A question about formalized theories that may be both consistent and w-consistent Let T be a first order set theory formalized in the language L(ZF) of ZF, which has "membership"
and "=" as its only atomic predicates. For each positive integer n, let P(n) be the sentence which
expresses that "there exists a set having exactly n elements". P(n) can be formalized in L(ZF) and
is an axiom of T for each positive integer n. Note that L(ZF) does not need to contain any terms
that are constants for this to be possible. Now let Q be the sentence stating that "there exists
a finite set (in Tarski's sense of finite) which can be mapped onto every non-empty set". Q is
also an axiom of T that can be formalized in L(ZF) and should be consistent with every finite
collection of sentences of the form P(n). However with infinitely many axioms it would seem 
appropriate to call T at least a w-inconsistent theory. But could T still be consistent? The
answer is not clear to me since it would depend upon what other axioms T has. T must have some
other axioms (such as the axiom of pairing) in order to define the mapping of one set onto
another as the sentence Q describes. We must be careful that these additional axioms are not
inconsistent with Q. We should probably not want a power set axiom, for example. But, with a
proper choice of the new axioms could we end up with a consistent T? If so, T would be an
example of a consistent but w-inconsistent formalized theory whose language need contain no
constants and no formulae of the form N(x) intended to be interpreted as "x is a positive integer".
 A: If we replace your axiom $Q$ by the stronger-seeming assertion
$Q^+$ that asserts: "there is a largest set, which contains all
other sets as subsets, but which is Tarski finite", then the
corresponding theory $T$, asserting every $P(n)$ and also $Q^+$,
is consistent, but not $\omega$-consistent.
To see this, simply consider the ultrapower model $M=\Pi \langle
V_m,{\in}\rangle/U$, where $U$ is a nonprincipal ultrafilter on
$\mathbb{N}$ and $V_m$ consists of the sets of rank less than $m$
in the von Nuemann hierarchy, defined by iterating the power set
via $V_0=\emptyset$ and $V_{m+1}=P(V_m)$. Each statement $P(n)$ is
true in all but finitely many $V_m$, and hence true in $M$.
Similarly, each $V_{m+1}$ satisfies $Q^+$, since the object $V_m$
is largest in $V_{m+1}$ and Tarski finite there, and so $Q^+$ also
is true in $M$.
Alternatively, one could consider $V_N$ for nonstandard $N$ inside
a nonstandard model of finite set theory $\text{ZFC}^{\neg\infty}$.
These models furthermore satisfy some nice axioms, such as
extensionality, union, foundation, axiom of choice, collection,
replacement, separation, pairing restricted to sets of non-maximal
rank, power set restricted to sets of non-maximal rank and induction
on the von Neumann natural numbers. And so one can place all these axioms
also into $T$.
This theory, however, is clearly not $\omega$-consistent. 
