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Harvey Friedman is well known for investigating concrete mathematical statements that requires strong assumptions, i.e. those that can only be interpreted in a strong extension of $\text{ZF(C)}$. My first question:

  1. Is there any known example of a concrete mathematical statement that requires the strength of an extension of $\text{ZF}$ that can prove the existence of a Reinhardt cardinal?

My second question is:

  1. If we suppose such an example had been put forth, would that constitute a reason to reject Choice?

The idea is that if 1 is true, then this might open the door for many examples of concrete mathematical statements that can only be interpreted in extensions of $\text{ZF}$ that can prove the existence of a Reinhardt cardinal or higher cardinals, but it is known that all such extensions negate Choice. And by then Choice would appear as a restrictive assumption, since it would restrict interpret-ability of concrete mathematical statements, which is the primary goal of foundation after all.

  1. What makes us believe in an axiom? is it its formalization ability (i.e. its interpret-ability strength) or its intuitive appeal? so in a situation like the above, which would we favor?
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You're leaving yourself too much wiggle room for "concrete mathematical statement".

Is this going to be something like TREE-related statements, or about Laver tables? Is this going to be something like AD? Is this going to be something like "There exists a topological space such that something"?

Yes, there is an issue with defining a "concrete mathematical statement", and arguably $\sf\operatorname{Con}(ZF+\exists\kappa\text{ Reinhardt})$ is not an example of a statement that you're looking for. But nevertheless, it's unclear what is a statement that you are looking for.

Now, to your actual question, I don't quite know the answer. I don't think that there has been much research towards this either. You could argue that he failure of the HOD Conjecture requires the consistency of a Reinhardt cardinal, and that the HOD Conjecture is a concrete mathematical statement. But not that much more has been said and done on the topic.

For your second question, would you say that you should accept $\sf CH$ because of one of many "weird" constructions? or consequences from Forcing Axioms ($\sf PFA$ or $\sf MM$) should motivate you to reject $\sf CH$?

If your answer is yes to any of the above, then a Reinhardt cardinal may or may not be a reason to reject choice, depending on the type of statement you found, how compelling you found it, and whether or not you want to reject choice. Ultimately, the only reason to reject choice is that you want to reject choice.

Being an anti-Platonist, I can only suggest rejecting the whole view of "accepting or rejecting something".

For your third question, it's a combination of plausibility, usability, and our education. Your mileage may vary.

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    $\begingroup$ Your statement that "it's clear to everyone" is not true, since I find it perfectly reasonable and even mundane to say that a consistency assertion is a concrete mathematical statement, of complexity $\Pi^0_1$. Perhaps this issue is like the debates about "natural" mathematical statements. I haven't ever seen a robust or even a satisfactory notion of what counts as "natural", and the reality is that people often disagree about what counts as natural. Too often, some people end up rejecting as unnatural mathematical statements arising in an area with which they are simply unfamiliar. $\endgroup$
    – Joel David Hamkins
    Commented May 21, 2018 at 18:05
  • $\begingroup$ Joel, I think that we're interpreting "concrete" in different ways, and I interpret it as "natural" in your "quote-unquote" sense. But I also feel that the OP does too. I fear that this is a barrier that you, as a native speaker, is on the other side of the issue for once. :) $\endgroup$
    – Asaf Karagila ♦
    Commented May 21, 2018 at 18:06
  • $\begingroup$ Its not about Platonism really, I'm speaking from a more general foundational point of view. Because of Godel we cannot interpret all mathematics in one theory, but we can interpret all of it in a line of theories of ascending strength, where each theory extends the prior one, now this line of theories is not effectively generated, but still if I come to know that there is an axiom in the theory at the bottom of that line that would impose a limit on interpret-ability of mathematical statements in the high up theories, then I'd reject it. $\endgroup$
    – Zuhair Al-Johar
    Commented May 21, 2018 at 18:08
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    $\begingroup$ @Asaf I am rejecting the idea that there is a meaningful notion of "concrete mathematical statement", just as I reject the notion that there is meaningful notion of "natural mathematical statement". We can measure the complexity of mathematical statements in various precise ways, such as in the arithmetic hierarchy, and by this measure, consistency assertions come out as being comparatively concrete, if not extremely concrete. $\endgroup$
    – Joel David Hamkins
    Commented May 21, 2018 at 18:13
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    $\begingroup$ My main point in commenting was simply to point out that the "it's clear to everyone" statement is over-stated, since in fact there is debate about this kind of thing. $\endgroup$
    – Joel David Hamkins
    Commented May 21, 2018 at 18:18

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