Create a well ordering for functions "being $0$ near limits"

Question

Let $\gamma$ be an ordinal and let $A$ = {$\alpha_i$|$i$ < $\gamma$ }. Define $P$ to be $\Pi${$\alpha_i$|$i$$\in$ $\gamma$ }. We can also view $P$ as the set of all the functions $f$ pointing from $\gamma$ to $\cup$ {$\alpha_i$|$\alpha_i$ < $A$} such that $f(i)\in \alpha_i$ for all $i\in\gamma$.

We here give a new definition for a function $f$ $\in$ $P$. We say $f$ is "$0$ near limits" if for every limit ordinal $\lambda$ with $\lambda$ $\leqslant$ $\gamma$, there exists an $\mu$ < $\lambda$ such that $f(x) = 0$ whenever $\mu$ < $x$ < $\lambda$.

Define $P_0$ = {$f$ $\in$ $P$ | $f$ is $0$ near limits}. Now the question asks me to find a well-ordering for the set $P_0$.

My original attempt is to write all of these functions $f$ in $P_0$ into a sequence. Given $f_1, f_2$ from $P_0$, if they are different, then there exists a smallest element which belongs to $\gamma$, say $l$, such that $f_1(l) \neq f_2(l)$. In general, this is not enough for a well-ordering but let us try to remember this $l$ for a while.

Then, beside this first level of ordering, I turn to compare that $\mu$ given in the definition. Again for $f_1$ and $f_2$ and a limit ordinal $\lambda_0$ which is $\leqslant$ than $\gamma$, I can find $\mu_1$ for $f_1$ and $\mu_2$ for $f_2$ such that for any x between $\mu_1$ and $\lambda_0$, $f_1(x) = 0$ and for any y between $\mu_2$ and $\lambda_0$, $f_2(y) = 0$. After these two steps, I have an $l$, $\lambda_0$, $\mu_1$ and $\mu_2$.

Now, I want to create a order like the following:

If $f_1(l) < f_2(l)$, then I will directly conclude that f_1 is smaller than f_2.

Otherwise: For each limit ordinal $\lambda_0$ bigger than $l$, I can get a new $\mu_1$ and $\mu_2$ (resp.). First, given the smallest limit ordinal which is bigger than $l$, say $\lambda_{0, 0}$, I can obtain $\mu_{1, 0}$ and $\mu_{2, 0}$ (resp.) for $f_1$ and $f_2$(resp.) and then I will compare $\mu_{1, 0}$ and $\mu_{2, 0}$. If, WLOG, $\mu_{1, 0}$ is smaller, then I will conclude that $f_1$ is smaller than $f_2$. If not, then turn to the smallest ordinal which is bigger than $\lambda_{0, 0}$, say $\lambda_{0, 1}$ and then obtain $\mu_{1, 1}$ and $\mu_{2, 1}$ and compare them.

Continue this algorithm and if I can ever get a bigger $\mu_{1, i}$ or $\mu_{2, i}$, then stop. Otherwise, if $\gamma$ is also a limit ordinal and the corresponding $\mu_1$ and $\mu_2$ are the same, then I can conclude $f_1$ is the same as $f_2$. If $\gamma$ is not a limit ordinal, after the largest limit ordinal which is smaller than $\gamma$, say $\lambda_{\gamma}$, I only need to consider all ordinals between $\lambda_{\gamma}$ and $\gamma$ and there will only be finitely many of them.

I mainly have difficulty in applying this algorithm to all the functions in $P_0$ (if it is correct ...). My main purpose is to well order those functions block by block, which means well order their coordinates from $\lambda_{0, j}$ to $\lambda_{0, j + 1}$ and then from $\lambda_{0, j + 1}$ to $\lambda_{0, j + 2}$ and so on. However, I do not know how to construct the proof. Could anyone please tell me if my idea is correct? If you have ideas about how to write down this proof, I would like to hear some hints from you.

Thank you all viewers' help in advance.

Answer 1

Here is an outline of how you can wellorder $P_0$.

1. Prove that for $f \in P_0$, $\mathrm{sup}(f) := \{ x < \gamma \mid f(x) \neq 0 \}$ is finite.
2. Let $[\mathrm{Ord}]^{< \omega}$ be the class of all finite sets of ordinals. For $a,b \in [\mathrm{Ord}]^{< \omega}$ let $$ a <^* b \iff \max (a \Delta b) \in b, $$ where $a \Delta b$ is the symmetric difference of $a,b$. Prove that $<^*$ is a strict wellorder of $[\mathrm{Ord}]^{< \omega}$.
3. Recall that there is a definable bijection $$ \mathrm{Ord} \times \mathrm{Ord} \to \mathrm{Ord}, (\alpha, \beta) \to \langle \alpha, \beta \rangle, $$ (e.g. Gödel's pairing function).
4. For $f,g \in P_0$ let $$ f^* := \{ \langle \alpha, \beta \rangle \mid f(\alpha) = \beta \wedge \alpha \in \mathrm{sup}(f) \} $$ and $$ f < g \iff f^* <^* g^*. $$ Prove that this is a wellorder of $P_0$.