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.