What is the Length of the maximal antichain in a Poset of cardinality n? Referring to the Antichain page http://mathworld.wolfram.com/Antichain.html

Sperner (1928) proved that the maximum size (and hence the width of
  the partial order) of an antichain containing $n$ elements is
  $$w_{\text{max}} = {{n}\choose{[\frac{n}{2}]}}, $$

where $[]$ is the floor function (i.e. smallest integer less than or equal to).
I do no understand how the cardinality (size) of the maximal antichain can be larger than the size of the set. Also, I think there is some ambiguity in the wording here because it seems to suggest that the antichain has $n$ elements, which is certainly not true. Can someone clarify this result?

This antichain is in the poset of subsets of $\{1,2,\ldots,n\}$. There
are $2^n$ elements in this poset (so plenty enough).
When $n$ is even, the sets of size $n/2$ form an antichain of size $\binom{n}{n/2}$. For odd $n$, you can choose all the sets of size
$\frac12(n-1)$ or all the sets of size $\frac12(n+1)$.
