In set theory, a branch of mathematics, a reflection principle says that it is possible to find sets that, with respect to any given property, 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 Zermelo–Fraenkel set theory) due to , 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

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.

Reflection principles are associated with attempts to formulate the idea that no one notion, idea, or statement can capture our whole view of the universe of sets. Kurt Gödel described it as follows:

Georg Cantor expressed similar views on absolute infinity: All cardinality properties are satisfied in this number, in which held by a smaller cardinal.

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 X, which can be thought of as a model of some set theory. But in light of this view, V is not exhaustible by a handful of operations, otherwise it would be easily describable from below, this principle is known as inexhaustibility (of V). As a result, V is larger than X. Applying the methods in A to the set X itself would also result in a collection smaller than V, as V is not exhaustible from the image of X under the operations in A. Then 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". After adding this principle to A, V is still not exhaustible by the operations in this new A. This process may be repeated further and further, adding more and more operations to the set A and obtaining larger and larger models X. Each X resembles V in the sense that it shares the property with V of being closed under the operations in A.

, called reflection theorems. Alternatively we can use this argument to motivate introducing new axioms for set theory, such as some axioms asserting existence of large cardinals.

In ZFC

, 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" all of which state that we can find a set that is almost a model of ZFC.

of the cumulative hierarchy. This is known as the Lévy–Montague reflection principle, or the Lévy reflection principle, principally investigated in and . Another version of this reflection principle says that for any finite in the cumulative hierarchy such that all the formulas in the set are absolute resembles the universe of all sets, at least as far as the given finite number of formulas is concerned.

is a theorem schema that can be described as follows: be a formula with at most free variables proves that

denotes the relativization (that is, replacing all quantifiers, respectively).

says that for any finite we can find a countable transitive model satisfying these axioms. (In particular this proves that, unless inconsistent 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 second incompleteness theorem.) This version of the reflection theorem is closely related to the Löwenheim–Skolem theorem.

is a strong inaccessible cardinal, then there is a closed unbounded subset is an elementary substructure.

As new axioms

Large cardinals

Reflection principles are connected to and can be used to motivate large cardinal axioms. Reinhardt gives the following examples, using Cantor's informal notion of absolute infinity in place of the universe of sets:

1. It may be helpful to give some informal arguments illustrating the use of reflection principles.

2. The simplest is perhaps: the universe of sets is inaccessible (i.e., satisfies the replacement axiom), therefore, therefore; therefore).

Bernays class theory

Paul Bernays used a reflection principle as an axiom for one version of set theory (not Von Neumann–Bernays–Gödel set theory. 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, and a class theory like Morse–Kelley set theory normally has to be used. The consistency of Bernays's reflection principle is implied by the existence of an ω-Erdős cardinal.

More precisely, the axioms of Bernays' class theory are:

1. extensionality

2. class specification

3. subsets

4. foundation

5. choice

denotes the powerset.

According to Akihiro Kanamori, in a 1961 paper, Bernays considered the reflection schema

is transitive, Bernays just with this schema established pairing, union, infinity, and replacement.

Others

Some formulations of Ackermann set theory,

Peter Koellner showed that a general class of reflection principles deemed "intrinsically justified" are either inconsistent or weak, in that they are consistent relative to the Erdös cardinal. However, there are 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. An example of this is the wholeness axiom, which implies the existence of super-n-huge cardinalss for all finite n and its consistency is implied by an I3 rank-into-rank cardinal.

Add an axiom saying that Ord is a Mahlo cardinal for every closed unbounded class of ordinals C (definable by a formula with parameters), there is a regular ordinal in C. This allows one to derive the existence of strong inaccessible cardinals and much more over any ordinal.

For arithmetic

.

Soundness

denote Peano arithmetic denote the set of true sentences in the language of PA in the arithmetical hierarchy. Mostowski-definable, this must be expressed as a theorem schema.p. 4 These soundness principles are sometimes referred to as syntactic reflection principles, in contrast to the satisfaction-based varieties mentioned above, which are called semantic reflection principles.p. 1

.p. 205

is the union of the sets of Gödel-numbers is the partial truth predicate formulas.p. 205

Model reflection

, a \beta_k-modelth level of the analytical hierarchy-model of a subsystem of second-order arithmetic-model.Theorem VII.2.16

formulas.Theorem VII.7.6

-model reflection has connections to set-theoretic reflection, for example over the weak set theory KP formulas.

References

Citations

External links

• Mizar system proof: http://mizar.org/version/current/html/zf_refle.html