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\label{prop:no-comprehension}
Category~$\tau$ has no comprehension scheme.
The only set-formation principle is the
\textbf{bounded powerset}
$\mathcal{P}_\tau(\underline{x})
= \{\underline{a} : \underline{a} \mid \underline{x}\}$
(Definition~\ref{def:bounded-powerset}),
which is finite for every $\underline{x}$.
There is no mechanism to form
``the set of all objects satisfying property~$\varphi$''
for an arbitrary predicate~$\varphi$.

In $\tau$-set theory,
sets are natural numbers in $\tau$-Idx,
and membership is divisibility
(Chapter~\ref{ch:membership-divisibility}).
The powerset of $\underline{x}$ is the set of divisors
of $\underline{x}$, which is finite
(Proposition~\ref{prop:powerset-cardinality}).
To form a subset by comprehension,
one would need a mechanism
that takes a predicate $\varphi$
and returns a natural number $\underline{y}$
whose divisors are exactly those $\underline{a}$
satisfying $\varphi(\underline{a})$.
But the divisors of any natural number
are determined by its prime factorization ---
a fixed, finite, arithmetic structure ---
and there is no operation in $\tau$
that converts an arbitrary logical predicate
into a prime factorization.
The $\tau$-axioms provide only earned operations:
$\rho$ (successor), addition, multiplication,
exponentiation, tetration.
None of these constructs subsets from predicates.
Hence no comprehension schema exists in~$\tau$.