I'm trying to undestand two proofs of the Hilbert's Nullstellensatz. One is the model-theoretic version given by Nicolas Ford and the other version is the one given by Terry Tao on his blog. The formulations of the theorem goes as follow
(Nicolas Ford) For $S \subseteq K[\overline{x}]$, let $V(S) = \{ \overline{a} \in K^{n} : f(\overline{a}) = 0, \forall f \in S \}$. If $I \subset J$ are radical ideals, then $V(J) \subset V(I)$
(Terry Tao's Blog) Let $P_1, P_2, \cdots, P_m \in F[X]$ where $X = (x_1, x_2, \cdots, x_d)$, then only one of the following statements holds: i) The system of equations $P_1(x) = \cdots = P_m(x) = 0$, $R(x) \neq 0$ has a solution on $x \in F^{d}$ ii) There exists polynomials $Q_1, \cdots, Q_m \in F[x]$ and a natural number $r$ such that $P_1Q_1 + \cdots + P_mQ_m = R^{r}$
I'm having trouble understanding how are this two formulations telling me similar (the same) things. I would really appreciate if someone could enlighten me. Thanks beforehand.