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I am hesitant to attempt this, so If this makes no sense, or is not in keeping with what you are looking for I'm sorry.

We will be playing a trick on ZF, namely we will be using the fact that NBG is a conservative extension of ZF.

Firstly, let $R_1, R_2 \subset \kappa\times\kappa$, be the two well-orderings of $\kappa$ we are considering. Then, as $On \kappa \times On$ \kappa$has a well-ordering$\lt_O$and in fact is which has order isomorphic type with$On$in NBG, \kappa$ via some isomorphism $P:On P:\kappa \times On \kappa \rightarrow On$, \kappa$, with for each$(\alpha,\beta)\in On \kappa \times On$:\kappa$:

$\gamma = max(\alpha,\beta) \implies P((\alpha,\beta)) = P(\alpha,\beta) \subset in P''((\gamma+1)\times(\gamma+1)).$

(This is a modification of the global ordering from Set Theory and The Continuum Hyp. by R. M. Smullyan and M. Fitting, p. 119).

The

This ordering is defined as follows: $(\alpha,\beta)\lt_O(\gamma,\delta)$ iff one of the following holds

• $max(\alpha,\beta) \lt max(\gamma,\delta)$, or
• $max(\alpha,\beta) = max(\gamma,\delta)$, and $\alpha \lt \gamma$, or
• $max(\alpha,\beta) = max(\gamma,\delta)$, and $\beta \lt \delta$

As such, we may form the following construction: Let $r^k_0=min_{\lt_O} (R_k)$ and for $\alpha \lt \kappa$ define $r^k_\alpha = min_{\lt_O} (R_k \backslash \{r^k_\gamma :\gamma \lt \alpha\} )$ for $k=1,2.$

Next following along with the answer from, Alessandro Sisto, we may define the sequence $u_\alpha$

$u_\alpha = min\{ \beta \in P''(\alpha+1)\times(\alpha+1): \forall \gamma \lt \alpha ( P^{-1}(u_{\gamma}) \lt_O r^1_{\beta} \text{ and } P^{-1}(u_\gamma) \lt_O r^2_\beta) \}.$

Then $u_\alpha$ will be defined for every $\alpha \lt \kappa$, and so preforming the same bit that Alessandro did, with $y_\alpha = min \{ \beta \in \kappa: \exists \gamma \lt \alpha( (\eta,\beta) = P^{-1}(y_\gamma)) \}$ should produce the result.

Since the result follows from NBG, we have that the result follows from ZF.

1

I am hesitant to attempt this, so If this makes no sense, or is not in keeping with what you are looking for I'm sorry.

We will be playing a trick on ZF, namely we will be using the fact that NBG is a conservative extension of ZF.

Firstly, let $R_1, R_2 \subset \kappa\times\kappa$, be the two well-orderings of $\kappa$ we are considering. Then, as $On \times On$ has a well-ordering $\lt_O$ and in fact is order isomorphic with $On$ in NBG, via some isomorphism $P:On \times On \rightarrow On$, with for each $(\alpha,\beta)\in On \times On$:

$\gamma = max(\alpha,\beta) \implies P((\alpha,\beta)) = P(\alpha,\beta) \subset P''((\gamma+1)\times(\gamma+1)).$

(This is from Set Theory and The Continuum Hyp. by R. M. Smullyan and M. Fitting, p. 119).

The ordering is defined as follows: $(\alpha,\beta)\lt_O(\gamma,\delta)$ iff one of the following holds

• $max(\alpha,\beta) \lt max(\gamma,\delta)$, or
• $max(\alpha,\beta) = max(\gamma,\delta)$, and $\alpha \lt \gamma$, or
• $max(\alpha,\beta) = max(\gamma,\delta)$, and $\beta \lt \delta$

As such, we may form the following construction: Let $r^k_0=min_{\lt_O} (R_k)$ and for $\alpha \lt \kappa$ define $r^k_\alpha = min_{\lt_O} (R_k \backslash \{r^k_\gamma :\gamma \lt \alpha\} )$ for $k=1,2.$

Next following along with the answer from, Alessandro Sisto, we may define the sequence $u_\alpha$

$u_\alpha = min\{ \beta \in P''(\alpha+1)\times(\alpha+1): \forall \gamma \lt \alpha ( P^{-1}(u_{\gamma}) \lt_O r^1_{\beta} \text{ and } P^{-1}(u_\gamma) \lt_O r^2_\beta) \}.$

Then $u_\alpha$ will be defined for every $\alpha \lt \kappa$, and so preforming the same bit that Alessandro did, with $y_\alpha = min \{ \beta \in \kappa: \exists \gamma \lt \alpha( (\eta,\beta) = P^{-1}(y_\gamma)) \}$ should produce the result.

Since the result follows from NBG, we have that the result follows from ZF.