Typical reflection principles in set theory are concerned with the height of the universe, or the relative height of certain stages. The resemblance between stages or between the universe itself and some of these stages is a very useful guiding principle that serves us to motivate large cardinal statements and many consequences of forcing axioms.
It is natural to wonder about similar reflection principles concerned instead with the width of the universe. In our paper Inner-model reflection principles, Neil Barton, Gunter Fuchs, Joel David Hamkins, Jonas Reitz, Ralf Schindler and I consider precisely this kind of reflection. Say that the inner-model reflection principle holds if and only if for any set 
We establish the consistency of the principle relative to ZFC. In fact, we build a model of the stronger ground-model reflection principle, where we further require that any such first-order 
The paper studies the principle under large cardinals and forcing axioms, and compares it with other statements considered in recent years, such as the maximality principle or the inner model hypothesis. The most technically involved and interesting results in the paper show that inner-model reflection and even ground-model reflection hold in certain fine-structural inner models but also that this requires large cardinals, and that the large cardinal requirements differ for both principles (precisely a proper class of Woodin cardinals is needed for ground-model reflection).
Curiously, the paper started as a series of informal exchanges in response to a question on math.stackexchange.
See also here. The paper will appear in Studia Logica. Meanwhile, it can be accessed on the arXiv, or in my papers page.
Posted by andrescaicedo 
is a countable ordinal and 
, then there is a countable ordinal 
such that
has order type 
.
, except for when 
, in which case we can take 
as well, this is Ramsey’s theorem.
is 
, in which case a stronger result holds, namely 
. In fact, more is true: the homogeneous set 
can be taken to be a stationary subset of 
can be required to be closed, meaning that its 
holds for any countable ordinal 
for specific values of 
and 
. 
-separable spaces.
Implies 
.
and 
, where the latter is (first-order) Peano arithmetic, and the former is finite set theory, the result of replacing in 
the axiom of infinity with its negation (and with foundation formulated as the schema of 
-induction). The reference is
of hereditarily finite sets and 
and verifies that the translation extends to a definable translation of the relations, functions and constants of the language of each structure in a way that 
and 
. Recall that 
, that is, it is the collection of sets resulting from iterating the power-set operation (any finite number of times) starting from the empty set.
by
that injects (codes) 
became known thanks to the work of Peter Aczel, and it is his formulation that I recall below, although it was originally introduced in work of Forti and Honsell from 1983, where they call it 
. Aczel’s presentation appears in the excellent book
, its field 
is the union of its domain and codomain. A decoration of 
satisfying
. When 
such that 
, for instance. Under 
. The axiom is sometimes described as saying that it provides solutions to many “equations” among sets. For instance, consider the system of equations 
and 
. Under 
as its unique solution. Note that assuming 
satisfying
?
is transcendental for all non-well-founded hereditarily finite 
. The function 
. 
A kind of library 