Today I started reading Maddy's Believing the axioms. As I knew beforehand, it includes some discussion of ZFC axioms. However, I really hoped for a more extensive discussion of axiom of foundation/regularity.
Apparently, the reason why we usually take it is because it makes sets well-founded and makes $\in$-induction work, or because it puts all sets into a hierarchy (namely $V$). However, these reasons sound to me more like "we take this, because it's convenient". Another reason commonly given is "It's difficult to think of a set which is an element of itself". This is not a good reason, because many things are difficult to think of, and also one could argue that a set represented by $\{\{\{…\}\}\}$ should do the trick.
That brings me to my question:
Are there any "philosophical" reasons to believe that the axiom of regularity holds?
I understand that this question is quite vague and maybe too broad, but I will be thankful for any responses.
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Best Answer
Regularity (aka Foundation) can be seen philosophically as an axiom of restriction. It is not necessarily saying “all the things you consider as sets must be well-founded”. It can be read saying “for the purposes of this set theory, we restrict our universe of discourse to just the well-founded objects”. It’s clarifying what we mean by sets, in a similar way as the extensionality axiom does.
You may find this explanation unsatisfying, since it’s fairly similar to what Maddy gives. But the point is that if you are philosophically unsure about it, the question to ask is not “Are all sets really well-founded?” but “Is it really convenient/harmless/natural to restrict attention to the well-founded sets?”
A precise statement which can be seen as justifying this is the fact that within (ZF – Regularity), one can prove that the class of well-founded objects is a model of ZF.
Edit: see this followup question and its answer for:
a rather stronger sense in which regularity is harmless, in the presence of choice: ‘Over (ZFC – regularity), regularity has no new purely structural consequences’
a counter-observation that in the absence of choice, over (ZF – regularity), it’s not so clearly harmless; it has consequences that can be stated in purely structural terms, such as ‘every set is isomorphic to the set of the children of some element in some well-founded extensional relation’.