Reflection principle

In 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" 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 the cumulative hierarchy such that ${\displaystyle V_{\alpha }\models \varphi (x_1,\dots ,x_n)}$. This is known as the Lévy-Montague reflection principle, or the Lévy reflection principle, principally investigated in Lévy (1960) and Montague (1961). Another version of this reflection principle says that for any finite 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 are absolute 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: Let ${\displaystyle \varphi }$ be a formula with at most free 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 the relativization of ${\displaystyle \varphi }$ to ${\displaystyle M}$ (that is, replacing all quantifiers 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 any finite set of axioms of ZFC we can find a countable transitive 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 the Löwenheim–Skolem theorem.

If ${\displaystyle \kappa }$ is a strong inaccessible cardinal, then there is a closed unbounded subset ${\displaystyle C}$ of ${\displaystyle \kappa }$, such that for every ${\displaystyle \alpha \in C}$, ${\displaystyle V_{\alpha }}$ is an elementary substructure of ${\displaystyle V_{\kappa }}$.

As new axioms

Large cardinals

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

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., ${\displaystyle \kappa ={\theta }_{\kappa }}$).

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, 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 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 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 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: 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 the powerset.

According to Akihiro Kanamori, 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 ${\displaystyle x}$ is transitive. 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 established pairing, union, infinity, and replacement, in effect achieving a remarkably economical presentation of ZF.

Others

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

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

Peter Koellner showed that a general class of reflection principles deemed "intrinsically justified" are 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 cardinals 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

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

Soundness

Let ${\displaystyle \mathsf{P}\mathsf{A}}$ denote Peano arithmetic, and ${\displaystyle {\mathsf{P}\mathsf{A}}_k}$ denote the set of true sentences in the language of PA that are ${\displaystyle {\mathrm{\Sigma }}_k}$ in the arithmetical 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.^{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}

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 ${\displaystyle T}$.^{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 of Gödel-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 partial truth predicate for ${\displaystyle {\mathrm{\Sigma }}_n^0\cup {\mathrm{\Pi }}_n^0}$ formulas.^{p. 205}

Model reflection

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 the analytical hierarchy. A countable ${\displaystyle {\beta }_k}$-model of a subsystem of second-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.^{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.^{Theorem VII.7.6}

${\displaystyle \beta }$-model reflection has connections to set-theoretic reflection, for example over the weak set theory KP, 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.