Inset theory, a branch ofmathematics, areflection principle says that it is possible to find sets that, with respect to any given property, resemble theclass 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 ofZF set theory due toMontague (1961), 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.

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 ofthe universe of sets.^[1]Kurt Gödel described it as follows:^[2]

The universe of all sets is structurally indefinable. One possible way to make this statement precise is the following: The universe of sets cannot be uniquely characterized (i.e., distinguished from all its initial segments) by any internal structural property of the membership relation in it which is expressible in any logic of finite ortransfinite type, includinginfinitary logics of anycardinal number. This principle may be considered a generalization of theclosure principle.

— 8.7.3, p. 280

All the principles for setting up theaxioms of set theory should be reducible toAckermann's principle: TheAbsolute is unknowable. The strength of this principle increases as we get stronger and stronger systems of set theory. The other principles are only heuristic principles. Hence, the central principle is the reflection principle, which presumably will be understood better as our experience increases. Meanwhile, it helps to separate out more specific principles which either give some additional information or are not yet seen clearly to be derivable from the reflection principle as we understand it now.

— 8.7.9, p. 283

Generally I believe that, in the last analysis, everyaxiom of infinity should be derivable from the (extremely plausible) principle thatV is indefinable, where definability is to be taken in [a] more and more generalized and idealized sense.

— 8.7.16, p. 285

Georg Cantor expressed similar views onabsolute 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 collectionA of methods for forming sets (for example, takingpowersets,subsets, theaxiom of replacement, and so on). We can imagine taking all sets obtained by repeatedly applying all these methods, and form these sets into a classX, which can be thought of as amodel of some set theory. But in light of this view,V is not be exhaustible by a handful of operations, otherwise it would be easily describable from below, this principle is known as inexhaustibility (ofV).^[3] As a result,V is larger thanX. Applying the methods inA to the setX itself would also result in a collection smaller thanV, asV is not exhaustible from the image ofX under the operations inA. 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 collectionA is also a set". After adding this principle toA,V is still not exhaustible by the operations in this newA. This process may be repeated further and further, adding more and more operations to the setA and obtaining larger and larger modelsX. EachX resemblesV in the sense that it shares the property withV of being closed under the operations inA.

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, such as someaxioms asserting existence of large cardinals.^[3]

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 propertiesA (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. In contrast to stronger reflection principles, these are provable in ZFC.

One of the most common reflection principles for ZFC is a theorem schema that can be described as follows: for any formula${\displaystyle \varphi (x_1,\dots ,x_n)}$ with parameters, if${\displaystyle \varphi (x_1,\dots ,x_n)}$ is true (in the set-theoretic universe${\displaystyle V}$), then there is a level${\displaystyle V_{\alpha }}$ of thecumulative hierarchy such that${\displaystyle V_{\alpha }\models \varphi (x_1,\dots ,x_n)}$. This is known as the Lévy-Montague reflection principle,^[4] or the Lévy reflection principle,^[5] principally investigated inLévy (1960) andMontague (1961).^[6] Another version of this reflection principle says that for anyfinite number of formulas of ZFC we can find a set${\displaystyle V_{\alpha }}$ in the cumulative hierarchy such that all the formulas in the set areabsolute for${\displaystyle V_{\alpha }}$ (which means very roughly that they hold in${\displaystyle V_{\alpha }}$ if and only if they hold in the universe of all sets). So this says that the set${\displaystyle V_{\alpha }}$ resembles the universe of all sets, at least as far as the given finite number of formulas is concerned.

Another reflection principle for ZFC is a theorem schema that can be described as follows:^[7][8] Let${\displaystyle \varphi }$ be a formula with at mostfree variables${\displaystyle x_1,\dots ,x_n}$. Then ZFC proves that

${\displaystyle (\mathrm{\forall }N)(\mathrm{\exists }M\supseteq N)(\mathrm{\forall }{x}_1,\dots ,x_n\in M)(\varphi (x_1,\dots ,x_n)\leftrightarrow {\varphi }^M)}$

where${\displaystyle {\varphi }^M}$ denotes therelativization of${\displaystyle \varphi }$ to${\displaystyle M}$ (that is, replacing allquantifiers appearing in${\displaystyle \varphi }$ of the form${\displaystyle \mathrm{\forall }x}$ and${\displaystyle \mathrm{\exists }x}$ by${\displaystyle \mathrm{\forall }x\in M}$ and${\displaystyle \mathrm{\exists }x\in M}$, respectively).

