EMBRACING THE GROUNDLESSNESS OF THINGS 193

of objects, *X*_{0}, could have been anything. We may take *X*_{𝟂} or *X*_{𝟉} to be subsets
of *X*_{0}. Indeed, we may just take *X*_{0} to be the universal set, *V*, which contains
everything.^{25} In this case, the members of *X*_{𝟂} or *X*_{𝟉} have already been analysed.

And now, finally, to return to the problem with which we started this part of
the investigation: the worry was that the structure which provides loci is itself a
free-standing object.We now see that we do not need to assume this.The overall
structure is *X*_{𝟂} or *X*_{𝟉} (depending how you look at it); and this has turned out to
be as empty as anything else. Indeed, we may suppose that the whole Net of Indra
is a node in its own network. The whole network is one jewel amongst many.^{26}

In this chapter we have looked at two objections to emptiness. The first was that it gives rise to a vicious regress. The second was that even to say that something is empty presupposes that some things are not thus. Both charges are, as we have seen, without foundation. The legitimacy of non-well-founded structures, including non-well-founded sets, has played an important role in this discussion. Buddhist philosophers who have taken the Madhyamaka and Huayan insight to heart have always insisted on the ontological groundlessness of things. They did not have the resources of contemporary non-well-founded mathematics to employ in their theorisation, of course. However, I have no doubt that they would have been absolutely delighted to learn of such a possibility!

So much for the coherence of emptiness. In the next few chapters, we will look at some of its implications, starting, in the next one, with implications concerning language and its limits.

^{25}
There is no universal set in standard non-well-founded set theory—or contradiction would
arise. However, naive (paraconsistent) set theory has a universal set, V, as well as non-well-founded
sets; indeed, the universal set is one of them, since V ∈ V. (See Priest (1987), chs. 10 and 18.) V is the
totality of all (empty) objects. In Madhyamaka terms, one might think of this as emptiness (ultimate
reality) itself. And since V ∈ V, emptiness is itself empty, as Madhyamaka has it.

^{26}
In non-well-founded set theory, we can construct a tree which is its own root, as follows. Take
a tree, *t* (which is a set of ordered pairs) with root, *r*. The set, *x*, which is a solution to the equation
*x* = {(*x, r*)} ∪ *t*, is a tree which is its own root. (See, for example, Barwise and Etchemendy (1987),
ch. 3, or Aczel (1988), ch. 1.)