  The argument here may be reconstructed as follows. Its overall  structure is: If there are many things, then there must be finitely  many things; and if there are many things, then there must be  infinitely many things. The assumption that there are many things is  thus supposed to have been shown to lead to contradiction, namely,  that things are both finitely and infinitely many. The particular  argument for the first arm of the antinomy seems to be simply: If  there are many things, then they must be just so many as they are. If the many things are just so many as they are, they must be finitely  many. Therefore, if there are many things, then there must be  finitely many things. Simplicius somewhat loosely describes the  antinomy’s second arm as demonstrating numerical infinity through  dichotomy (Simp. in Ph. 140.33–4). In fact, the argument  depends on a postulate specifying a necessary condition upon two  things being distinct, rather than on division per se, and  it may be reconstructed as follows: If there are many things, they  must be distinct, that is, separate from one another. Postulate: Any  two things will be distinct or separate from one another only if  there is some other thing between them. Two representative things,  x1 and x2, will be distinct  only if there is some other thing, x3, between  them. In turn, x1 and x3 will be distinct only if there is some other thing,  x4, between them. Since the postulate can be  repeatedly applied in this manner unlimited times, between any two  distinct things there will be limitlessly many other things.  Therefore, if there are many things, then there must be limitlessly  many things.