# Reflection principle

﻿
Reflection principle

In set theory, a branch of mathematics, a reflection principle says that it is possible to find sets that resemble the class of all sets. There are several different forms of the reflection principle depending on exactly what is meant by "resemble". Weak forms of the reflection principle are theorems of ZF set theory, while stronger forms can be new and very powerful axioms for set theory.

The name "reflection principle" comes from the fact that properties of the universe of all sets are "reflected" down to a smaller set.

Motivation for reflection principles

A naive version of the reflection principle states that "for any property of the universe of all sets we can find a set with the same property". This leads to an immediate contradiction: the universe of all sets contains all sets, but there is no set with the property that it contains all sets. To get useful (and non-contradictory) reflection principles we need to be more careful about what we mean by "property" and what properties we allow.

To find non-contradictory reflection principles we might argue informally as follows. Suppose that we have some collection "A" of methods for forming sets (for example, taking powersets, subsets, the axiom of replacement, and so on). We can imagine taking all sets obtained by repeatedly applying all these methods, and form these sets into a class "V", which can be thought of as a model of some set theory. But now we can introduce the following new principle for forming sets: "the collection of all sets obtained from some set by repeatedly applying all methods in the collection "A" is also a set". If we allow this new principle for forming sets, we can now continue past "V", and consider the class "W" of all sets formed using the principles "A" and the new principle. In this class "W", "V" is just a set, closed underall the set-forming operations of "A". In other words the universe "W" contains a "set" "V" which resembles "W" in that it is closed under all the methods "A".

We can use this informal argument in two ways. We can try to formalize it in (say) ZF set theory; by doing this we obtain some theorems of ZF set theory, called reflection theorems. Alternatively we can use this argument to motivate introducing new axioms for set theory.

The reflection principle as a theorem of ZFC

In trying to formalize the argument for the reflection principle of the previous section in ZF set theory, it turns out to be necessary to add some conditions about the collection of properties "A" (for example, "A" might be finite). Doing this produces several closely related "reflection theorems" of ZFC all of which state that we can find a set that is almost a model of ZFC.

One form of the reflection principle in ZFC says that for any finite set of axioms of ZFC we can find a countable transitive model satisfying these axioms. (In particular this proves that ZFC is not finitely axiomatizable, because if it were it would prove the existence of a model of itself, and hence prove its own consistency, contradicting Gödel's theorem.) This version of the reflection theorem is closely related to the Löwenheim-Skolem theorem.

Another version of the reflection principle says that for any finite number of formulas of ZFC we can find a set "V"α in the cumulative hierarchy such that all the formulas in the set are absolute for "V"α (which means very roughly that they hold in "V"α if and only if they hold in the universe of all sets). So this says that the set "V"α resembles the universe of all sets, at least as far as the given finite number of formulas is concerned.

For any natural number n, one can prove from ZFC a reflection principle which says that given any ordinal α, there is an ordinal &beta;>α such V&beta; satisfies all first order sentences of set theory which are true for V and contain fewer than n quantifiers.

Reflection principles as new axioms

Bernays used a reflection principle as an axiom for one version of set theory (not Gödel-Bernays set theory, which is a weaker theory). His reflection principle stated roughly that if "A" is a class with some property, then one can find a set "u" such that "A&cap;u" has the same property when considered as a subset of the "universe" "u". This is quite a powerful axiom and implies the existence of several of the smaller large cardinals, such as inaccessible cardinals. (Roughly speaking, the class of all ordinals in ZFC is an inaccessible cardinal apart from the fact that it is not a set, and the reflection principle can then be used to show that there is a set which has the same property, in other words which is an inaccessible cardinal.) The consistency of Bernays's reflection principle is implied by the existence of a measurable cardinal.

There are many more powerful reflection principles, which are closely related to the various large cardinal axioms. For almost every known large cardinal axiom there is a known reflection principle that implies it, and conversely all but the most powerful known reflection principles are implied by known large cardinal axioms.

If V is a model of ZFC and its class of ordinals is regular, i.e. there is no cofinal subclass of lower order-type, then there is a closed unbounded class of ordinals, C, such that for every αεC, the identity function from Vα to V is an elementary embedding.

References

*
*
*citation|id=MR|0997881|title=Higher order reflection principles|first=M. Victoria|last= Marshall R.
journal=The Journal of Symbolic Logic |volume= 54|issue= 2 |year=1989|pages= 474-489

*citation|id=MR|0401475
last=Reinhardt|first= W. N.
chapter=Remarks on reflection principles, large cardinals, and elementary embeddings. |title=Axiomatic set theory |series=Proc. Sympos. Pure Math.|volume= XIII, Part II|pages= 189--205|publisher= Amer. Math. Soc.|publication-place= Providence, R. I.|year= 1974

Wikimedia Foundation. 2010.

### Look at other dictionaries:

• Schwarz reflection principle — This article is about the reflection principle in complex analysis. For the reflection principles of set theory, see Reflection principleIn mathematics, the Schwarz reflection principle is a way to extend the domain of definition of an analytic… …   Wikipedia

• Reflection — or reflexion may refer to:Computers* in computer graphics, the techniques for simulating optical Reflection. * Reflection, a programming language feature for metaprogramming * Reflection , a piece of installation art by Shane Cooper also called… …   Wikipedia

• Reflection seismology — (or seismic reflection) is a method of exploration geophysics that uses the principles of seismology to estimate the properties of the Earth s subsurface from reflected seismic waves. The method requires a controlled seismic source of energy,… …   Wikipedia

• Reflection (physics) — Reflection is the change in direction of a wave front at an between two different media so that the wave front returns into the medium from which it originated. Common examples include the reflection of light, sound and water waves. Law of… …   Wikipedia

• Principle of least action — This article discusses the history of the principle of least action. For the application, please refer to action (physics). In physics, the principle of least action or more accurately principle of stationary action is a variational principle… …   Wikipedia

• Principle of virtual velocities — Virtual Vir tu*al (?; 135), a. [Cf. F. virtuel. See {Virtue}.] 1. Having the power of acting or of invisible efficacy without the agency of the material or sensible part; potential; energizing. [1913 Webster] Heat and cold have a virtual… …   The Collaborative International Dictionary of English

• Fermat's principle — In optics, Fermat s principle or the principle of least time is the idea that the path taken between two points by a ray of light is the path that can be traversed in the least time. This principle is sometimes taken as the definition of a ray of …   Wikipedia

• Total internal reflection — The larger the angle to the normal, the smaller is the fraction of light transmitted, until the angle when total internal reflection (blue line) occurs. (The color of the rays is to help distinguish the rays, and is not meant to indicate any… …   Wikipedia

• Uncertainty principle — In quantum physics, the Heisenberg uncertainty principle states that locating a particle in a small region of space makes the momentum of the particle uncertain; and conversely, that measuring the momentum of a particle precisely makes the… …   Wikipedia

• Chamber of Reflection — A Chamber of Reflection is a small darkened room adjoining a Masonic Lodge. Its sombre appearance and the gloomy emblems with which it is furnished are calculated to produce serious meditations.[1] It is a small darkened room or chamber, with the …   Wikipedia