I assume the whole thing is too much for anyone to want to read, so I'll make it easier. Does anyone see a problem with the following definitions for function $t$ and the functions $(\phi_{\alpha})_{2\leq\alpha<\omega_1}$?

Does anyone familiar with Fodor's lemma care to discuss whether there must be a minimum element $\kappa \in \omega_1$ where $\kappa \in \phi_{\omega}(\kappa)$?

Let $t(\alpha)$ equal a doublet of variables $(a,b)$ if $\alpha = 2$, a triplet of variables $(a,b,c)$ if $\alpha = 3$, a quadrulplet of variables $(a,b,c,d)$ if $\alpha = 4$, and so on, for any ordinal $\alpha$. Note that this notation is used to avoid confusion as $a=b$ does not imply $(a,b) = (b,a) = (a) = (b)$, whereas it does imply $\{a,b\} = \{b,a\} = \{a\} = \{b\}$ in general.

Let each element of $(\phi_{\alpha})_{2 \leq \alpha < \omega_1}$ be

1) $\phi_{\alpha} : \omega_1 \setminus \{0\} \rightarrow \{ t(\alpha) : a,b,c,\dots \in t(\alpha) \implies a,b,c,\dots < \omega_1 \}$ is bijective,

2) $a,b,c,\dots \leq \kappa$ for each $a,b,c,\dots \in \phi_{\alpha}(\kappa)$, and

3) $\zeta < \alpha \implies \min\{ \phi_{\zeta}^{-1}(b) : \exists k \in b \text{ where } k \geq \phi_{\zeta}^{-1}(b)\} < \min\{ \phi_{\alpha}^{-1}(b) : \exists k \in b \text{ where } k \geq \phi_{\alpha}^{-1}(b)\}$.

Does anyone familiar with Fodor's lemma care to discuss whether there must be a minimum element $\kappa \in \omega_1$ where $\kappa \in \phi_{\omega}(\kappa)$?

**Define $t(\alpha)$ for any ordinal $\alpha \geq 2$:**Let $t(\alpha)$ equal a doublet of variables $(a,b)$ if $\alpha = 2$, a triplet of variables $(a,b,c)$ if $\alpha = 3$, a quadrulplet of variables $(a,b,c,d)$ if $\alpha = 4$, and so on, for any ordinal $\alpha$. Note that this notation is used to avoid confusion as $a=b$ does not imply $(a,b) = (b,a) = (a) = (b)$, whereas it does imply $\{a,b\} = \{b,a\} = \{a\} = \{b\}$ in general.

**Define $(\phi_{\alpha})_{2 \leq \alpha < \omega_1}$:**Let each element of $(\phi_{\alpha})_{2 \leq \alpha < \omega_1}$ be

*almost regressive*such that:1) $\phi_{\alpha} : \omega_1 \setminus \{0\} \rightarrow \{ t(\alpha) : a,b,c,\dots \in t(\alpha) \implies a,b,c,\dots < \omega_1 \}$ is bijective,

2) $a,b,c,\dots \leq \kappa$ for each $a,b,c,\dots \in \phi_{\alpha}(\kappa)$, and

3) $\zeta < \alpha \implies \min\{ \phi_{\zeta}^{-1}(b) : \exists k \in b \text{ where } k \geq \phi_{\zeta}^{-1}(b)\} < \min\{ \phi_{\alpha}^{-1}(b) : \exists k \in b \text{ where } k \geq \phi_{\alpha}^{-1}(b)\}$.

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