According to the Wikipedia definition (current revision), the width of a poset is the cardinality of any maximum antichain, where "maximum antichain" here means an antichain of maximal cardinality. Perhaps I'm missing something, but there is no guarantee that such an antichain exists. In these cases, do we say that the width is not defined, or take the supremum instead?
To give an example, if I have done it correctly the poset $\Bbb N\times\Bbb N$ with the product order has a sequence of maximal antichains given by $A_n=\{(i,j)\mid i+j=n\}$, so since $|A_n|=n+1$ there are arbitrarily large antichains; but conversely if $(i,j)\in A$ is an antichain then $|A|\le i+j+1$ because there is at most one representative of each column $<i$ and each row $<j$, so there are no infinite antichains and no maximum antichains. So is the width of this poset undefined, or is it $\aleph_0$?
One can ask a similar question about the height (the cardinality of a maximum cardinality chain), for example with the order on $\{(i,j)\in\Bbb N\times\Bbb N\mid i\le j\}$ given by $(i,j)\le(i',j')\leftrightarrow i\le i'\land j=j'$ (which is just the disjoint union of chains of the form $\{(0,n),(1,n),\dots,(n,n)\}$).
I think this question is similar to whether $0$ belongs to the natural numbers or not. In some areas, it is convenient to include $0$ and in other areas it is not. Similarly here. If you deal with finite antichains only, you may consider the maximum. If you deal with infinite (or unbounded) antichains you want to consider the supremum.
In Set Theory for instance, where most posets are infinite, people consider the least cardinal $\kappa$ such that there is no antichain of size $\kappa$. In your example, $\kappa$ would be $\omega$. In these cases, the notion to consider is $\kappa$-chain condition.
See https://en.wikipedia.org/wiki/Countable_chain_condition