Question Regarding Ordinals

Question

Where $\mu, \beta, \gamma, \alpha,$ and $\kappa$ are ordinals and $\alpha, \kappa > 0$ does there exist a function $\phi$ such that $\phi(\alpha) = (\mu, \beta, \gamma)$ where:

1) $\mu, \beta, \gamma < \alpha$, and

2) $\alpha \neq \kappa \implies \phi(\alpha) \neq \phi(\kappa)$

We can restrict all variables to being less than $\omega_1$ (or any arbitrary regular uncountable cardinal) so as to apply Fodor's lemma if that helps, though we are able to consider $\phi$ as a class function too.

Where there are $\omega_1^3$ triplets $(\mu, \beta, \gamma)$ that can be comprised from the elements of $\omega_1$ given $\mu, \beta, \gamma < \omega_1$, we generally have $\phi : \omega_1 \rightarrow \omega_1^3$ (or, if a class function, $\phi: Ord \rightarrow Ord^3$). This is a little different than Fodor's lemma, which considers only $\phi : \omega_1 \rightarrow \omega_1$. That is not to say Fodor's lemma may not prove valuable in answering the question, as the cardinality of $\omega_1$ and $\omega_1^3$ are equal, etc.

I personally am asking because I am working with different models for what I have been calling $T$ sequences (sequences generated by starting with some initial finite segment of ordinals and then considering all the triplets, quadruplets, quintuplets, and so on, that can be made from the initial segment so as to generate rules that add additional ordinals to the sequence based on the availability of the triplets, quadruplets, etc., over $\omega$ initial segments generated using an iterative process).

Answer 1

Here is a solution, in two parts:

1. First, there is such a function $\phi$, as long as we restrict its domain to $\delta\smallsetminus\{0\}$ for some countable ordinal $\delta$. Indeed, we can define $\phi(n+1)=(0,n,0)$ for $n\in\omega$ and if $\delta$ is infinite, fix a bijection $g$ between $\delta$ and $\omega\times\omega$, and let $\phi(\alpha)=(1,g(\alpha))$ for $\omega\le\alpha<\delta$.

2. There is no such function if we want its domain to contain $\omega_1\smallsetminus\{0\}$. Indeed, the result follows from Fodor's lemma for $\omega_1$, which states that if $f$ is a regressive function defined on a stationary subset of $\omega_1$, then $f$ is constant on a stationary subset of its domain. Apply Fodor's lemma to the first coordinate of $\phi\upharpoonright\omega_1\smallsetminus\{0\}$. There is a stationary set $S_1$ where this first coordinate is fixed, with value (say) $\alpha$. Now apply Fodor's lemma to the second coordinate of $\phi\upharpoonright S_1$. Again, there is a stationary subset $S_2$ of $S_1$ where this second coordinate is fixed, say equal to $\beta$. One last application, now to the last coordinate of $\phi\upharpoonright S_2$ completes the proof: There is a stationary set $S_3\subset\omega_1$, and countable ordinals $\alpha,\beta,\gamma$, such that $\phi(\rho)=(\alpha,\beta,\gamma)$ for all $\rho\in S_3$.