In
set theory
Set theory is the branch of mathematical logic that studies Set (mathematics), sets, which can be informally described as collections of objects. Although objects of any kind can be collected into a set, set theory – as a branch of mathema ...
, a branch of
mathematical logic
Mathematical logic is the study of Logic#Formal logic, formal logic within mathematics. Major subareas include model theory, proof theory, set theory, and recursion theory (also known as computability theory). Research in mathematical logic com ...
, an inner model for a
theory
A theory is a systematic and rational form of abstract thinking about a phenomenon, or the conclusions derived from such thinking. It involves contemplative and logical reasoning, often supported by processes such as observation, experimentation, ...
''T'' is a
substructure of a
model
A model is an informative representation of an object, person, or system. The term originally denoted the plans of a building in late 16th-century English, and derived via French and Italian ultimately from Latin , .
Models can be divided in ...
''M'' of a
set theory
Set theory is the branch of mathematical logic that studies Set (mathematics), sets, which can be informally described as collections of objects. Although objects of any kind can be collected into a set, set theory – as a branch of mathema ...
that is both a model for ''T'' and contains all the ordinals of ''M''.
Definition
Let ''L'' = ⟨∈⟩ be the language of set theory. Let ''S'' be a particular set theory, for example the
ZFC axioms and let ''T'' (possibly the same as ''S'') also be a theory in ''L.''
If ''M'' is a model for ''S,'' and ''N'' is an such that
# ''N'' is a substructure of ''M,'' i.e. the
interpretation ∈
''N'' of ∈ in ''N'' is ∈
''M'' ∩ ''N''
2
# ''N'' is a model of ''T''
# the domain of ''N'' is a
transitive class of ''M''
# ''N'' contains all
ordinals in ''M''
then we say that ''N'' is an inner model of ''T'' (in ''M''). Usually ''T'' will equal (or subsume) ''S'', so that ''N'' is a model for ''S'' 'inside' the model ''M'' of ''S''.
If only conditions 1 and 2 hold, ''N'' is called a
standard model
The Standard Model of particle physics is the Scientific theory, theory describing three of the four known fundamental forces (electromagnetism, electromagnetic, weak interaction, weak and strong interactions – excluding gravity) in the unive ...
of ''T'' (in ''M''), a ''standard submodel'' of ''T'' (if ''S'' = ''T'' and) ''N'' is a ''set'' in ''M''. A model ''N'' of ''T'' in ''M'' is called ''transitive'' when it is standard and condition 3 holds. If the
axiom of foundation is not assumed (that is, is not in ''S'') all three of these concepts are given the additional condition that ''N'' be
well-founded
In mathematics, a binary relation is called well-founded (or wellfounded or foundational) on a set (mathematics), set or, more generally, a Class (set theory), class if every non-empty subset has a minimal element with respect to ; that is, t ...
. Hence inner models are transitive, transitive models are standard, and standard models are well-founded.
The assumption that there exists a standard submodel of
ZFC (in a given universe) is stronger than the assumption that there exists a model. In fact, if there is a standard submodel, then there is a smallest standard submodel
called the ''
minimal model'' contained in all standard submodels. The minimal submodel contains no standard submodel (as it is minimal) but (assuming the
consistency
In deductive logic, a consistent theory is one that does not lead to a logical contradiction. A theory T is consistent if there is no formula \varphi such that both \varphi and its negation \lnot\varphi are elements of the set of consequences ...
of ZFC) it contains
some model of ZFC by the
Gödel completeness theorem. This model is necessarily not well-founded otherwise its
Mostowski collapse would be a standard submodel. (It is not well-founded as a relation in the universe, though it
satisfies the
axiom of foundation so is "internally" well-founded. Being well-founded is not an absolute property.
[, Page 117])
In particular in the minimal submodel there is a model of ZFC but there is no standard submodel of ZFC.
Use
Usually when one talks about inner models of a theory, the theory one is discussing is
ZFC or some extension of ZFC (like ZFC + "a
measurable cardinal
In mathematics, a measurable cardinal is a certain kind of large cardinal number. In order to define the concept, one introduces a two-valued measure (mathematics), measure on a cardinal ''κ'', or more generally on any set. For a cardinal ''κ'', ...
exists"). When no theory is mentioned, it is usually assumed that the model under discussion is an inner model of ZFC. However, it is not uncommon to talk about inner models of
subtheories of ZFC (like
ZF or
KP) as well.
Related ideas
Kurt Gödel
Kurt Friedrich Gödel ( ; ; April 28, 1906 – January 14, 1978) was a logician, mathematician, and philosopher. Considered along with Aristotle and Gottlob Frege to be one of the most significant logicians in history, Gödel profoundly ...
proved that any model of ZF has a least inner model of ZF, the
constructible universe
In mathematics, in set theory, the constructible universe (or Gödel's constructible universe), denoted by L, is a particular Class (set theory), class of Set (mathematics), sets that can be described entirely in terms of simpler sets. L is the un ...
, which is also an inner model of ZFC +
GCH.
There is a branch of set theory called
inner model theory that studies ways of constructing least inner models of theories extending ZF. Inner model theory has led to the discovery of the exact
consistency strength of many important set theoretical properties.
See also
*
Countable transitive models and generic filters
References
{{DEFAULTSORT:Inner Model
Inner model theory