Another form of the reflection principle in ZFC says that for anyfinite set of axioms of ZFC we can find a countabletransitive model satisfying these axioms. (In particular this proves that, unless inconsistent, 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 second incompleteness theorem.) This version of the reflection theorem is closely related to theLöwenheim–Skolem theorem.

If${\displaystyle \kappa }$ is astrong inaccessible cardinal, then there is aclosed unbounded subset${\displaystyle C}$ of${\displaystyle \kappa }$, such that for every${\displaystyle \alpha \in C}$,${\displaystyle V_{\alpha }}$ is anelementary substructure of${\displaystyle V_{\kappa }}$.

Reflection principles are connected to and can be used to motivate large cardinal axioms. Reinhardt gives the following examples:^[9]

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

The simplest is perhaps: the universe of sets is inaccessible (i.e., satisfies the replacement axiom),therefore there is an inaccessible cardinal. This can be elaborated somewhat, as follows. Let${\displaystyle {\theta }_{\nu }}$ enumerate the inaccessible cardinals. By the same sort of reasoning,${\displaystyle {\theta }_{\nu }}$ is not bounded; the Cantor absolute${\displaystyle \mathrm{\Omega }}$ (all ordinals) is an inaccessible above any proposed bound${\displaystyle \beta }$,therefore there is an inaccessible cardinal above${\displaystyle \beta }$. Clearly, then, there are${\displaystyle \mathrm{\Omega }}$ inaccessibles above below${\displaystyle \mathrm{\Omega }}$;therefore there is an inaccessible${\displaystyle \kappa }$ such that there are${\displaystyle \kappa }$ inaccessibles below it (i.e.,${\displaystyle \kappa ={\theta }_{\kappa }}$).

Paul Bernays used a reflection principle as an axiom for one version of set theory (notVon Neumann–Bernays–Gödel set theory, which is a weaker theory). His reflection principle stated roughly that if${\displaystyle A}$ is a class with some property, then one can find a transitive set${\displaystyle u}$ such that${\displaystyle A\cap u}$ has the same property when considered as a subset of the "universe"${\displaystyle u}$. This is quite a powerful axiom and implies the existence of several of the smallerlarge cardinals, such asinaccessible cardinals. (Roughly speaking, the class of allordinals 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 that has the same property, in other words that is an inaccessible cardinal.) Unfortunately, this cannot be axiomatized directly in ZFC, and a class theory likeMorse–Kelley set theory normally has to be used. The consistency of Bernays's reflection principle is implied by the existence of anω-Erdős cardinal.

More precisely, the axioms of Bernays' class theory are:^[10]

1. extensionality
2. classspecification: for any formula${\displaystyle \varphi }$ without${\displaystyle a}$ free,${\displaystyle \mathrm{\exists }a\mathrm{\forall }b(b\in a\leftrightarrow \varphi \wedge b\text{ is a set})}$
3. subsets:${\displaystyle b\subseteq a\wedge a\text{ is a set}\to b\text{ is a set}}$
4. reflection: for any formula${\displaystyle \varphi }$,${\displaystyle \varphi (A)\to \mathrm{\exists }u(u\text{ is a transitive set}\wedge {\varphi }^{\mathcal{P}u}(A\cap u))}$
5. foundation
6. choice

where${\displaystyle \mathcal{P}}$ denotes thepowerset.

According toAkihiro Kanamori,^[11]: 62 in a 1961 paper, Bernays considered the reflection schema

${\displaystyle \varphi \to \mathrm{\exists }x(\text{transitive}(x)\wedge {\varphi }^x)}$

for any formula${\displaystyle \varphi }$ without${\displaystyle x}$ free, where${\displaystyle \text{transitive}(x)}$ asserts that${\displaystyle x}$ istransitive. Starting with the observation that set parameters${\displaystyle a_1,\dots ,a_n}$ can appear in${\displaystyle \varphi }$ and${\displaystyle x}$ can be required to contain them by introducing clauses${\displaystyle \mathrm{\exists }y(a_i\in y)}$ into${\displaystyle \varphi }$, Bernays just with this schema establishedpairing,union,infinity, andreplacement, in effect achieving a remarkably economical presentation ofZF.

