A conjecture involving a universal set, and a train of thought for formulating a kosher equivalent

[PseuTonym]

The status quo of mathematics today seems to be reliance upon an underlying set theory that doesn’t permit the existence of a universal set. However, that doesn’t mean that when we think of a conjecture that involves reference to a universal set, that we should simply cross out the conjecture, throw away the paper that it was written on, and drop the paper down a memory hole of Orwell’s novel 1984. There may be a train of thought that will transform the conjecture so that we get some equivalent conjecture that is kosher.

Does anybody know of an example of such a conjecture and of such a train of thought for transforming it into something kosher?

 

[Just_Lurking]

V = universe of all sets
L = universe of constructible sets
Conjecture: V = L

See here for Wikipedia article.

 

[Paddy_Walker]

.maybe you could give an example.

 

[PseuTonym]

V = universe of all sets
L = universe of constructible sets
Conjecture: V = L

I’m not familiar with the concept of a universe of sets in set theory.

Is the following equivalent to your suggestion? Alternatively, is it a parody, straw man attack, or other misrepresentation of your idea?

My version of your suggestion:

V = the class of all sets
L = the class of sets that can be constructed
The conjecture: V = L

In my version or your suggestion, the conjecture is kosher, so there’s no need for a train of thought to formulate a kosher equivalent of the conjecture. Thus, if my version is equivalent to your suggestion, then your suggestion helps to demonstrate the difference between hitting the broad side of a barn, and actually hitting the topic on the nose, but isn’t strictly speaking on-topic. If my version is equivalent to your suggestion, then your suggestion is a contribution to the meta-discussion, and a contribution to the pedagogy, but isn’t strictly speaking on topic from the point of view of people who have had enough sleep and don’t require extra tutoring.

 

[Just_Lurking]

It is possible to extend the language of set theory to include additional types of objects such as proper classes, in which case V and L are classes, and V = L is a legitimate expression just as it stands.

Alternatively, one can ask what the equivalent of V = L is in standard “sets only” set theory, where V and L are not directly allowable expressions. In this case, the approach is to translate the pseudo-expression V = L into another pseudo-expression that is closer to allowable set theory, and keep on doing that until one obtains an expression that is directly in the allowable language of set theory. For example,

V = L becomes (for all x) ((x in V) if and only iff (x in L))

(x in V) becomes x = x

(x in L) becomes x in (union over ordinals a of L_a) becomes (exists ordinal a) (x in L_a)

and so on.