Talk:Morse–Kelley set theory

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1 needs work
2 redone
- 2.1 another issue in rewrite
3 relationship to NBG
4 second-order language???
5 Kelley's original axioms
6 thanks
7 Axiom of Pairs redundant

[edit] needs work

Doesn't say anything about what a "property" is (actually we want one definable by a formula in the second-order language of set theory). Doesn't mention what the other axioms are, relationship to ZFC, consistency following from an inaccessible cardinal. --Trovatore 15:05, 4 October 2005 (UTC)

[edit] redone

I have given a full account of this theory, though the form of limitation of size due to von Neumann may be viewed as eccentric. KM is not the same as second-order ZF.

Randall Holmes 16:23, 15 December 2005 (UTC)

Thanks; this article definitely needed work. I note though that you removed the point about KM being in the second-order language of set theory. I don't think that should have been removed. You're quite right of course that KM is a first-order theory, but it is expressed in second-order LST. It also should perhaps be noted that if you take exactly the same collection of axioms and give them a true second-order semantics (as opposed to the two-sorted first-order semantics that correspond to the first-order theory) you get something categorical up to the first inaccessible. --Trovatore 19:44, 15 December 2005 (UTC)

[edit] another issue in rewrite

I don't quite follow the claim that "second-order ZF" "proves the same theorems in practice" as KM. First, I'm not sure what it means for a second-order theory to "prove" something, but if it means logical implication (something is proved if true in all models of the theory), then second-order ZF proves lots of things KM doesn't (for example, Con(KM), and either CH or ¬CH, depending on which one is true). --Trovatore 06:35, 17 December 2005 (UTC)

see how you like the new language on the relationship between Morse-Kelley and second-order ZF. Randall Holmes 17:25, 17 December 2005 (UTC)

I've modified it slightly; the formulation involving definability was a little problematic. Suppose for example there were exactly five inaccessibles in the real world (of course that's nonsense, there's really proper-class many, but it's just an example). Then "the largest inaccessible" would be a definable rank of the von Neumann hierarchy. Now if we call the second inaccessible k2 and third k3, then V_k2 and V_k3 are both models of second-order ZF, but they disagree about the theory of the stage whose height is the largest inaccessible (though only because they disagree about what the largest inaccessible is). Granted, the new formulation is a little too weak, but at least it avoids this problem, which is awkward to find good language to work around.

[edit] relationship to NBG

This has now grown into a nice article.

MK can be viewed as a strengthened version of von Neumann-Bernays-Gödel’s set theory (NBG). Perhaps this relationship can be described?

the reference was already there, but not explicit enough. I have added a direct reference. Randall Holmes 19:57, 20 December 2005 (UTC)

Michael Meyling 15:56, 20 December 2005 (UTC)

[edit] second-order language???

I didn't know the phrase "second-order language". I read this as "second-order logic". Is this phrase well known? Perhaps it can be replaced or explained further after the following passage:

"Although this is a first-order theory, it is a common practice (not followed here) to use second-order language, that is, to distinguish between class variables (often capitalized) and set variables."

Perhaps something like this:

"This is can be done in the following way: $\forall x_i \ \phi(x_i)$ is shorthand for $\forall X_i \ (\mbox{set} X_i \rightarrow \phi(X_i))$ and analogous for the existence quantifier."

Michael Meyling 15:55, 20 December 2005 (UTC)

It's not really my phrase, but the result of a conversation with someone else. All I'm saying is that it is the custom of some authors to use set variables and class variables, which makes the language appear like that of second-order logic (though it is not really a two-sorted theory). I don't want to clarify it because I am not actually using this convention here. Randall Holmes 19:52, 20 December 2005 (UTC)

further, I made a small change to make it clear that I'm not using some peculiar technical terminology "second-order language". Randall Holmes 19:58, 20 December 2005 (UTC)

Well, but isn't it the correct technical terminology? The set variables are intended to be interpreted by objects, and the class variables by predicates. That's second-order language. I think this nomenclature is completely standard, though I don't have any handy reference for it. (It's not second-order logic simply because that would imply some way of making inferences beyond those of first-order logic, and we aren't using any such strengthened inference rules.) --Trovatore 21:58, 20 December 2005 (UTC)

I am interested in any reference you find. --Michael Meyling 04:37, 21 December 2005 (UTC)

I'm not sure it is: second-order logic is two-sorted and KM is not. Even if it is correct it isn't universally recognized; notice also that I simply paraphrased it as "language like that of second-order logic" (though not quite the same!) so that (one hopes) it would be clear what I meant. Randall Holmes 23:45, 20 December 2005 (UTC)

I'm not very convinced by this "not two-sorted" thing. I certainly think of the sets and classes as being different sorts. I don't see any reason the sorts have to be disjoint. --Trovatore 00:03, 21 December 2005 (UTC)

They aren't disjoint in the common nomenclature. In E. Mendelson's "Introduction to Mathematical Logic" or J. D. Monk's "Introduction to Set Theory" lower case letters just indicate classes that are sets and capital letters stand for classes (see above). Being a proper class is just a predicate. Or did I miss the point? --Michael Meyling 04:37, 21 December 2005 (UTC)

I'm not saying that the domains of sets and classes in KM are or should be disjoint (they aren't and shouldn't be); the domains of objects and predicates/sets of objects in second-order logic should be disjoint (more precisely, equations between objects of different sorts do not make sense). Randall Holmes 17:25, 21 December 2005 (UTC)

