# What is the consistency strength of F accessibility?

Let's assume all axioms of $\text{Z}- \text{Infinity}$.

Now let $F$ be any function that is definable over the whole universe of discourse by a formula in the language of $\text{Z}$.

Now we define $F$ related accessibility $ACC^F"$ as:

$$X \ ACC^F \alpha \iff \alpha <X \wedge \exists \beta < X \ \big{[}F \big{(}\bigcup(\alpha \cup \beta)\big{)} \geq X \big {]}$$

Where: $x < y \iff \exists f (f:x\to y \wedge f \text{ is an injection}) \wedge \not \exists g (g: y \to x \wedge g \text{ is an injection})$

and: $x \geq y \iff \exists f (f: y \to x \wedge f \text { is an injection} )$

Define $F$ related linear accessibility $LACC^F"$ as:

$$X \ LACC^F \alpha \iff \alpha <X \wedge \exists \beta <X \ \big{[} \beta \ ACC^F \alpha \wedge F \big{(}\bigcup (\alpha \cup \beta)\big{)} \geq X \big{]}$$

$\text{Axiom schema of Accessibility:}$ if $F$ is a binary function symbol, and $\phi(y)$ is a formula in which $x$ doesn't occur free and $y$ occurs free and only free, then all closures of:

$$\exists \alpha \forall y [\phi(y) \to y \ LACC^F \alpha] \to \exists x \forall y \ [y \in x \leftrightarrow transitive(y) \wedge \phi(y) ]$$

are axioms.

Where: $transitive(y) \iff \forall m \in y \ (m \subset y)$

Now $\text{Z}-\text{INF.} + \text{Accessibility}$ would prove $\text{Con(ZF)}$, I assume.

Should this theory be consistent, what is the consistency strength of $\text{Z}-\text{INF.}+ \text{Accessibility}$?

Does it prove Replacement?

Afternote: a continuation of the line of thought along this theory that might help salvage this approach is present at:

What is the consistency strength of Z+ Accessibility?

• In the last line, what is "Rank," and should "ZF" be "Z"? Mar 11, 2018 at 19:13
• The way it is now, the text is extremely confusing, I believe not only for me. Seems like mixture of something obsolete with something replacing it. Is not it possible to leave only the current state of the question? Mar 12, 2018 at 11:41
• OK, I'll post the corrections in another question Mar 12, 2018 at 11:43
• Good. I suppose the followup is "What is the consistency strength of Z+ Accessibility?"? Could you add a link to it in this question? Mar 14, 2018 at 2:34

Take $$F(x)=\bigcup_{y\in x}(\mathcal{P}(y)).$$ Then for all $X$ with more than one element and any $a$ with $a<X$ we have $X$ ACC$^F$ $a$: taking $b=\{\{X\}\}$, we have $\mathcal{P}(X)\subseteq F(\bigcup (a\cup b))$.
Now consider $\varphi(y)$ to be the statement "$y$ has more than two elements," and let $a=2$ ...