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from $\sf L-S$ theorems we can have an external bijection $j$ from a set to its powerset. Obviously, $j$ cannot be fully internalized (used in all instances of Replacement). However, partial internalization is possible. The question here is about if $j$ can be partially internalized in the particular manner described below:

$\exists \alpha: \text{ limit} (\alpha) \land j: V_\alpha \to V_{\alpha +1} \land j \text{ is a bijection }$

$ \forall A \ \exists B: B=\{\{j(m),j(n)\} \mid \{n,m\} \in A\}$

$ \forall A \ \exists B: B=\{\{n,m\} \mid \{j(m),j(n)\} \in A\}$

In the other words, is it consistent to add to the language of $\sf ZF$ a partial function symbol $j$, and add [to axioms of $\sf ZF$] the above three sentences as axioms? Provided, of course, that Replacement doesn't use $j$.

From $\sf L-S$ theorems we can have an external bijection $j$ from a set to its powerset. Obviously, $j$ cannot be fully internalized (used in all instances of Replacement). However, partial internalization is possible. The question here is about if $j$ can be partially internalized in the particular manner described below:

$\exists \alpha: \text{ limit} (\alpha) \land j: V_\alpha \to V_{\alpha +1} \land j \text{ is a bijection }$

$ \forall A \ \exists B: B=\{\{j(m),j(n)\} \mid \{n,m\} \in A\}$

$ \forall A \ \exists B: B=\{\{n,m\} \mid \{j(m),j(n)\} \in A\}$

In the other words, is it consistent to add to the language of $\sf ZF$ a partial function symbol $j$, and add [to axioms of $\sf ZF$] the above three sentences as axioms? Provided, of course, that Replacement doesn't use $j$.

from $\sf L-S$ theorems we can have an external bijection $j$ from a set to its powerset. Obviously, $j$ cannot be fully internalized (used in all instances of Replacement). However, partial internalization is possible. The question here is about if $j$ can be partially internalized in the particular manner described below:

$\exists \alpha: \text{ limit} (\alpha) \land j: V_\alpha \to V_{\alpha +1} \land j \text{ is a bijection }$

$ \forall A \ \exists B: B=\{\{j(m),j(n)\} \mid \{n,m\} \in A\}$

$ \forall A \ \exists B: B=\{\{n,m\} \mid \{j(m),j(n)\} \in A\}$

In the other words, is it consistent to add to the language of $\sf ZF$ a partial function symbol $j$, and the abve three sentences as axioms? Provided, of course, that Replacement doesn't use $j$.

from $\sf L-S$ theorems we can have an external bijection $j$ from a set to its powerset. Obviously, $j$ cannot be fully internalized (used in all instances of Replacement). However, partial internalization is possible. The question here is about if $j$ can be partially internalized in the particular manner described below:

$\exists \alpha: \text{ limit} (\alpha) \land j: V_\alpha \to V_{\alpha +1} \land j \text{ is a bijection }$

$ \forall A \ \exists B: B=\{\{j(m),j(n)\} \mid \{n,m\} \in A\}$

$ \forall A \ \exists B: B=\{\{n,m\} \mid \{j(m),j(n)\} \in A\}$

In the other words, is it consistent to add to the language of $\sf ZF$ a partial function symbol $j$, and add [to axioms of $\sf ZF$] the above three sentences as axioms? Provided, of course, that Replacement doesn't use $j$.

from $\sf L-S$ theorems we can have an external bijection $j$ from a set to its powerset. Obviousely, $j$ cannot be fully internalized (used in all instances of Replacement). However, partial internalization is possible. The question here is about if $j$ can be partially internalized in the particular manner described below:

$\exists \alpha: \text{ limit} (\alpha) \land j: V_\alpha \to V_{\alpha +1} \land j \text{ is a bijection }$

$ \forall A \ \exists B: B=\{\{j(m),j(n)\} \mid \{n,m\} \in A\}$

$ \forall A \ \exists B: B=\{\{n,m\} \mid \{j(m),j(n)\} \in A\}$

In the other words, is it consistent to add to the language of $\sf ZF$ a partial function symbol $j$, and the abve three sentences as axioms, provided, of course, that Replacement doesn't use $j$.

from $\sf L-S$ theorems we can have an external bijection $j$ from a set to its powerset. Obviously, $j$ cannot be fully internalized (used in all instances of Replacement). However, partial internalization is possible. The question here is about if $j$ can be partially internalized in the particular manner described below:

$\exists \alpha: \text{ limit} (\alpha) \land j: V_\alpha \to V_{\alpha +1} \land j \text{ is a bijection }$

$ \forall A \ \exists B: B=\{\{j(m),j(n)\} \mid \{n,m\} \in A\}$

$ \forall A \ \exists B: B=\{\{n,m\} \mid \{j(m),j(n)\} \in A\}$

In the other words, is it consistent to add to the language of $\sf ZF$ a partial function symbol $j$, and the abve three sentences as axioms? Provided, of course, that Replacement doesn't use $j$.