Unconstructed set of constructed sets? Grok

 

Can there be an unconstruc t ed set of all constructible sets. If so, define it

No, there cannot be any set—whether constructible or unconstructible (non-constructible)—that consists of all constructible sets. In standard ZFC set theory, Gödel's constructible universe $L$ is the proper class of all constructible sets, meaning it is not itself a set. If there were a set containing all constructible sets, it would contradict the fact that $L$ is a proper class (e.g., via the Burali-Forti paradox, since $L$ contains all ordinals, which form a proper class). This holds regardless of assumptions like $V = L$ or $V \neq L$; no such set exists in either case.

Well, outside of ZFC or NGB

In set theories that allow a universal set (a set containing all sets, including itself), such as Quine's New Foundations (NF) or its variant NFU (New Foundations with Urelements), or positive set theories (e.g., those with generalized positive comprehension), the entire universe of sets is itself a set, often denoted $V$.

In these theories, one can define a notion analogous to Gödel's constructible universe $L$ by restricting comprehension to stratified definable subsets (in NF/NFU) or positive formulas, yielding a subclass of "constructible" sets.

Since the full universe $V$ is a set, this constructible subclass is also a set—specifically, the set of all constructible sets.

For example:

• In NF(U), stratified comprehension allows building a cumulative hierarchy of stratified-definable sets, resulting in a set $L$ (or similar) containing exactly the constructible sets.
• Positive set theories explicitly permit a universal set and comprehension for positive formulas, enabling a bounded constructible hierarchy that forms a set.

These theories avoid Russell's paradox via stratified (NF) or positive (positive set theories) restrictions on comprehension, rather than the cumulative hierarchy of ZFC.

Note that such theories differ significantly from ZFC: they may lack the axiom of choice, have different cardinalities (e.g., the universe can be well-orderable or not), and their consistency strength relative to ZFC remains open in some cases (though NFU is known to be consistent relative to ZF).

In class-set theories like von Neumann–Bernays–Gödel (NBG) or Morse–Kelley (MK), the class of all constructible sets remains a proper class, just as in ZFC.

Thus, a set of all constructible sets exists precisely in those alternative theories admitting a universal set. It can be defined as the set obtained by closing the empty set under the restricted definability operations corresponding to the theory's comprehension axiom.

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Morse-kelley set theory

Morse-Kelley set theory (often abbreviated MK or KM) is a powerful axiomatic set theory with classes, developed in the mid-20th century by Anthony Morse and John L. Kelley (with earlier ideas from Hao Wang and others). It is presented in the appendix of Kelley's 1955 book General Topology and in Morse's 1965 book A Theory of Sets.

Key Features

MK distinguishes between sets (which can be elements of other collections) and classes (collections that may be "too large" to be sets, called proper classes). The universe includes a universal class $V$ (the class of all sets), which is a proper class—no set contains all sets, avoiding Russell's paradox.

The main axioms include:

• Extensionality for classes
• Foundation
• Pairing, union, power set (restricted to ensure results are sets)
• Infinity
• Choice (often global choice for classes)
• Replacement
• Limitation of size (proper classes are those not in bijection with any set)

The crucial difference from similar theories is the impredicative Class Comprehension schema: For any formula $\phi(x, \vec{p})$ (possibly quantifying over classes), there exists a class $\{x \mid \phi(x, \vec{p})\}$. This impredicativity (allowing quantification over all classes, including the one being defined) makes MK very strong.

Comparison to Other Theories

Theory	Classes?	Class Comprehension	Conservative over ZFC?	Consistency Strength	Finitely Axiomatizable?
ZFC	No	N/A	Yes (itself)	Baseline	No
NBG (von Neumann–Bernays–Gödel)	Yes	Predicative (quantifiers over sets only)	Yes	Equiconsistent with ZFC	Yes
MK	Yes	Impredicative (quantifiers over classes)	No	Strictly stronger than ZFC; proves Con(ZFC)	No

MK is much stronger than NBG/ZFC because its impredicative comprehension allows defining a first-order truth predicate for sets, proving the consistency of ZFC (and NBG) within MK.

Models and Consistency

A standard model of MK is $\langle V_\kappa, \in, V_{\kappa+1} \rangle$, where $\kappa$ is an inaccessible cardinal in the background universe (e.g., from ZFC + inaccessible). Thus:

• If ZFC + "there exists an inaccessible cardinal" is consistent, then MK is consistent.
• MK's consistency strength lies between ZFC and ZFC + inaccessible (strictly above ZFC, below one inaccessible).

Relevance to Constructible Sets

In MK (like NBG or ZFC), the Gödel constructible universe $L$ (the class of all constructible sets) is a proper class, not a set. There is no set containing all constructible sets, as that would lead to paradoxes similar to those with ordinals or the universal class.

MK remains a class-set theory without a universal set (in the sense of a set of all sets); the class $V$ is proper.

If you're asking in the context of our previous discussion about a "set of all constructible sets," MK does not allow one—$L$ is a proper class.

MK is useful for foundations of category theory (handling large categories as classes), proving strong reflection principles, and studying impredicative theories. It's less commonly used as a foundation than ZFC due to its greater strength and non-finite axiomatizability.

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