Some formulations ofAckermann set theory use a reflection principle. Ackermann's axiom states that, for any formula${\displaystyle \varphi }$ not mentioning${\displaystyle V}$,^[2]

${\displaystyle a\in V\wedge b\in V\to \mathrm{\forall }x(\varphi \to x\in V)\to \mathrm{\exists }u\in V\mathrm{\forall }x(x\in u\leftrightarrow \varphi )}$

Peter Koellner showed that a general class of reflection principles deemed "intrinsically justified" are eitherinconsistent or weak, in that they are consistent relative to theErdös cardinal.^[12] 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.^[10] An example of this is thewholeness axiom,^[13] which implies the existence ofsuper-n-huge cardinals for all finiten and its consistency is implied by an I3rank-into-rank cardinal.

Add an axiom saying thatOrd is aMahlo cardinal — for every closed unbounded class of ordinalsC (definable by a formula with parameters), there is aregular ordinal inC. This allows one to derive the existence of stronginaccessible cardinals and much more over any ordinal.

Reflection principles may be considered for theories of arithmetic which are generally much weaker than ZFC.

Let${\displaystyle \mathsf{P}\mathsf{A}}$ denotePeano arithmetic, and${\displaystyle {\mathsf{P}\mathsf{A}}_k}$ denote theset of true sentences in the language of PA that are${\displaystyle {\mathrm{\Sigma }}_k}$ in thearithmetical hierarchy. Mostowski's reflection theorem is that for each natural number${\displaystyle k}$,${\displaystyle PA}$ proves the consistency of${\displaystyle {\mathsf{P}\mathsf{A}}_k}$. As each set${\displaystyle {\mathsf{P}\mathsf{A}}_k}$ is${\displaystyle {\mathrm{\Sigma }}_k}$-definable, this must be expressed as a theorem schema.^{[14]p. 4} These soundness principles are sometimes referred to assyntactic reflection principles, in contrast to the satisfaction-based varieties mentioned above, which are calledsemantic reflection principles.^{[15]p. 1}

The local reflection principle${\displaystyle Rfn(T)}$ for a theory${\displaystyle T}$ is the schema that for each sentence${\displaystyle \varphi }$ of the language of${\displaystyle T}$,${\displaystyle {\mathrm{P}\mathrm{r}\mathrm{o}\mathrm{v}}_T(\varphi )⟹\varphi }$. When${\displaystyle Rf{n}_{\mathrm{\Gamma }}(T)}$ is the restricted version of the principle only considering the${\displaystyle \varphi }$ in a class of formulas${\displaystyle \mathrm{\Gamma }}$,${\displaystyle \mathrm{C}\mathrm{o}\mathrm{n}(T)}$ and${\displaystyle Rf{n}_{{\mathrm{\Pi }}_1^0}(T)}$ are equivalent over${\displaystyle T}$.^{[16]p. 205}

The uniform reflection principle${\displaystyle RFN(T)}$ for a theory${\displaystyle T}$ is the schema that for each natural numbers${\displaystyle n}$,${\displaystyle \mathrm{\forall }(⌜\varphi ⌝\in {\mathrm{\Sigma }}_n^0\cup {\mathrm{\Pi }}_n^0)\mathrm{\forall }(y_0,\dots ,y_m\in \mathbb{N})({\mathrm{P}\mathrm{r}}_T(⌜\varphi (y_0,\dots ,y_n{)}^{\ast }⌝⟹{\mathrm{T}\mathrm{r}}_n(⌜\varphi (y_0,\dots ,y_n{)}^{\ast }⌝))}$, where${\displaystyle {\mathrm{\Sigma }}_n^0\cup {\mathrm{\Pi }}_n^0}$ is the union of the sets ofGödel-numbers of${\displaystyle {\mathrm{\Sigma }}_n^0}$ and${\displaystyle {\mathrm{\Pi }}_n^0}$ formulas, and${\displaystyle \varphi (y_0,\dots ,y_n{)}^{\ast }}$ is${\displaystyle \varphi }$ with its free variables${\displaystyle y_0,\dots ,y_m}$ replaced with numerals${\displaystyle \underset{y_0}{\underbrace{S\dots S}}0}$, etc. in the language of Peano arithmetic, and${\displaystyle {\mathrm{T}\mathrm{r}}_n}$ is the partialtruth predicate for${\displaystyle {\mathrm{\Sigma }}_n^0\cup {\mathrm{\Pi }}_n^0}$ formulas.^{[16]p. 205}

