Vopěnka's principle (VP) is the statement that, given any proper class $\{\mathcal{A}_\eta: \eta\in ON\}$ of first-order structures in the same language, there are some $\alpha\not=\beta$ with $\mathcal{A}_\alpha$ elementarily embeddable into $\mathcal{A}_\beta$. There are several other equivalent formulations of VP - see http://ncatlab.org/nlab/show/Vop%C4%9Bnka%27s+principle.

VP is a very strong large cardinal axiom - in particular, it implies the existence of a proper class of extendible cardinals. However, this needn't stop us from showing that many nice classes of structures aren't counterexamples to VP:

• Say a class of structures $\mathcal{C}$ satisfies VP if, for any subclass $\mathcal{D}\subseteq\mathcal{C}$, there are distinction $\mathcal{A}_0,\mathcal{A}_1\in\mathcal{D}$ with $\mathcal{A}_0$ elementarily embeddable in $\mathcal{A}_1$.

(Note: to avoid annoyance, let's work in some theory like $NGB$ which can directly treat classes.)

My question is:

(Q1) What are some classes which we can prove - in ZFC (maybe + additional assumptions whose consistency strength is much weaker than full VP) - satisfy VP?

A trivial example is the class of pure sets; an easy, but not quite trivial, example is the class of ordinals (viewed as linear orders). But in general this seems a very hard problem. For example,

(Q2) Does the class of linear orders have VP?

I suspect the answer to this smaller question is yes, via some clever argument (perhaps using Laver's theorem), but I don't see it.

An interesting side question is whether, by restricting our attention to certain classes of structures, we can find principles of intermediate strength:

(Q3) Are there "natural" (say, $\mathcal{L}_{\omega_1\omega}$-definable) classes of structures $\mathcal{C}$ such that "$\mathcal{C}$ satisfies VP" has nontrivial consistency strength over ZFC (yet still much weaker consistency strength than full VP)?

Vopěnka's principle (VP) is the statement that, given any proper class $\{\mathcal{A}_\eta: \eta\in ON\}$ of first-order structures in the same language, there are some $\alpha\not=\beta$ with $\mathcal{A}_\alpha$ elementarily embeddable into $\mathcal{A}_\beta$. There are several other equivalent formulations of VP - see http://ncatlab.org/nlab/show/Vop%C4%9Bnka%27s+principle.

VP is a very strong large cardinal axiom - in particular, it implies the existence of a proper class of extendible cardinals. However, this needn't stop us from showing that many nice classes of structures aren't counterexamples to VP:

• Say a class of structures $\mathcal{C}$ satisfies VP if $\mathcal{C}$ is a proper class and, for any sub-proper class $\mathcal{D}\subseteq\mathcal{C}$, there are distinction $\mathcal{A}_0,\mathcal{A}_1\in\mathcal{D}$ with $\mathcal{A}_0$ elementarily embeddable in $\mathcal{A}_1$.

(Note: to avoid annoyance, let's work in some theory like $NGB$ which can directly treat classes.)

My question is:

(Q1) What are some classes which we can prove - in ZFC (maybe + additional assumptions whose consistency strength is much weaker than full VP) - satisfy VP?

A trivial example is the class of pure sets; an easy, but not quite trivial, example is the class of ordinals (viewed as linear orders). But in general this seems a very hard problem. For example,

(Q2) Does the class of linear orders have VP?

EDIT: after Joel's answers below, Q2 is the only question which remains open.

I suspect the answer to this smaller question is yes, via some clever argument (perhaps using Laver's theorem), but I don't see it.

An interesting side question is whether, by restricting our attention to certain classes of structures, we can find principles of intermediate strength:

(Q3) Are there "natural" (say, $\mathcal{L}_{\omega_1\omega}$-definable) classes of structures $\mathcal{C}$ such that "$\mathcal{C}$ satisfies VP" has nontrivial consistency strength over ZFC (yet still much weaker consistency strength than full VP)?

