Inner model

Inner model

In mathematical logic, suppose "T" is a theory in the language

:L = langle in angle

of set theory.

If "M" is a model of L describing a set theory and "N" is a class of "M" such that

: langle N, in_M, cdots angle

is a model of "T" containing all ordinals of "M" then we say that "N" is an inner model of "T" (in "M").

This term "inner model" is sometimes applied to models which are proper classes; the term set model is used for models which are sets.

A model of set theory is called standard if the element relation of the model is the actual element relation restricted to the model. A model is called transitive when it is standard and the base class is a transitive class of sets. A model of set theory is often assumed to be transitive unless it is explicitly stated that it is non-standard. Inner models are transitive, transitive models are standard, and standard models are well-founded.

The assumption that there exists a standard model of ZFC (in a given universe) is stronger then the assumption that there exists a model. In fact, if there is a standard model, then there is a smallest standard model called the minimal model contained in all standard models. It is countable and satisfies V=L. The minimal model contains no standard model (as it is minimal) but (assuming the consistence of ZFC) it containssome model of ZFC by the Godel completeness theorem. This model is necessarily not well founded otherwise its Mostowski collapse would be a standard model. (It is not well founded as a relation in the universe, though itsatisfies the axiom of foundation so is "internally" well founded. Being well founded is not an absolute property.)In particular in the minimal model there is a model of ZFC but there is no standard model of ZFC.


Usually when one talks about inner models of a theory, the theory one is discussing is ZFC or some extension of ZFC (like ZFC+exists a measurable cardinal). 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

It was proved by Kurt Gödel that any model of ZF has a least inner model of ZF (which is also an inner model of ZFC + GCH), called the constructible universe, or L.

There is a branch of set theory called inner model theory which 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.

ee also

*Countable transitive models and generic filters

Wikimedia Foundation. 2010.

Look at other dictionaries:

  • Inner model theory — In set theory, inner model theory is the study of certain models of ZFC or some fragment or strengthening thereof. Ordinarily these models are transitive subsets or subclasses of the von Neumann universe V , or sometimes of a generic extension of …   Wikipedia

  • Model Town, Lahore — Model Town Lahore is one of residential suburbs of Lahore, Punjab, Pakistan. It is located next to Faisal Town, Gulberg, Garden Town and Township suburbs of Lahore. Contents 1 History 1.1 Model Town Society 1.2 Evacuee property …   Wikipedia

  • Inner sphere electron transfer — Inner sphere or bonded electron transfer[1] proceeds via a covalent linkage between the two redox partners, the oxidant and the reductant. In Inner Sphere (IS) electron transfer (ET), a ligand bridges the two metal redox centers during the… …   Wikipedia

  • Model dwellings company — Model Dwellings Companies (MDCs) were a group of private companies in Victorian Britain that sought to improve the housing conditions of the working classes by building new homes for them, at the same time receiving a competitive rate of return… …   Wikipedia

  • inner speech model for verbal auditory hallucinations —    Also known as misattribution model for verbal auditory hallucinations. The two names are used interchangeably to denote a hypothetical model attributing the mediation of some types of verbal auditory hallucination (VAH) to a disorder of inner… …   Dictionary of Hallucinations

  • Inner class — In object oriented programming (OOP), an inner class or nested class is a class declared entirely within the body of another class or interface. It is distinguished from a subclass. Contents 1 Overview 2 Types of inner classes 3 Why can t an… …   Wikipedia

  • Inner core — Earth cutaway from core to exosphere. Partially to scale. The inner core of the Earth, its innermost hottest part as detected by seismological studies, is a primarily solid ball about 1,220 km (760 mi) in radius,[1] …   Wikipedia

  • inner speech —    Also known as inner language, internalized speech, inner thought, and self talk. All five terms are used interchangeably to denote speech spoken by oneself without vocalization (also referred to as verbal thought or thinking in words ). The… …   Dictionary of Hallucinations

  • Inner city — The inner city is the central area of a major city or metropolis. In the United States, Canada, United Kingdom and Ireland, the term is often applied to the poorer parts of the city centre and is sometimes used as a euphemism with the connotation …   Wikipedia

  • Model (physical) — A physical model is a smaller or larger physical copy of an object. The object being modelled may be small (for example, an atom) or large (for example, the Solar System).The geometry of the model and the object it represents are often similar in …   Wikipedia