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 $C = B\_1+B\_2+B\_3+B\_4+B\_5+B\_6+B\_7$


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. 



Seemingly complex logic/set-theoretic puzzle
First, observe that you can get around the difficulty that
you don't know if high means yes or low in the following
way. If you really want to ask the question $\varphi$, you
should instead ask the question *"high means yes for this round if and
only if $\varphi$"*. If high means yes, then this is the
same as asking $\varphi$. But if high means no, then it is like asking $\neg\varphi$, and so we may interpret a high anwer to this question as yes to $\varphi$. This
transformation therefore ensures that we can in effect know
that high means yes. (I mentioned a similar trick in [this
MO
answer](https://mathoverflow.net/questions/32269/guess-a-number-with-at-most-one-wrong-answer/32337#32337)
about guessing a number, when there can be wrong answers.)


For the lying issue, let me assume that by lying, you mean
that Adam first decides whether to lie or tell the truth,
and then calculates what a truthful answer would be, and
then when telling the truth plays the appropriate tone, but if lying
plays the opposite tone. With this interpretation, a
similar trick allows us to extract the desired information.
Namely, if you want to ask $\psi$, instead ask *"you have decided to be truthful for this question iff $\psi$"*. If Adam decides to
be truthful, then this question is answered the same as
$\psi$. If he decides to lie, then he calculates what a
truthful answer would be, given that he has already decided
to lie, which is the opposite of $\psi$, and so he says the
opposite of this. In this way, the double negation of the
transformation allows us to get the desired information.


Combining the two transformations allows us to get answers
to any desired question.


Now, we simply proceed as follows. Since it seems
permissible in the world of your question, let us enumerate
all the elements of what you call the universe of ordinary
mathematics, and ask of each such element whether it is in
Adam's set, using the transformations above. In this way,
we find out exactly the set of which he is thinking.


The end result is $\beth\_\omega$ many questions. This is the optimal in the sense that any smaller bound on the number of questions would be less than $\beth\_n$ for some $n$, with only $\beth\_{n+1}$ many possible patterns of answers, but there are $\beth\_{\omega+1}$ many sets that Adam might be considering. 


Incidently, what you call the universe of ordinary mathematics
is closely related to what is known in set theory as
$V\_{\omega+\omega}$, which is a model of the Zermelo axioms
of set theory, one of the first axiomatizations of set theory. The $V$ hierarchy begins with $V\_0$ being
the empty set, and $V\_{\alpha+1}=P(V\_\alpha)$ and
$V\_\lambda=\bigcup\_{\alpha\lt\lambda} V\_\alpha$ for limit
ordinals $\lambda$. Your universe is contained within
$V\_{\omega+\omega}$, but is actually missing huge parts of
$V\_{\omega+1}$, because you started only with the natural
numbers, rather than the hereditary finite sets. For
example, the set $\{\ a\_k\mid k\in\mathbb{N}\ \}$, where
$a\_k=\{\{\{\cdots\}\}\}$ has depth $k$, is missing
from your universe, but exists in $V\_{\omega+1}$. It follows that your world of mathematics does not have the set HF consisting of all hereditary finite sets, or any similar set with unbounded finite depths. From this, it follows that your world does not satisfy some of the very elementary axioms of set theory, which would allow you to construct HF from the natural numbers. For example, the set mentioned above is the result of a very simple induction on finite-depth finte sets.


The fact that $V\_{\omega+\omega}$ itself has no sets of size $\beth\_\omega$ is precisely what led to the realization that the Zermelo axioms are too weak to prove even that $\beth\_\omega$ exists. This realization led directly to the 
addition of the Replacement axiom to the axioms of set theory, resulting in the theory now known as ZFC.