For${\displaystyle k\ge 1}$, a${\displaystyle {\beta }_k}$-model is a model which has the correct truth values of${\displaystyle {\mathrm{\Pi }}_k^1}$ statements, where${\displaystyle {\mathrm{\Pi }}_k^1}$ is at the${\displaystyle k+1}$th level of theanalytical hierarchy. A countable${\displaystyle {\beta }_k}$-model of a subsystem ofsecond-order arithmetic consists of a countable set of sets of natural numbers, which may be encoded as a subset of${\displaystyle \mathbb{N}}$. The theory${\displaystyle {\mathrm{\Pi }}_1^1{\mathsf{-}\mathsf{C}\mathsf{A}}_0}$ proves the existence of a${\displaystyle {\beta }_1}$-model, also known as a${\displaystyle \beta }$-model.^{[17]Theorem VII.2.16}

The${\displaystyle {\beta }_k}$-model reflection principle for${\displaystyle {\mathrm{\Sigma }}_n^1}$ formulas states that for any${\displaystyle {\mathrm{\Sigma }}_n^1}$ formula${\displaystyle \theta (X)}$ with${\displaystyle X}$ as its only free set variable, for all${\displaystyle X\subseteq \mathbb{N}}$, if${\displaystyle \theta (X)}$ holds, then there is a countable coded${\displaystyle {\beta }_k}$-model${\displaystyle M}$ where${\displaystyle X\in M}$ such that${\displaystyle M\models \theta (X)}$. An extension${\displaystyle {\mathrm{\Sigma }}_k^1{\mathsf{-}\mathsf{D}\mathsf{C}}_0}$ of${\displaystyle {\mathsf{A}\mathsf{C}\mathsf{A}}_0}$ by a schema of dependent choice is axiomatized. For any${\displaystyle 0\le k}$, the system${\displaystyle {\mathrm{\Sigma }}_{k+2}^1{\mathsf{-}\mathsf{D}\mathsf{C}}_0}$ is equivalent to${\displaystyle {\beta }_{k+1}}$-reflection for${\displaystyle {\mathrm{\Sigma }}_{k+4}^1}$ formulas.^{[17]Theorem VII.7.6}

${\displaystyle \beta }$-model reflection has connections to set-theoretic reflection, for example over the weak set theoryKP, adding the schema of reflection of${\displaystyle {\mathrm{\Pi }}_n}$-formulas to transitive sets (${\displaystyle \varphi ⟹\mathrm{\exists }z(\text{transitive}(z)\wedge {\varphi }^z)}$ for all${\displaystyle {\mathrm{\Pi }}_n}$ formulas${\displaystyle \varphi }$) yields the same${\displaystyle {\mathrm{\Pi }}_4^1}$-consequeneces as${\displaystyle \mathsf{A}\mathsf{C}\mathsf{A}\mathsf{+}\mathsf{B}\mathsf{I}}$ plus a schema of${\displaystyle \beta }$-model reflection for${\displaystyle {\mathrm{\Pi }}_{n+1}^1}$ formulas.^[18]

• Jech, Thomas (2002),Set theory, third millennium edition (revised and expanded), Springer,ISBN 3-540-44085-2
• Kunen, Kenneth (1980),Set Theory: An Introduction to Independence Proofs, North-Holland,ISBN 0-444-85401-0
• Lévy, Azriel (1960),"Axiom schemata of strong infinity in axiomatic set theory",Pacific Journal of Mathematics,10:223–238,doi:10.2140/pjm.1960.10.223,ISSN 0030-8730,MR 0124205
• Montague, Richard (1961), "Fraenkel's addition to the axioms of Zermelo", in Bar-Hillel, Yehoshua; Poznanski, E. I. J.; Rabin, M. O.;Robinson, Abraham (eds.),Essays on the foundations of mathematics, Hebrew Univ., Jerusalem: Magnes Press, pp. 91–114,MR 0163840
• Reinhardt, W. N. (1974), "Remarks on reflection principles, large cardinals, and elementary embeddings.",Axiomatic set theory, Proc. Sympos. Pure Math., vol. XIII, Part II, Providence, R. I.: Amer. Math. Soc., pp. 189–205,MR 0401475

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

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