I am considering a Principle of Infinitude, expressed as follows - for a class theory where precisely the elements are sets - with the aid of set abstracts:

For $\alpha(y,z)$ a first order condition so that $\forall y(\exists w(y\in w)\to \exists w (\{z|\alpha(y,z)\}\in w))$:

$\forall v(\exists t (v\in t)\to\exists t(\{w|\forall x(v\in x\wedge \forall y(y\in x\to \{z|\alpha(y,z)\}\in x)\to w\in x)\}\in t))$

The set abstracts can be eliminated with the following Mendelsonian abstraction schema:

$\forall x(x\in \{x|\alpha\}\leftrightarrow\exists y(x\in y)\wedge\alpha)$

It is immediate that we will get the existence of $\omega$, as well as the least transitive closure of all sets; moreover, several further instances of replacement will hold.

May Z with Infinitude, instead of just the existence of an infinite set, justify the Recursion Theorem?

I am considering a Principle of Ubiquity, expressed as follows - for a class theory where precisely the elements are sets - with the aid of set abstracts:

For $\alpha(y,z)$ a first order condition so that $\forall y(\exists w(y\in w)\to \exists w (\{z|\alpha(y,z)\}\in w))$:

$\forall v(\exists t (v\in t)\to\exists t(\{w|\forall x(v\in x\wedge \forall y(y\in x\to \{z|\alpha(y,z)\}\in x)\to w\in x)\}\in t))$

The set abstracts can be eliminated with the following Mendelsonian abstraction schema:

$\forall x(x\in \{x|\alpha\}\leftrightarrow\exists y(x\in y)\wedge\alpha)$

It is immediate that we get a theorem of infinity, as well as the least transitive closure of all sets; moreover, several further instances of replacement will hold, though with countable co-finality.

May Z with Ubiquity, instead of just the Axiom of Infinity, justify the Recursion Theorem?