There's something I didn't catch above: it is not necessarily the case that the classes of KM are to be interpreted as predicates. That seems to go better with the predicative class theory NBG. The classes of KM are more likely to be viewed realistically (as actual large collections which are not necessarily all definable as predicates). That is certainly the way I think of it. (Note that this is not a counterargument to second-order logic being applicable: second-order objects can be arbitrary collections as easily as predicates). Re the immediately preceding remark, equality between variables of different sorts is at best problematic; I prefer to think of KM as a one-sorted first order theory. But I think this is all in danger of becoming a quibble; the relationship with second order logic is clearly acknowledged at several points in the article, isn't it? Randall Holmes 00:33, 21 December 2005 (UTC)

further on equality: the basic rule for equality is that if a=b and P(a) then P(b): but this will not work well if a and b are of different sorts (and so cannot freely be substituted for one another in all contexts). Randall Holmes 00:41, 21 December 2005 (UTC)

Well, I didn't necessarily mean definable predicates. More that they shouldn't be completed totalities. If they're completed totalities, after all, why aren't they sets? But this is part of the neither-fish-nor-flesh problem with interpreting KM. So mostly I tend to think of an interpretation of KM as having some fixed inaccessible κ specified in advance; then its sets are objects of rank less than κ and its classes are subsets of V_κ. Trying to interpret it so that the individuals can be sets of arbitrary rank seems philosophically problematic. --Trovatore 01:11, 21 December 2005 (UTC)

I don't disagree with anything here... Randall Holmes 01:30, 21 December 2005 (UTC)

[edit] Kelley's original axioms

I have added the full text of Kelley's original axioms to the article. It is worth noting that Kelley does not use "second-order language"; for him the sets and classes evidently have the same status exactly. Randall Holmes 06:18, 30 December 2005 (UTC)

Having fun, Randall? I'm afraid that Kelley's axioms do imply the limitation of size in MK, but not necessarily in MKU (with urelements).

I am having fun (here, if not in relation (mathematics)). Randall Holmes 07:10, 25 January 2006 (UTC)

counterexample -- loosely, start with a model of set theory with at least 2 inaccessibles, κ₀ < κ₁. Let U be a "set" of urelements with cardinality κ₁, and construct V as the "set" of all sets which are hereditarily of cardinality < κ₀ over U. Classes are arbitrary subsets of "the universe".

As On^(V) is κ₀, while |V| is κ₁, there cannot be a bijection between On^(V) and V.

proof (in "K", that is, MK) -- again, loosely --

Lemma There is a function W on V (instead of U, for reasons which seem clear), such that for each set x, W(x) is a bijection from an ordinal onto x.

Define, by transfinite recursion,

$W(x)(\alpha) = c \left( x - \left\{W(x)(\beta) \mid \beta < \alpha \right\} \right)$

Then define, by transfinite recursion,

$V_\alpha = \bigcup_{\beta < \alpha} P (V_\beta)$

For any set x, define the rank of x by:

ρ(x) =

the first

α

such that $x \in V_\alpha$ .

Define R, by

x R y iff $\rho(x) < \rho(y) \or (\rho(x) = \rho(y) \wedge (W(V_\rho(x))^{-1}(x) < (W(V_\rho(x))^{-1}(y))$

R is a well-ordering of the universe with order type On (See WE 4S in the reference below.) Finally, for any proper class X, R restricted to X is also a well-ordering of X with order type On, so that any two proper classes are equivalent.

(More detailed proofs of the implication can be found in Equivalents of the Axiom of Choice, H. Rubin & J. Rubin, with the additional note that any model of MK is also a model of NBG; Kelley's choice is approximately AC 1S in that book, while the axiom of limitations is approximately P 1S and clearly follows from WE 5S.

I don't know if this is something we want to move up to the main body, or not. Arthur Rubin | (talk) 00:41, 25 January 2006 (UTC)

Careful study shows that κ₁ need not be inaccessible, so the "counterexample" can be constructed relative to (MK+1 inaccessible), rather than (MK+2 inaccessibles). Arthur Rubin | (talk) 15:30, 25 January 2006 (UTC)

my guess would be that (the way the article is put together now) the fact that Kelley's axioms prove Limitation of Size would be of interest to the reader, but the proof can be left here. The fact that it doesn't hold with urelements is fun but perhaps too technical? Randall Holmes 17:39, 25 January 2006 (UTC)

[edit] thanks

thanks for the comments; I'll look. I didn't say anything about what could happen with urelements, and I'm not surprised if weird things happen :-) If the Kelley axioms imply Limitation of Size, I'm pleased (I don't think I say that they do or don't, do I?) Randall Holmes 01:02, 25 January 2006 (UTC)

[edit] Axiom of Pairs redundant

Is the Axiom of Pairs necessary? It seems to me that Limitation of Size, Class Comprehension, and Infinity suffice to imply it. For any sets x and y, we have from Class Comprehension a class containing only x and y; from Limitation of Size, this will be a proper class only if the universe contains no more than two sets; and from Infinity, the universe contains more than two sets. Therefore, the pair set {x, y} must exist. Just a minor piddling thing, but I was curious if I missed something here, since the article leaves out the Axiom of the Empty Set and notes why that can be done, but does not do so with regard to the Axiom of Pairs. -Chinju 15:12, 13 June 2006 (UTC)

Your argument seems correct to me. But sometimes it is better for clarity to include redundant axioms rather than try to make do with a minimalist set. Personally, I would include the axiom of empty set also rather than derive it. If a theorem is more obviously true than the "axioms" from which it is deduced, it would make sense to just declare it an axiom in its own right. JRSpriggs 03:26, 14 June 2006 (UTC)

I tend to agree with that line of thought (especially insofar as it makes for a more "modular" design; i.e., it allows us to then consider alternative systems where, say, Infinity has been removed without losing Empty set and Pairs, or such things). However, just given that the article as given seemed to be attempting a minimal axiomatization and falling clearly short, I thought I'd make sure I wasn't missing anything. -Chinju 16:43, 14 June 2006 (UTC)

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