People love to wonder why the universe exists. But usually when such questions are asked they are asked in such a way that they incoherent, confused, or inherently unanswerable. So before we can address why something exists instead of nothing we must first clear away the confusions that may prevent the issue from being clearly addressed.
The first confusion is expecting a causal answer to the question, as reveal by such phasing as “why did the universe begin?” or “what is the cause of the universe?”. This is a confusion because the notion of causation is necessarily connected to that of time, as causation is a way of talking about how a event follows from previous ones, in time. And time is part of the universe. So clearly nothing can be the cause of the universe under a reasonable understanding of causation, because that would imply that the cause of the universe is itself in time. But if it is in time it is in the universe. So either the universe is self-causing or it is uncaused. Such an answer may be informative if you are interested in the nature of causation and whether an event being uncaused is possible. But it doesn’t address why the universe exists as a whole.
Another confusion is to think that somehow nothing is a default state that must be transformed into a universe that exists. There are actually two problems with thinking along these lines, the first of which is obviously with the idea that there can be an order of events, or change, outside of the universe. Obviously both of those require time, and time only exists within universes and only relates parts of that universe to each other. The other problem is with the assumption that nothingness is somehow the natural order of affairs, and that we must explain why it is that things deviate from it, such that the universe exists. Of course we may still want an explanation, I am just trying to point out that our explanation doesn’t have to struggle uphill. It is like explaining why the square root of a number has a particular value, your explanation doesn’t have to counter the possibility that it has some other value.
A third confusion is to think of nothingness as itself something. This generally comes in two varieties. The first is a linguistic confusion, where nothingness is mistakenly thought of as an object rather than as a syntactic device meaning “no particular thing”. For example, it would be a mistake to conclude from the true claim that nothing is the cause of everything that nothingness itself causes everything. Surprisingly this is a mistake that is made more often than you would expect. The second is thinking of nothingness as like a vacuum. Consider two universes (as the existence of a single universe doesn’t prevent existence of other universe, so long as they don’t interact). Often when people think of such a situation they imagine two universe “bubbles” floating next to each other, separated by nothingness. But this isn’t the case. Two universes cannot be spatially related to each other, as that would make them different parts of a single universe. Nor are they separated by nothingness, as that would make nothingness some kind of thing or expanse, as if you could get to nothing by simply going “outside” the universe.
In fact thinking of the universe in physical terms is often misleading. The universe is most naturally visualized as a bubble of space-time. This is sometimes useful, but leaves us with the false impression that space and time extend beyond the universe. To avoid this it is better to think of the universe as a set of facts. Think of the fundamental physical constituents as logical objects which various properties and relations can be predicated of. The physical laws are then like axioms, saying that if certain facts hold, if there is a certain arrangement of particles at a certain time, then certain other facts about those particles at later times must also hold. The set of facts that represents the universe is of course closed under these axioms (they can’t generate facts not already part of our set of facts) because our set of facts is meant to capture all the facts at all times, not just at a particular moment. And also note that in this set of facts every object is related to every other object by a number of relations (for example, the pair of relations “have the same charge” and “have different charges” relate every object between them). Now another universe would simply be another set of objects properties and relations with its own axioms. Obviously the objects of this universe aren’t related in any way to the objects of the first (because different properties and relations are involved). And thus to represent the state of affairs of both these universes existing we can simply consider the union of these two sets.
With this kind of thinking we can now address the original question. We can represent the non-existence of a single universe by considering it to be the empty set instead of the set containing all its facts, in terms of considering how it contributes to the state of everything. And the state of everything then is simply the union of all possible universes. Now there are an infinite number of internally consistent, and thus possible, universes that we can describe in this fashion. So the state of everything can only be the empty set, i.e. nothing exists, only if each of the infinite number of possible universes fails to exist. But, as established before, we shouldn’t consider the default state of a universe to be non-existence. While we don’t know how to estimate the probability that a given universe exists we have no justification for saying that probability is zero. Given that we estimate the probability of any given universe existing as greater than zero, and that there are an infinite number of such universes, we can conclude that the state of everything is necessarily not the empty set. And thus we can conclude that, given our current state of ignorance regarding why a universe exists or doesn’t (or whether there are any universes besides the one we find ourselves in), that it is impossible for nothing to exist, and thus that something must exist.