**Motivation:**


This question is motivated by wondering to what extent "natural" theories are linearly ordered (or at least ordered in a [directed](https://en.wikipedia.org/wiki/Directed_set) manner) by their (first-order) arithmetic consequences, in analogy to the phenomenon that "natural" theories seem to be linearly ordered by consistency strength.


In an answer to [this question](https://mathoverflow.net/questions/236408/does-the-consistency-strength-hierarchy-coincide-with-the-arithmetic-consequenc?noredirect=1&lq=1), Joel David Hamkins gave several examples illustrating how these two hierarchies -- arithmetic consequence versus consistency strength - may differ, and indeed his examples also illustrate that the ordering by arithmetic consequence is not linear, even for arguably "natural" theories.


But it so happens that these examples largely involved playing games with consistency statements, so the departure from linearity feels "small". One way of making this a bit more precise is that it seems, as far as I can tell, to still be possible that the partial ordering of theories is still *directed*, at least for "natural" theories. That is, what I know about the matter is consistent with the idea that there really is one "true arithmetic" with which all of our "natural" or "serious" theories (apart from things like $T + \neg Con(T)$ for various $T$) are consistent.


In order to have a hope of challenging this view, it seems one needs an alternate "coherent" picture of the mathematical universe. The only example that comes to mind for me is $ZF + AD$, which is inconsistent with ZFC, but nonetheless its consistency strength is well-calibrated (and nontrivial), and seems to have some sort of "inner logic" to it which could potentially yield a different picture of the world even at the level of arithmetic. AD [paints a different picture of the universe](http://cantorsattic.info/Axiom_of_determinacy#Implications_of_the_axiom_of_determinacy) at least if we include non-arithmetic statements in our scope. But I'm not sure it does anything unusual at the level of arithmetic.


**Questions:**


1. Are the arithmetic consequences of $ZF+AD$ *consistent* with standard theories like $ZFC + L$ for various large cardinal axioms $L$?
2. Are the arithmetic consequences of $ZF+AD$ *implied* by large cardinals?
3. More broadly, is there any good candidate out there for a theory $T$ whose arithmetic consequences are inconsistent with (or at least not implied by) ZFC + large cardinals, such that $T$ has some kind of "inner coherence" (in analogy to how large cardinals are said to have "inner coherence" by virtue of inner model theory -- as opposed to theories of the form $T + \neg Con(T)$ and the like)?



Does ZF+AD have any unusual arithmetic consequences?
For (1) and (2), the key point here is **absoluteness**. For example, suppose $V$ is a model of ZFC + enough large cardinals. Then $L(\mathbb{R})^V$ satisfies ZF + DC + AD. But $L(\mathbb{R})^V$ and $V$ have the same natural numbers, hence satisfy the same arithmetic sentences. So if $(\*)$ is a "big" large cardinal axiom - namely one which is strong enough to imply that determinacy holds in some inner model - then any arithmetic consequence of determinacy is also a consequence of $(\*)$. 


* Indeed, as far as I know all such axioms imply that determinacy holds in $L(\mathbb{R})$ - and that gives us not just the arithmetic facts, but the *projective* facts as well: every second-order consequence of determinacy is a consequence of any large cardinal axiom implying that determinacy holds in any model containing all the reals, in particular $L(\mathbb{R})$.


This addresses (1) and (2). The same reasoning provides a strong heuristic (in my opinion) that the answer to (3) will be "no" - any such theory would have to somehow be fundamentally unrelated to outer/inner models, to put it mildly, and we don't know any such candidate yet (ignoring things like ZFC + $\neg$ Con$(\*)$). The situation, annoyingly, remains:



> 
> So far, we know of no situation where different set-theoretic axioms (ignoring "pathological" examples like inconsistency statements) yield contradictory arithmetic consequences.
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> 
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