Initial caveat: some misapprehension seems to have arisen over my reference to physical sets. But in this, I am trying to follow the language of modern topology, which seems to be applied everywhere without hesitation in physics (as with quantum field theory). I should also mention Augenstein (c.f. Svozil), which I saw linked to over on the MathOverflow (in a post asking about connections between set theory and physics). —Now, I suppose there is a problem with my question being posed on the PhilosophySE if it just so happens that the more helpful responses will have to come from people who are well-versed in both philosophy and physics; I optimistically assume there are respondents meeting those criteria, however (migrating my question to the PhysicsSE would probably trigger a "fringe science" VTC, I pessimistically expect).
Without further ado... For the sake of specific argument, let strong physicalism be the thesis that the universe is physically closed. (Weak physicalism would be more like methodological naturalism, or then an epistemological thesis, which I have no strong objections to (AKA instead of "all possible objects are physical" we would just say "all strongly knowable objects are physical," although as to what "strong knowledge" would be, well...).) Now closure is a matter of quantification via predication, i.e. quantifying over satisfiers of predicates, whereby we quantify over a closed set of satisfiers, and since we are speaking of a universe, then we are speaking of universal closure. It seems then that:
- If the universe is the set of all physical objects, then if this set is well-founded, then the universe is not itself a physical object. So it would be (A) a physical non-object, (B) a non-physical object, or (C) a non-physical non-object.
- If the universe is the set of all physical objects and is a physical object itself, then it is not a well-founded set.
- If the universe is the set at the head of an infinitely descending membership chain, then the regress of physical objects doesn't terminate in smallest physical objects, so subatomic particles would not be "indivisible" after all and there would never be a limit to particle decomposition.
None of these options seems unproblematic. (1) ends up with hyperphysical objects, physical hyperobjects, or hyperphysical hyperobjects, any of which seem perhaps fancifully named (or outright outlandish); mystics and theists might be happy with such a conclusion, but not so much would physicalists, I suppose.☆ (2) would mean that the objective world has circular phenomena, in fact is a circular phenomenon "as a whole," which raises the specter of allowing circularity among the parts too, and I know there are those who would be upset with this.☆☆ (3) would irritate believers in the absolutely small, and might raise the issue of hyperstuff too (though the predicate inversion function on hypersets works a little differently compared to the one for well-founded sets).
Can Kant's antinomies be reformulated as tension/conflict between not just two, but three, lines of reasoning (per antinomy)?
☆Denying that the universe is an object seems equivalent to denying that the universe is predicatively closed in the first place (the default concept of an object is that which predicates are true of, so if the universe is not an object, then the universe is not a boundary of predication, but is open, and then physicalism is defeated at least insofar as we can't epistemically rule out the existence of hyperphysical predicates and objects beyond the physical domain).
☆☆Closed timelike curves, for example, though, seem admissible among physicists.
First corollary: or put the issue in terms of physical laws. If the set of all physical laws is well-founded, then this set does not itself represent a physical law, so it represents (A) a physical non-law, (B) a non-physical law, or (C) a non-physical non-law. Or if this set is an element of itself, then the quintessential law(s) of the entire universe will be derived by circular reasoning. Or if there is a highest physical law that is not well-founded, then there will be no fundamental physical laws and nuclear physics will prove no more fundamental than chemistry turned out to be. —Or consider temporal sets: if there is a well-founded set of all temporal sets, then this set is not itself temporal; or time encloses itself; or time goes backwards forever without a beginning.
Second corollary: the case of causal sets, and so the Third Antinomy, seems fraught with peculiarities. (I would qualify that phrase "causal sets" as "generically causal sets," to avoid bringing in the details of the actual causal-set program in the metatheory of quantum gravity.) To wit:
- A well-founded set of all causes would not be a cause, and therefore would have no effect. A well-founded set of all effects would not be an effect, and therefore would have no cause.
- A circular set of all causes would have some effect, and a circular set of all effects would have some cause. I'm not sure the universal generically causal set would be a contributing cause, directly, to every effect, but maybe it would be the cause of the universal generic effect-set specifically.
- Hyperfounded generically causal/effected sets might pertain to backwards causation/quantum retrocausality, but for now I'm kind of just guessing about that.