Suppose you have a list of postulates/properties P1, P2, P3 etc. that you would like a (possibly not yet defined) mathematical object to satisfy. Is it possible to prove that the properties P1, P2, etc. are self-consistent without explicitly constructing a mathematical object satisfying the required properties?
The reason for asking this is that I am currently looking at ways of defining the homogenous spatial Poisson point process on (a sub region of) $\mathbb{R}^2$. I am aware of explicit constructions of this process. However, Diggle's book on "Statistical Analysis of Spatial and Spatio-temporal Point Patterns" avoids these constructions and instead starts from two postulates, called PP1 and PP2. PP1 states that, for some $\lambda>0$ and for any planar region $A$, the number of points in $A$ is Poisson distributed with mean $\lambda$ (i.e. $N(A)\sim Po(\lambda)$. PP2 states that, given $N(A)=n$, the points in $A$ form an independent random sample from the uniform distribution on $A$.
The text goes on to demonstrate "self-consistency" of PP1 and PP2 (pages 61-62 in the third edition)...
This is done by first showing that PP1 and PP2 together imply PP3, which states that for disjoint regions $A$ and $B$, $N(A)$ and $N(B)$ are independent. Then various arguments are given, the substance of which is something like "if PP2 holds for $A$ then it also holds for any subregion of $A$" and "if PP1 and PP2 hold for disjoint $A$ and $B$ then they also hold for $A\cup B$". I don't see how such an argument, with no explicit construction of the point process, can show that PP1 and PP2 are self-consistent. Any ideas what is going on here?