Generalizing Silver's Theorem Part 1

I should probably start this off with what Silver's theorem is and why it matters. In invariant descriptive set theory, quotients of the real line and other Polish spaces are studied. They are compared to each other via Borel reductions, which are maps that inject one quotient space into another. In this sense, different sizes are generated by taking quotients. For instance, 1 is achieved by identifying all points in the real line with each other. 2 could be formed by identifying all negative numbers together and all non-negative numbers together. In similar ways, all of the finite numbers are constructed. To construct ω, one could identify all points in $[n,n+1)$ together for all n. Finally, the real line itself comes from using the identity equivalence relation. But what about other, more complicated sets? Can larger ordinals be constructed as quotients of the real line? Are there quotients which fit in between ω and the continuum?

One way that ω₁ can be created is to consider reals (now as elements of ω^ω) as coding partial orders on the naturals. Those that code well-orders can be assigned the corresponding ordinal that well-order is in bijection with. Give reals which don't code well-orders a formal rank of ∞. Then identify reals together if they have the same rank. The resulting quotient is in bijection with ω₁. This construction heavily used the set of reals which code well-orders, which is a prime example of non-Borel set. It is in fact co-analytic. That makes this equivalence relation much more complicated than Borel. It is in fact more complicated than analytic or co-analytic, as seeing if two reals are equivalent requires searching a co-analytic set. So if we allow equivalence relations which are more than co-analytic, complicated sets can arise as quotients. What about co-analytic sets themselves? Silver's theorem tells us that if a quotient space is created a co-analytic equivalence relation, then either the quotient is countable or the real line embeds into the quotient. Basically, for co-analytic equivalence relations, we have obtained a form of the continuum hypothesis.

In these posts I am going to prove Silver's theorem twice, and then I will prove it's generalization, the Harrington-Shelah theorem. Silver's theorem is commonly proved topologically, using the topology generated by the lightface analytic sets (sets which can be written as the projection of a recursive tree on ω$\times$ω). Since that is easily found, I am first going to prove in the language of forcing with the lightface analytic sets as conditions. This is the way that almost generalizes, but doesn't quite. The proof to come next time.