance of the concept it refers to. Although in this instance it is more practical, since it allows the things that relate to the meaning of $1$ to translate more directly to how we have written it: e.g. a set has one element if and only if there is a bijection from that set to $1$. (that is, to the set we have chosen to write $1$ as)
Usually, we would write this definition instead as $1 = \{ 0 \}$, given that we have previously defined $0 = \{ \}$.
