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(This question was originally asked and bountied at MSE - with different notation, some more explicit arguments, and topology in place of forcing.) Suppose $\mathcal{R}=(R_i)_{i\in\mathbb{N}}$ is a ...

Take the Hilbert Basis Theorem over the rational numbers in this form in the language of Second Order Arithmetic:  For every $n\in N$ every ideal of the polynomial ring $\mathbb{Q}[x_1,\dots,x_n]$ is ...