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Stoy, pages 117-122, discusses how to represent functions $f:\mathscr{P\omega\to P\omega}$ as elements $\mbox{graph}(f)\in\mathscr{P\omega}$, definition 7.5 page 120.

What would $\mbox{graph}(\mbox{id})\in\mathscr{P\omega}$ for the identity function be in this representation?

In the likely case you don't have Stoy's book handy, here are the relevant Pages 117-122. They'll introduce his notation, outline the procedure, etc, but assumes all necessary prerequisites...

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--- Edit ---
I should probably elaborate that after doing a little work on this myself (but not enough to find the actual answer, which seems a little trickier than I first imagined:), it became clear that the answer wasn't going to be what I'd hoped, that $\mbox{graph(id)}=\mathbb{N}$.

Towards the end of that section, on pages 121-122, Stoy mentions that various other representations are possible, and he gives the constraints that any such valid representation must satisfy. So my underlying/followup question here is: find a representation of $\mathscr{P\omega}\sim[\mathscr{P\omega\to P\omega}]$ such that $\mbox{graph(id)}=\mathbb{N}$. Or at least suggest how to go about finding such a representation, i.e., how would you set up the problem?

John Forkosh
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  • Trying to open you link to the pages in question gives me Error 403. – Leo163 Jun 03 '19 at 11:59
  • @Leo163 I clicked through to it when I first posted the question, and it worked okay. And I tried again after reading your comment, and it still works fine. Anybody else having a problem with it??? If so, maybe you can just directly download the pdf from http://www.forkosh.com/stoy.pdf and then use any pdf viewer you like. Does that work for you, Leo??? – John Forkosh Jun 03 '19 at 12:02
  • Why would you hope to get $\mathbb{N}$ from $id$? – Noah Schweber Jun 03 '19 at 12:07
  • Questions should be self-contained - please write out at least a sketch of what Stoy's construction is. In particular, something must be wonky here: the set of maps from $\mathcal{P}(\omega)$ to $\mathcal{P}(\omega)$ has cardinality $2^{2^{\aleph_0}}$, so can't be in bijection with $\mathcal{P}(\omega)$. EDIT: it looks like Stoy is talking about continuous* function(al)s, not arbitrary ones, so this actually works.* – Noah Schweber Jun 03 '19 at 12:09
  • @NoahSchweber I'm looking for a poset ordering, where for my purposes $\mathbb{N}$ would be least. – John Forkosh Jun 03 '19 at 12:09
  • @JohnForkosh OK, why would $id$ be least in any sense? Why not the map sending everything to $\emptyset$? – Noah Schweber Jun 03 '19 at 12:10
  • @NoahSchweber Re wonky, the tag says "domain-theory", and that $[.\to.]$ denotes >>continuous<< function space (wrt Scott topology), which solves the cardinality issue. And along those lines, a synopsis of the procedure is meaningless without tons of prerequisite material. – John Forkosh Jun 03 '19 at 12:12
  • @NoahSchweber I don't really want to go into the details of what I'm trying to work on. But I am looking for an ordering on the function space where id would be least (and $\mathbb{N}$ is correspondingly least in the usual domain theory sense that it gives you "no information"). – John Forkosh Jun 03 '19 at 12:16
  • @NoahSchweber Oh, I just noticed you edited "continuous" into your own comment. As a history note (if you're interested) all of domain theory began as an attempt to give a set-theoretical meaning/interpretation of the lambda calculus, whose wff's can take other wff's as arguments. And that means it's isomorphic to its own function space, whereby it was considered meaningless for decades. Until Scott came along in 1969 with his $D_\infty$ construction to demonstrate how "continuous" (wrt Scott's T0 topology) solves the problem. So it's interesting how the same cardinality issue occurred to you. – John Forkosh Jun 03 '19 at 12:37
  • @JohnForkosh Yes, I'm somewhat familiar with that material albeit in a different context (continuous/countable functionals a la Kleene/Kreisel); I just didn't recognize the book here. – Noah Schweber Jun 03 '19 at 13:07

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