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It is well known that one can represent sets as digraphs. What is the proper digraph representation of $\mathscr P$($\omega$)? I ask this because $\mathscr P$($\omega$) is $\Pi_1$ in the Levy hierarchy and this is taken to mean that the value of |$\mathscr P$($\omega$)|, say, is not absolute, that is, it can change relative to the model of ZFC it is found in.

What does this mean, relative to $\mathscr P$($\omega$)'s representation as a digraph? It would seem that, if one takes the relativity of |$\mathscr P$($\omega$)| seriously, it would mean that $\mathscr P$($\omega$) is to be represented by a proper class of graphs ($\mathscr P$($\omega$) could not represent a set of such graphs, otherwise that set would itself be representable by a digraph, and a form of the Burali-Forti paradox would ensue) each element of the proper class formed by an addition of, say, a Cohen real to the previously formed digraph.

On the other hand, because each digraph representation of $\mathscr P$($ \omega$) for some model $\mathfrak M$ (where $\mathfrak M$ is a (proper) class of digraphs satisfying the graph-theoretical analogues of the ZFC axioms, if such exist....) is a unique mathematical object, it would seem that each digraph representing $\mathscr P$($\omega$) is an incomplete object because one can add (by a ccc forcing) to any $\mathscr P$($\omega$) in some model $\mathfrak M$ a Cohen real creating the forcing extension $\mathfrak M$[G] of $\mathfrak M$. Taking the digraph perspective of this situation, it would seem that an infinite digraph representing $\mathscr P$($\omega$) is determined in some sense by the class of digraphs $\mathfrak M$ in which it lies. If this is false, then it would seem that each digraph representation of $\mathscr P$($\omega$) is, in fact, incomplete. Is this a correct rendering of the situation?

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In each model of set theory, there is a concrete set that is $\mathcal P(\omega)$, and this set can, if one wants, be represented as a directed graph.

Two different models will, in general, have their own $\mathcal P(\omega)$ sets, and each of these sets will have a graph representation. It is not the intention that there should be a single thing that, at once, represents the $\mathcal P(\omega)$s of every model graphically.

It is not fruitful to think of "$\mathcal P(\omega)$" as a single Platonic thing with different manifestations in different models. Rather, "being $\mathcal P(\omega)$" is merely a property an object in a model can have -- and incidentally the axioms of set theory imply that exactly one object in each model will have this property -- but it is the individual objects in particular models that have graph representations, not the property as such.

(If you want, you can say that there is a single Platonic $\mathcal P(\omega)$, namely the one in the "intended interpretation" of set theory, whatever that might mean. But even so, that "true" $\mathcal P(\omega)$ is just one set, and has nothing in particular to do with the $\mathcal P(\omega)$s that exist in other non-standard models of the axioms).

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  • $\begingroup$ I don't think of $\mathscr P$($\omega$) as "a single Platonic thing with different manifestations in different models", rather, I think of $\mathscr P$($\omega$) as the proper class of all digraph representations of $\mathscr P$($\omega$) in the ground model $\mathtt L$ and all successive ccc-forcing extensions $\mathtt L$[G] formed by adding a Cohen real. The question is, for the digraph representation of $\mathscr P$($\omega$)$\cap$$\mathtt L$, what is the digraph representation for 'adding a Cohen real'? $\endgroup$ – Thomas Benjamin Nov 21 '14 at 8:01
  • $\begingroup$ That is, what are you adding to the digraph representation of $\mathscr P$($\omega$)$\cap$$\mathtt L$ to make it a digraph representation of $\mathscr P$($\omega$)$\cap$$\mathtt L$[G] where G is the generic filter for adding one Cohen real? $\endgroup$ – Thomas Benjamin Nov 21 '14 at 8:19

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