I got this puzzle some time ago and it has been bugging me since, I cant solve it - but it is supposedly solvable, I am interested in a solution or any tips on how to proceed.

In front of you is an entity named Adam. Adam is a solid block with a single speaker, through which he hears and communicates. For all propositions (statements that are either true or false) $p$, if $p$ is true and logically knowable to Adam, then Adam knows that $p$ is true. Adam is confined to his physical form, cannot move, and only has the sense of hearing. The only sounds Adam can make are to play one of two pre-recorded audio messages. One message consists of a very high note played for one second, and the other one a very low note played for one second.

Adam has mentally chosen a specific subset of the Universe of ordinary mathematics. The Universe of ordinary mathematics is defined as follows:

Let $S_0$ be the set of natural numbers:

$$S_0 = \{1,2,3,\ldots\}$$

$S_0$ has cardinality $\aleph_0$, the smallest and only countable infinity.

The power set of a set $X$, denoted $2^X$, is the set of all subsets of $X$. The power set of a set always has a cardinality larger than the set itself, $$|2^X| = 2^{|X|}$$

Let $S_1 = S_0 \cup 2^{S_0}$. $S_1$ has cardinality $2^{\aleph_0} = \beth_1$.

Let $S_2 = S_1 \cup 2^{S_1}$. $S_2$ has cardinality $2^{\beth_1} = \beth_2$.

In general, let $S_{n+1} = S_n \cup 2^{S_n}$. $S_{n+1}$ has cardinality $2^{\beth_n} = \beth_{n+1}$.

The Universe of ordinary mathematics is defined as $$\bigcup_{i=0}^\infty S_i$$

This Universe contains all sets of natural numbers, all sets of real numbers, all sets of complex numbers, all ordered $n$-tuples for all $n$, all functions, all relations, all Euclidean spaces, and virtually anything that arises in standard analysis.

The Universe of ordinary mathematics has cardinality $\beth_\omega$.

Your goal is to determine the subset Adam is thinking of, while Adam is trying to prevent you from doing so. You are only allowed to ask Adam yes/no questions in trying to accomplish your task. Adam must respond to each question, and does so by playing a single note. After Adam hears your question, he either chooses the low note to mean yes and the high note to mean no, or the high note to mean yes and the low note to mean no, for that question only. He also decides to either tell the truth or lie for each question after hearing it. If at any time you ask a question which cannot be answered by Adam without him contradicting himself, Adam will either play the low note or the high note, ignoring the question entirely.

Adam has given you an infinite amount of time to accomplish your task. More specifically, the set of both questions asked by you and notes played by Adam can be of any cardinality. If in your strategy this set is uncountably large, for any number of possibilities of Adam's chosen subset, you must describe the order that the elements of this set take place in as completely as possible.

During your questioning, you are keeping track of the following numbers:

$B_1 = $ The number of questions in which Adam had the option of truthfully responding in the affirmative. (This number and the following numbers can of course be cardinal numbers.)

$B_2 = $ The number of questions in which Adam had the option of truthfully responding in the negative.

$B_3 = $ The number of questions in which Adam had the option of falsely responding in the affirmative.

$B_4 = $ The number of questions in which Adam had the option of falsely responding in the negative.

$B_5 = $ The number of questions in which Adam responded with the high note.

$B_6 = $ The number of questions in which Adam responded with the low note.

$B_7 = $ The number of questions.

Let $C_1 = 2^{B_1}$, $C_2 = 2^{2^{B_2}}$, and so on (the length of the tower defining $C_k$ from $B_k$ is $k$). Denote $$C = C_7^{C_6^{C_5^{C_4^{C_3^{C_2^{C_1}}}}}}$$ (Exponentiation is calculated from the top down.)

A strategy exists which will eventually allow you to determine Adam's chosen subset. Describe such a strategy in which $C$ is as small as possible, for all possibilities of Adam's chosen subset.