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 due to Montague (1961), while stronger forms can be new and very powerful axioms for set theory.
[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 or transfinite type, including infinitary logics of any cardinal number.
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.Generally I believe that, in the last analysis, every axiom of infinity should be derivable from the (extremely plausible) principle that V is indefinable, where definability is to be taken in [a] more and more generalized and idealized sense.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).
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 (of V).
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.
Each X resembles V in the sense that it shares the property with V of being closed under the operations in A.
[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 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
[6] Another version of this reflection principle says that for any finite number of formulas of ZFC we can find a set
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
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.
is a strong inaccessible cardinal, then there is a closed unbounded subset
Reflection principles are connected to and can be used to motivate large cardinal axioms.
(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.)
The consistency of Bernays's reflection principle is implied by the existence of an ω-Erdős cardinal.
According to Akihiro Kanamori,[11]: 62 in a 1961 paper, Bernays considered the reflection schema for any formula
, Bernays just with this schema established pairing, union, infinity, and replacement, in effect achieving a remarkably economical presentation of ZF.
Some formulations of Ackermann set theory use a reflection principle.
,[2] 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.
[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.
Reflection principles may be considered for theories of arithmetic which are generally much weaker than ZFC.
denote the set of true sentences in the language of PA that are