Talk:Reflection principle

Reflection in complex variables

What about "reflection principle" as in complex variables?

To 24.199.94.94: Please sign your contributions to talk with four tildas, i.e. ~~~~. I am not familiar with complex analysis or any "reflection principle" in it. If the mathematicians in that area want to write such an article, no doubt, they would create a disambiguation page to distinguish it from this article. Have you tried looking for it by other names? JRSpriggs 06:03, 4 July 2006 (UTC)

Found it! Schwarz reflection principle -- added reciprocal disambig links on each page Zero sharp 22:02, 12 September 2007 (UTC)

Reflection principle in probability theory

Does anyone know whether there is an article on the reflection principle for Brownian motion yet? siℓℓy rabbit (talk) 11:48, 13 September 2008 (UTC)

Yes. Just created recently, and needs some work. See Reflection principle (Wiener process). — Arthur Rubin (talk) 20:43, 14 February 2013 (UTC)

Reflection theorems in number theory

"Reflection theorem" redirects to "Reflection Principle". There is a well established series of reflection theorems in algebraic number theory, beginning with Scholz in 1932 and continuing through current research by G. Gras. Does anyone know if these other "Reflection Principles" (set theory, complex variables, or Brownian motion) are sometimes referred to as "Reflection Theorems"? If not, then I'll go ahead and stop "Reflection Theorem" from redirecting here. Regardless, it sounds like "Reflection Principle" needs to become a disambiguation page. If I don't get a response soon, I'll just make this one big disambiguation page for both "reflection principles" and "reflection theorems".B2smith (talk) 00:19, 4 December 2008 (UTC)

I suggest that you add another hat-note, if necessary, rather than make this into a disambiguation page. I am not aware of how Brownian motion is involved in this. I suspect that the reflection theorems are related to this reflection principle, being the application of the same idea to a sub-theory of set-theory. JRSpriggs (talk) 03:30, 5 December 2008 (UTC)

Okay, I don't know about the Brownian motion either, so maybe it just deserves a hat-note. But I still want to know if "Reflection theorem" is often used as a substitute for "Reflection Principle" in any of the cases: set theory, complex variables, or Brownian motion. The "reflection theorems" in number theory are totally unrelated to the set theoretic "reflection principle". I'd like to stop the redirect from "reflection theorem" to "reflection principle" and give the page to the number theoretic stuff, unless "reflection theorem" is a common synonym for "reflection principle" in set theory. If it is only occasionally used, then I can devote "reflection theorem" to the number-theoretic stuff and put a hat note linking here for people interested in set theory.B2smith (talk) 20:45, 5 December 2008 (UTC)

Go ahead and change the redirect, reflection theorem. I am only concerned about this article, reflection principle. JRSpriggs (talk) 05:22, 6 December 2008 (UTC)

NBG

Article says:

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 is a little bit confusing, since NBG is finitely axiomatizable. Aren't theorems in ZFC supposed to also be true in NBG? 76.195.10.34 (talk) 15:31, 4 March 2009 (UTC)

Von Neumann–Bernays–Gödel set theory is a conservative extension of Zermelo–Fraenkel set theory. That is, any formula in the language of ZFC will be a theorem of ZFC if and only if it is a theorem of NBG. However, NBG achieves this by adding another type, class, which is not present in ZFC. So the proof used in NBG cannot in general be pushed down into ZFC. Indeed, many of the axioms of NBG cannot even be stated in the language of ZFC.

If you tried to convert NBG into a one-type theory, then you would find that the thing which was the class of all ordinals would become an inaccessible cardinal. It is well known that ZFC+inaccessible cardinal implies the consistency of ZFC, so there would be no problem. JRSpriggs (talk) 16:43, 4 March 2009 (UTC)

Is this the uniform reflection principle???

I don't understand this article, because I am not so familiar with set-theory.

The following article refers to this page (at the bottom):

Ω-consistent theory

I do understand the definition there. It is not limited to set theory.

If this is the same kind of reflection, then this article needs a rewrite, because this reflection is not limited to set theory.

If it is not the same, then a new page is needed. Lkruijsw (talk) 20:09, 7 December 2009 (UTC)

Unfortunately, I think that that article should not be linked to this one but to another one which does not yet exist. "Reflection" is used for many different purposes. JRSpriggs (talk) 06:59, 8 December 2009 (UTC)

Motivation section is incomplete

Current wording explains what reflection principle is,but not a motivation. I would expect something along the following lines

The motivation behind the principle is that the universe of sets should be completely undefinable, and therefore if it has a property, it should not be the first, second, nameable set/class with that property - it should be possible to exhibit a smaller set/class satisfying the same property (subject to limitations on allowable properties)

If this is not a valid motivation, the section should start "it was once thought that..." or "some have incorrectly asserted that...", with a wording similar to the above, followed by the currently agreed correct motivation. — Preceding unsigned comment added by 74.37.130.75 (talk) 15:44, 28 May 2013 (UTC)

Reference needed

"Bernays used a ..." where? 109.153.242.70 (talk) 10:20, 7 May 2016 (UTC)

Potentially unsourced paragraph

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.

This argument was introduced in an early edit in 2005, and revised in 2023 to incorporate the "Believing the Axioms" citation and the inexhaustibility justification, but the core argument of "extending V past V" with a "new principle for forming sets" does not seem to be adequately supported by "Believing the Axioms" or any good source I could find. Bbbbbbbbba (talk) 05:02, 24 July 2026 (UTC)

At the top of page 502 of Maddy, it says: "It depends on the widespread view that the universe of sets is too complex to be exhausted by any handful of operations, in particular by power set and replacement, the two given by the axioms of Zermelo and Fraenkel. Thus there must be an ordinal number after all the ordinals generated by replacement and power set. This is an inaccessible. Similarly, the universe above a given point should not be exhausted by these two operations, so there is another inaccessible, and so on. Versions of inexhaustibility can also be used to defend the various hyperinaccessibles and Mahlo cardinals. All of these are generated by thinking of processes that build up larger ordinals from below." (emphasis added) JRSpriggs (talk) 13:30, 24 July 2026 (UTC)

It still seems to me that this paragraph adds too much details (such as talking about a whole model X instead of just the ordinals, jumping from "V is larger than X" to "X is a set", using "the collection of all sets obtained from some set by repeatedly applying all methods in the collection A is also a set" as a new principle, and using this argument to support reflection (Maddy regards reflection as stronger than inexhaustibility, although in the context of ZF an inaccessible cardinal does imply that the axioms are reflected)). Since this is a philosophical argument, not a mathematical argument, I think we should take more care in declaring it as a more detailed presentation of the original. I will check the papers cited by Maddy and see if they support it. Bbbbbbbbba (talk) 09:00, 27 July 2026 (UTC)