Consider the function $f:n\mapsto k$, where the power set $2^{\aleph_n}$ has the form $\aleph_{\omega\beta+k}$ for some natural number $n$, consider k$. This function is everywhere defined, since the power set$2^{\aleph_n}$, which 2^{\aleph_n}$ must be $\aleph_\alpha$ for some ordinal $\alpha$, and every such ordinal can be expressed uniquely expressed in the form $\omega\beta+k$, where \omega\beta+k$. The number$k$is a natural number, simply the residue of$\alpha$modulo$\omega$, the finite part of$\alpha$sticking above its last limit. Consider the corresponding function$f:n\mapsto k$, where$2^{\aleph_n}=\aleph_{\omega\beta+k}$as above. This So this function is everywhere defined at each$n$. But meanwhile, but we cannot provably determine in ZFC say with certainty any particular value of it$f$. If the GCH holds, then$f(n)=n+1$, but if the GCH fails in complicated patterns, then$f$will be similarly complicated. And neither can we do so We cannot provably determine in ZFC---or even in much stronger theories , such as ZFC plus + large cardinals. cardinals---any particular value of$f$. Indeed, the particular values of$f(n)$are completely independent from one another, from the perspective of what is provable in ZFC or in ZFC+large cardinals, since by Easton's theorem it is consistent with ZFC that$f$is any function at all from$\mathbb{N}\to\mathbb{N}$. For any particular function$g:\mathbb{N}\to\mathbb{N}$, there is a forcing extension of the universe whose version of$f$coincides with$g$. So this would seem to be a fairly strong sense in which we do not know the values of$f(n)$for any particular$n$. 1 For each natural number$n$, consider the power set$2^{\aleph_n}$, which must be$\aleph_\alpha$for some ordinal$\alpha$, and every such ordinal can be expressed uniquely in the form$\omega\beta+k$, where$k$is a natural number, the residue of$\alpha$modulo$\omega$, the finite part of$\alpha$sticking above its last limit. Consider the corresponding function$f:n\mapsto k$, where$2^{\aleph_n}=\aleph_{\omega\beta+k}$as above. This function is everywhere defined, but we cannot provably determine in ZFC any particular value of it. And neither can we do so in much stronger theories, such as ZFC plus large cardinals. Indeed, the particular values of$f(n)$are completely independent from one another, from the perspective of what is provable in ZFC or in ZFC+large cardinals, since by Easton's theorem it is consistent with ZFC that$f$is any function at all from$\mathbb{N}\to\mathbb{N}$. For any particular function$g:\mathbb{N}\to\mathbb{N}$, there is a forcing extension of the universe whose version of$f$coincides with$g$. So this would seem to be a fairly strong sense in which we do not know the values of$f(n)$for any particular$n\$.