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In a prior posting, I've posed the idea of reducing set theory to an extended kind of second order ordinal arithmetic $\small \sf`` 2 oO A"$. The idea was to have a domain of ordinals and sets of ordinals where the set of all ordinals is well ordered by relation $<$ denoting ordinal strict smaller than, that have an ordered pairing function on ordinals to ordinals, and that liberally define sets of ordinals after any formula. Thus it can define relations between ordinals as sets of ordinals that are ordered pairs of ordinals. The ordinals are not limited to the naturals, they are extended by axioms of infinity, size and successor cardinals, so that it results in defining all kinds of ordinals that $\small \sf ZFC$ can define. The idea is that the known set theory $\small \sf ZFC$ can be interpreted in this system via building $\sf L$ inside it. Now I'm wondering about if this extended kind of second order ordinal arithmetic can be extended further to the same length $\small \sf ZFC$ set theory can be extended. Can for example the Wholeness axiom be expressed and have the same strength as it is with extending set theory? Thereby playing the same foundation role extensions of $\small \sf ZFC$ has as regards encoding large cardinal properties.

Wholeness axiom: there is a non-trivial elementary embedding $j$ from the set of all ordinals to itself, with respect to formulas that doesn't mention $j$.

If we assume that $\small \sf 2 oO A$ interprets $\small \sf ZFC$, then would $\small \sf 2oOA$ plus that wholeness axiom be equi-interpretable with $\small \sf ZFC$+ Wholeness axiom?

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    $\begingroup$ In general your language have the same expressive power as the usual language of set theory. Consider the well-founded graph translation $\tau$ of language of $ZFC$ into your language, which idea is to interpret sets by well-founded extensional binary relations $(A,R)$ with a fixed point $a\in A$ (by Mostowski's transitive collapse theorem there is a unique transitive set $X$ such that $(A,R)$ and $(X,\in)$ are isomorphic, the whole triple $(A,R,a)$ encodes the set $f(a)$, where $f$ is the isomorphism). The translations of $=$ and $\in$ are naturally defined according to this understanding. $\endgroup$ – Fedor Pakhomov Sep 10 '19 at 13:05
  • $\begingroup$ Then, in order to obtain an extension of your theory corresponding to some $T\subseteq ZFC$, you just add to your base theory all $\tau(\varphi)$, for axioms $\varphi$ of $T$. $\endgroup$ – Fedor Pakhomov Sep 10 '19 at 13:05
  • $\begingroup$ I want an extension of my theory corresponding to some $T \supseteq ZFC$ $\endgroup$ – Zuhair Al-Johar Sep 10 '19 at 15:42
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    $\begingroup$ It was a typo, should have been $T\supseteq ZFC$. $\endgroup$ – Fedor Pakhomov Sep 10 '19 at 15:47
  • $\begingroup$ @FedorPakhomov, I think I got what you mean, you interpret sets after graphs, and the later are encoded by sets of ordered pairs of ordinals, etc... I didn't work out the details, of course this way we get to extend this theory as far as ZF can go by brute translation force. However, I'm interested if we can extend this arithmetic naively without such explicit translations in mind, and whether those extensions can work to be as strong as those of ZFC. Look at the wholeness axiom, it doesn't mention any such translation, but it might turn to be equivalent to the one extending set theory. $\endgroup$ – Zuhair Al-Johar Sep 11 '19 at 11:03

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