Why do many philosophers posit that there are no members in the set of necessary beings? There seem only two explanations if they are correct: 1) Necessary beings are logically possible, but none exist in this world or 2) Necessary beings are logically impossible. Explanation 1 seems untenable since if a necessary being exists in one world (is logically possible), then it must exist in all worlds (and thus this one) by virtue of its necessity. But explanation 2 (which seems likely the more preferred one) seems to do no better, since the set of necessary beings is made a subset of the set of impossible beings. While perhaps this is merely a trivial case, it still seems unsettling, if not contradictory. Is the existence of at least one necessary being necessary? Or is there some other explanation for how none could exist?

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Just a quick comment on your remark about (2). If there are no necessary beings, then the set of necessary beings is empty. The empty set is a subset of every set. (Every element of the empty set is a member of any given set — since the empty set has no elements.) Hence, if there are no necessary beings, the set of necessary beings is indeed a subset of the set of impossible beings, just as it is a subset of any set. I'm not quite sure what an impossible being is, so I'm not quite sure what the set of impossible beings is. It sounds to me like it's another way of describing the empty set. But whatever the set of impossible beings is, we know that, if there are no necessary beings, then the set of necessary beings is a subset of it. I don't see any contradiction here.

There's another distinction that needs to be made here and that is relevant to the objection to explanation (1): We need to distinguishdifferent sorts of necessity. Nowadays, most philosophers and logicianswould agree that there is nothing whose existence is logically necessary, even the objects of mathematics, although their existence is mathematically and even metaphysically necessary. Even God's existence would not be regarded as logically necessary, even by philosophers who accept God's existence. Perhaps we should regard it as agreed, then, that, if God exists, God exists by metaphysical necessity. If so, then there is no contradiction in holding that it is logically possible that God should have existed. Whether it is consistent to hold that it is metaphysically possible that God should have existed depends upon whether one thinks the so-called Brouweresche axiom of modal logic holds for metaphysical necessity. (Axiom B says that, if it is possible that it is necessary that A, then A is true.) Most metaphysicians would, I think, accept B, but there is some controversy there.

In any event, as you speculate, the popular option is likely to be that it is not metaphysically possible that there should have been a metaphysically necessary being like God. (Many philosophers would suppose that there are plenty of things whose existence is metaphysically necessary: the natural numbers, for example.) As Alex says, there's no immediate contradiction in this view, and the apparent oddity of holding that the set of divine beings (say) is a subset of the set of impossible beings is merely apparent. The set of female US presidents is empty and so a subset of the set of impossible beings, but it doesn't even follow that there might not have been a female US president. It's just a reflex of the fact that the empty set is, in virtue of how the notion "subset" has been defined, a subset of every set.

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