Consider the polynomial ring of countable variables with coefficients in the real numbers, i.e, $S=\Bbb{R}[x_1,x_2,x_3,...,x_n,...]$. My Question is about the **depth** of this ring.

**Question**: *Could we find any maximal ideal in $S$ such that it contains an anti-chain with uncountable number of prime ideals*?

**PS**: We recall that a set $\{P_{\alpha}\} _{\alpha \in \mathcal{A}}$ is an anti-chain of prime ideals if for each $u ,v \in T$, neither $P_{u} \subseteq P_{v}$ nor $P_{v} \subseteq P_{u}$.

**PPS**: By this notation, I am seeking an anti-chain $\{P_{\alpha}\} _{\alpha \in \mathcal{A}}$ of prime ideals in $S$ for which $T$ is uncountable.