How many parameter-free definable subsets are there of the structure $(\mathbb{R}, *)$?

Question

This is related to a question where I asked how many subsets of $\mathbb{R}$ are parameter-free definable in the structure $(\mathbb{R}, +)$. Now, I am asking how many subsets of $\mathbb{R}$ are parameter-free definable in $(\mathbb{R}, *)$. I believe there are only 32 of them. Is this correct?

Answer 1

A good first step, which is almost always easier, is to count automorphism orbits. (Note that this is also how the answer to your previous question went.)

In the structure $(\mathbb{R};*)$, this is easy to do: the orbits are exactly $\{0\}$, $\{1\}$, $\{-1\}$, $\mathbb{R}_{>0}\setminus\{1\}$, and $\mathbb{R}_{<0}\setminus\{-1\}$. Checking that the latter two are in fact orbits is a bit annoying, but it's not hard. My suggestion is to first think about the orbits of the structure $(\mathbb{R};+)$, which is isomorphic to $(\mathbb{R}_{>0};*)$ and is easier to think about.

This means that there are exactly $32$ automorphism-closed subsets of $(\mathbb{R};*)$. Since every parameter-freely-definable set is automorphism-closed, this gives an upper bound for the OP; it's easy to check that in fact each of the five orbits above (and hence each automorphism-closed set) is parameter-freely definable, so we're done.