The enumeration of $\epsilon_0$ below can be used to demonstrate Fodorâ€™s lemma if, for $\alpha > 0$, we say:

$$\phi(\alpha) = \begin{cases}

(\mu,\beta,\gamma) & \text{if Rule } \alpha \text{ : } (\mu,\beta,\gamma) \implies \{\alpha\} \text{ and } \mu,\beta,\gamma < \alpha \\

(0,0,0) & \text{if Rule } \alpha \text{ : } (\mu,\beta,\gamma) \implies \{\alpha\} \text{ and } \mu,\beta, \text{ or } \gamma \geq \alpha \\

\end{cases}$$

Let $Ord$ denote the class of all ordinals and let each $Ord_{\alpha}$ be a subclass of ordinals such that $(Ord_{\alpha})_{\alpha \in Ord}$ forms a partition of $Ord$. The transfinite sequence $(Ord_{\alpha})_{\alpha \in Ord}$ is defined as follows:

$Ord_0$ is the class of ordinals that are not limit ordinals.

$Ord_{\alpha} = (K, <)$, where $<$ is the standard ordering and $K = \{ x_j : (x_j, <)_{j \in Ord} \text{ well orders } Ord \setminus \bigcup_{\kappa < \alpha} Ord_{\kappa} \text{ and } j \text{ is not a limit ordinal} \}$.

Note that where $\alpha \geq \omega$ we need $u,v < \alpha$ for each $(\alpha)_{u,v}$ to be successful.

$$\begin{matrix}

Ord_0 \text{ : } & 0_{0,0} & 1_{0,1} & 2_{0,2} & \dots & (\omega + 1)_{0,\omega} & (\omega + 2)_{0,\omega+1} & (\omega + 3)_{0,\omega+2} & \dots & (\omega \cdot 2 + 1)_{0,\omega \cdot 2} & (\omega \cdot 2 + 2)_{0,\omega \cdot 2 + 1} & (\omega \cdot 2 + 3)_{0,\omega \cdot 2 + 2} & \dots\\

Ord_1 \text{ : } & (\omega)_{1,0} & (\omega \cdot 2)_{1,1} & (\omega \cdot 3)_{1,2} & \dots & (\omega^2 + \omega)_{1,\omega} & (\omega^2 + \omega \cdot 2)_{1,\omega+1} & (\omega^2 + \omega \cdot 3)_{1,\omega+2} & \dots & (\omega^2 \cdot 2 + \omega)_{1,\omega \cdot 2} & (\omega^2 \cdot 2 + \omega \cdot 2)_{1,\omega \cdot 2 + 1} & (\omega^2 \cdot 2 + \omega \cdot 3)_{1,\omega \cdot 2 + 2} & \dots \\

Ord_2 \text{ : } & (\omega^2)_{2,0} & (\omega^2 \cdot 2)_{2,1} & (\omega^2 \cdot 3)_{2,2} & \dots & (\omega^3 + \omega^2)_{2,\omega} & (\omega^3 + \omega^2 \cdot 2)_{2,\omega + 1} & (\omega^3 + \omega^2 \cdot 3)_{2,\omega + 2} & \dots & (\omega^3 \cdot 2 + \omega^2)_{2,\omega \cdot 2} & (\omega^3 \cdot 2 + \omega^2 \cdot 2)_{2,\omega \cdot 2 + 1} & (\omega^3 \cdot 2 + \omega^2 \cdot 3)_{2,\omega \cdot 2 + 2} & \dots\\

\vdots \\

Ord_{\omega} \text{ : } & (\omega^{\omega})_{\omega,0} & (\omega^{\omega} \cdot 2)_{\omega,1} & (\omega^{\omega} \cdot 3)_{\omega,2} & \dots & (\omega^{\omega} \cdot \omega + \omega^{\omega})_{\omega,\omega} & (\omega^{\omega} \cdot \omega + \omega^{\omega} \cdot 2)_{\omega,\omega + 1} & (\omega^{\omega} \cdot \omega + \omega^{\omega} \cdot 3)_{\omega,\omega + 2} & \dots & (\omega^{\omega} \cdot \omega \cdot 2 + \omega^{\omega})_{\omega,\omega \cdot 2} & (\omega^{\omega} \cdot \omega \cdot 2 + \omega^{\omega} \cdot 2)_{\omega,\omega \cdot 2 + 1} & (\omega^{\omega} \cdot \omega \cdot 2 + \omega^{\omega} \cdot 3)_{\omega,\omega \cdot 2 + 2} & \dots\\

