Consider the polynomial ring of countable variables with coefficients in the real numbers, i.e, $S=\Bbb{R}[x_1,x_2,x_3,...,x_n,...]$. Is the following question true?

**Question:** *Is there any maximal ideal in $S$ such that it contains a chain with uncountable number of prime ideals*?

PS: We recall that a set $\{P_{\alpha}\} _{\alpha \in T}$ is a chain of prime ideals if for each $\alpha , \beta \in T$, either $P_{\alpha} \subseteq P_{\beta}$ or $P_{\beta} \subseteq P_{\alpha}$.

PPS: By this notation, i need a chain $\{P_{\alpha}\} _{\alpha \in T}$ of prime ideals in $S$ for which $T$ is uncountable.