# Analytic set in forcing extension

I am trying to understand the proof of the following lemma from the paper Can the fundamental group of a space be the rationals? by Saharon Shelah.

Let $$\mathcal{E}$$ be an analytic equivalence relation on $$\mathcal{P}(\mathbb{N})$$ such that when $$n\notin B$$ and $$A=B\cup\{n\}$$, $$A$$ and $$B$$ are not $$\mathcal{E}$$-equivalent. Then there is a perfect subset of $$\mathcal{P}(\mathbb{N})$$ of pairwise nonequivelent $$A\subseteq\mathbb{N}$$.

I am aware that Pawlikowski gave a neat proof of this using just basic descriptive set theory, but I want to understand the logic of this proof. The proof goes as follows (some phrases slightly altered).

Let $$M$$ be a countable elementary substructure of $$(H(\mathfrak{c}^+),\mathcal{E})$$ to which the real parameter in the definition of $$\mathcal{E}$$ belongs. Now if $$\langle A_1,A_2\rangle$$ is a pair of subsets of $$\mathbb{N}$$ which is Cohen generic over $$M$$, then they are $$\mathcal{E}$$-equivalent iff they are $$\mathcal{E}$$-equivalent in $$M[A_1,A_2]$$, by the absoluteness criterions. Now we show that they cannot be $$\mathcal{E}$$-equivalent in $$M[A_1,A_2]$$. Otherwise some finite information forces this, so for some $$n$$, if $$\langle A'_1,A_2\rangle$$ is Cohen generic over $$M$$ and $$A_1\cap\{0,1,...,n\}=A_1'\cap\{0,1,...,n\}$$, then $$A_1',A_2$$ are $$\mathcal{E}$$-equivalent in $$M[A'_1,A_2]$$. Let $$A_1'$$ differ from $$A_1$$ at $$n+1$$ only (if $$n+1\in A_1$$ then $$n+1\notin A_1'$$, and vice versa). Clearly $$\langle A'_1,A_2\rangle$$ is also Cohen generic, so they are $$\mathcal{E}$$-equivalent in $$M[A'_1,A_2]$$. By the above absoluteness we really have $$A'_1\sim A_2$$ and $$A_1\sim A_2$$, so $$A'_1\sim A_1$$, contradicting the property of $$\mathcal{E}$$.

I want to ask questions about pretty much every line, but here are the points that I feel most confused about:

1. What's the definition of $$N[G]$$ when $$N$$ is non-transitive? From Hamkins' answer to this question it seems essentially we have to take the collapse of $$N$$ and then do forcing, so $$N$$ is not a subset of $$N[G]$$. Is my understanding correct?

2. If $$N$$ contains the "real parameter defining $$\mathcal{E}$$", isn't $$\mathcal{E}$$ already an element of $$N$$, since $$\mathcal{E}$$ is definable in $$H(\mathfrak{c}^+)$$ from parameter and $$N$$ is an elementary substructure (at least that should be the case for the particular $$\mathcal{E}$$ in the paper)? What's the meaning of $$(H(\mathfrak{c}^+),\mathcal{E})$$ then?

3. What does "absoluteness criterions" refer to? Is it Shoenfield's Absoluteness? Does that apply to model without power set (plus $$\mathcal{P}(\mathbb{N})$$ exists)?

4. Where is it used that $$\mathcal{E}$$ is analytic?

• Cohen forcing is proper, which means the generic extension commutes with the Mostowski collapse (so to speak), so you can collapse to a transitive set, force over it, and then "uncollapse" in the obvious way. Moreover, since the generic for a Cohen real is a subset of $\omega$, it actually isn't moved by the collapse to begin with. Commented Sep 10, 2021 at 10:07
• As for the parameter, you want to make sure that not only $\cal E$ belongs to the model, but also that the definition that you're working with is interpreted correctly. Consider, for a fixed $r$, the equivalence relation that is either trivial if $r\in L$ or the identity if $r$ is Cohen generic over $L$. Depending on the parameter, the definition you use will produce a different relation, even if the relation, as a set, belongs to your model. Commented Sep 10, 2021 at 10:09
• @AsafKaragila But if $\mathcal{E}\in N$ and $N$ is elementary w.r.t. the language $\{\in\}$, isn't it also elementary in $\{\in,\mathcal{E}\}$?
– xXF
Commented Sep 10, 2021 at 14:11