\vdots \\

\exists Ord_{\alpha} \text{ : } & (\alpha)_{\alpha,0} \in Ord_{\alpha} & <------ \text{Oops!!!!}\\

\vdots

\end{matrix}$$

The following rules are for a $T$ sequence that enumerates $\epsilon_0$:

$$\text{Rule 1 : } \mu = 3, \beta = 2, \gamma = 1 \implies \{0\}$$

$$\text{Rule } \alpha \text{ : } \mu = 0, \beta = 1, \gamma = \alpha - 1 \implies \{\alpha\}, \text{ where } 2 \leq \alpha < \omega$$

$$\text{Rule } \alpha \text{ : } \mu = 0, \beta = u, \gamma = v \implies \{\alpha_{u,v}\}, \text{ where } \alpha \geq \omega$$

Each $Ord_{\alpha}$ can be associated with a normal function $f_{\alpha}$, the range of which is a club set $C_{\alpha} = Ord \setminus \bigcup_{\gamma \leq \alpha} Ord_{\alpha}$. For example, with respect to $Ord_0$, the set of countable limit ordinals $C_0 = \{ \omega \cdot \alpha : \alpha \in Ord \}$ and $f_0(\alpha) = \omega \cdot \alpha$ are the relative club set and normal function, respectively. For $Ord_1$, we have $C_1 = \{ \omega^2 \cdot \alpha : \alpha \in Ord \}$ and $f_1(\alpha) = \omega^2 \cdot \alpha$.

For any particular ordinal $\alpha$ where $0 < \alpha < \omega_1$, note that:

$$\alpha \not\in C_{\delta} \text{ for any particular } \delta < \alpha \implies (\alpha)_{\beta,\gamma} : \beta, \gamma < \alpha \implies (\text{Rule } \alpha \text{ is capable of extending the sequence by asserting } (0,\beta,\gamma) \implies \{\alpha\})$$

$$\alpha \in C_{\delta} \text{ for each } \delta < \alpha \implies (\alpha)_{\beta,\gamma} : \beta \geq \alpha \implies (\text{Rule } \alpha \text{ is not capable of extending the sequence by asserting }(0,\beta,\gamma) \implies \{\alpha\})$$

Note that $\epsilon_0$ is a fixed point of each $f_{\alpha}$ such that $\alpha < \epsilon_0$. In fact, it is the least fixed point that is a common fixed point across all the functions $(f_{\alpha})_{\alpha < \epsilon_0}$. Equivalently, $\epsilon_0$ is the least element of $C = \Delta_{\alpha < \omega_1}C_{\alpha}$ and the first element for which Fodor's lemma asserts a constant $\phi(\epsilon_0) = (0,0,0)$ must be assigned.

I just said you need to... study how things are done in math.

And just how are things done in math anyways? I have to ask... I assume we start with axioms and go on to prove theorems. The end, IMHO.

Maybe you believe in the old "we could train you to be a manager in 6 months or less but you should probably work as a teller for 15 years first just because the managers before you had to" philosophy. That is a really crappy philosophy for math though, as building on the work of others instead of having to relearn it all from scratch is the ideal that is accomplished via optimized means of communication (thank goodness for the internet!).

In any case, here is the effort I've put in. Via my

*silly* $T$ sequences, I have now learned about:

1) Ordinal arithmetic & Cantor normal form

2) Normal functions

3) Fixed points

4) Cofinality

5) Ackerman Ordinal

6) Veblen Hierarchy, Large and Small Veblen Ordinals

7) Church-Kleene Ordinal and Kleene's $\mathcal{O}$

8) Stationary sets

9) Club sets (closed and unbounded sets)

10) Diagonal intersections

11) Fodor's lemma

12) Some expanded philosophy

After all this, I still just want someone to answer my basic question:

https://math.stackexchange.com/questions/3401372/question-regarding-ordinals#3401372
Where my $\phi : \omega_1 \rightarrow \omega_1^3$, there may be a way for my $\phi$ to exist even though Fodor shows no regressive $\phi : \omega_1 \rightarrow \omega_1$ can exist.

PS - I use triplets, but we could use doublets, quadruplets, quintuplets, etc., until we got to $\omega$-lets for lack of a better way to put it, in which case we would be building towards a set that must be equal in cardinality to $\mathcal{P}(\mathbb{N})$ is how far I may choose to pursue this...