Talk:Morse–Kelley set theory

message: 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)
thread_title: needs work

message: 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)
thread_title:  relationship to NBG 

message: 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)
replies: text: 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)

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

text:  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)
replies: text: 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)
replies: text: 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)

text:   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)

text:   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)

text:  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)
replies: text: 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 Vκ. Trying to interpret it so that the individuals can be sets of arbitrary rank seems philosophically problematic. --Trovatore 01:11, 21 December 2005 (UTC)

text:  I don't disagree with anything here...  Randall Holmes 01:30, 21 December 2005 (UTC)
thread_title:  second-order language??? 

message: 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)
thread_title:  thanks 

message: M.K can be further simplified to:
M.K is the set of sentences entailed (from first order
logic with identity) by these axioms:
1) Extensionality: AxAy(x=y<->Az(zex<->zey)).
 
2) Regularity:Ax((Ez(zex))->Ey(yex & ~Ec(cey & cex ))).
 
3) Class comprehension: If P is a formula in which x doesn't occur free , then all closures of
 
ExAy(yex<->(P(y)&Ez(yez)))
are axioms.
 
 Definition 1) x=V <-> Ay(yex <-> Ez(yez)).
 Definition 2) x is a set <-> xeV 
 Definition 3) x is a proper class <-> ~Ey(xey).
 Definition 4) x=0 <-> Ay(~(yex)) 
Accordingly V is the proper class of all sets. 
4) Pairing:AaeVAbeVExeVAyeV(yex<->(y=a or y=b)).
 
5) Union:AaeVExeVAyeV(yex<->EzeV(zea&yez)).
6) Infinity:ENeV(0eN&(Ax(xeN->xU{x}eN))).
with 'U' and {x} having the usual definitions. 
7) limitation of size:
 
Ax((Ez(xez)) <-> x is subnumerous to V).
 
x is subnumerous to V <-> Af((f:x->V) ->(f is injective & ~ f is 
surjective)). 
8)Power: AaeVExeVAyeV(yex<->AzeV(zey->zea)).
Zaljohar 03:56, 4 March 2007 (UTC)
thread_title:  Writing Morse-Kelley 

message: There is a philosophical issue that I would like us to consider — should each separate axiom be formulated in: the strongest possible form, the weakest possible form, or some other form (simplest?)? In the strongest form, it would try to do its job without relying on help from other axioms. This would be advantageous for practical use of the axioms and for constructing sub-theories where some axioms are removed. In the weakest form, any redundant aspects (such as existence of the class for axioms other than class comprehension) would be removed. This would be useful for proving that some model satisfies certain axioms and, perhaps, establishing the independence of an axiom from the others (or the consistency of its negation with the others).
Specifically, the axioms of pairing, power set, union, and infinity each have at least two forms one of which is stronger than another. That is, they may assert the existence of the class which is the set in addition to its set-hood, or merely assert its set-hood. The axiom of infinity may also assert or not the existence of the empty class and successor classes; or merely say that if they exist and are sets, then they belong to the infinite set. JRSpriggs (talk) 14:25, 25 April 2008 (UTC)
thread_title:  Strong axioms versus weak axioms? 

message: It's a minor point, but why in 'domain f and range f denote the domain and range of the function f; this peculiarity has been carefully respected below;' is the word 'peculiarity' used?  — Preceding unsigned comment added by 213.122.37.54 (talk) 20:50, 18 October 2012 (UTC)
replies: text: Probably because it would be more usual to say "the domain X of the function f " in which "X" refers to the domain rather than the function. JRSpriggs (talk) 08:23, 19 October 2012 (UTC)
thread_title:  Peculiarity?