Vopenka's principle (VP) is the statement that, given any proper class $\{\mathcal{A}_\eta: \eta\in ON\}$ of first-order structures in the same language, there are some $\alpha\not=\beta$ with $\mathcal{A}_\alpha$ elementarily embeddable into $\mathcal{A}_\beta$. There are several other equivalent formulations of VP - see http://ncatlab.org/nlab/show/Vop%C4%9Bnka%27s+principle.

VP is a very strong large cardinal axiom - in particular, it implies the existence of a proper class of extendible cardinals. However, this needn't stop us from showing that many nice classes of structures aren't counterexamples to VP:

• Say a class of structures $\mathcal{C}$ satisfies VP if, for any subclass $\mathcal{D}\subseteq\mathcal{C}$, there are distinction $\mathcal{A}_0,\mathcal{A}_1\in\mathcal{D}$ with $\mathcal{A}_0$ elementarily embeddable in $\mathcal{A}_1$.

(Note: to avoid annoyance, let's work in some theory like $NGB$ which can directly treat classes.)

My question is:

(Q1) What are some classes which we can prove - in ZFC (maybe + additional assumptions whose consistency strength is much weaker than full VP) - satisfy VP?

A trivial example is the class of pure sets; an easy, but not quite trivial, example is the class of ordinals (viewed as linear orders). But in general this seems a very hard problem. For example,

(Q2) Does the class of linear orders have VP?

I suspect the answer to this smaller question is yes, via some clever argument (perhaps using Laver's theorem), but I don't see it.

An interesting side question is whether, by restricting our attention to certain classes of structures, we can find principles of intermediate strength:

(Q3) Are there "natural" (say, $\mathcal{L}_{\omega_1\omega}$-definable) classes of structures $\mathcal{C}$ such that "$\mathcal{C}$ satisfies VP" has nontrivial consistency strength over ZFC (yet still much weaker consistency strength than full VP)?

Vopěnka's principle (VP) is the statement that, given any proper class $\{\mathcal{A}_\eta: \eta\in ON\}$ of first-order structures in the same language, there are some $\alpha\not=\beta$ with $\mathcal{A}_\alpha$ elementarily embeddable into $\mathcal{A}_\beta$. There are several other equivalent formulations of VP - see http://ncatlab.org/nlab/show/Vop%C4%9Bnka%27s+principle.

VP is a very strong large cardinal axiom - in particular, it implies the existence of a proper class of extendible cardinals. However, this needn't stop us from showing that many nice classes of structures aren't counterexamples to VP:

• Say a class of structures $\mathcal{C}$ satisfies VP if, for any subclass $\mathcal{D}\subseteq\mathcal{C}$, there are distinction $\mathcal{A}_0,\mathcal{A}_1\in\mathcal{D}$ with $\mathcal{A}_0$ elementarily embeddable in $\mathcal{A}_1$.

(Note: to avoid annoyance, let's work in some theory like $NGB$ which can directly treat classes.)

My question is:

(Q1) What are some classes which we can prove - in ZFC (maybe + additional assumptions whose consistency strength is much weaker than full VP) - satisfy VP?

A trivial example is the class of pure sets; an easy, but not quite trivial, example is the class of ordinals (viewed as linear orders). But in general this seems a very hard problem. For example,

(Q2) Does the class of linear orders have VP?

I suspect the answer to this smaller question is yes, via some clever argument (perhaps using Laver's theorem), but I don't see it.

An interesting side question is whether, by restricting our attention to certain classes of structures, we can find principles of intermediate strength:

(Q3) Are there "natural" (say, $\mathcal{L}_{\omega_1\omega}$-definable) classes of structures $\mathcal{C}$ such that "$\mathcal{C}$ satisfies VP" has nontrivial consistency strength over ZFC (yet still much weaker consistency strength than full VP)?