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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 about guessing a number, when there can be wrong answers.)

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 about guessing a number, when there can be wrong answers.)

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 "you will beep high if and only if $\varphi$". If high means yes, then this is the same as asking $\varphi$. But if high means no, then an affirmative answer to $\varphi$ would result in a low note, which means Adam will beep high, and conversely. This transformation therefore ensures that we can in effect know that high means yes. (I mentioned a similar trick in this MO answer about guessing a number, when there can be wrong answers.)

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 about guessing a number, when there can be wrong answers.)

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 sets.

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.