[bibtex key=Hamkins:The-Vopenka-principle-is-inequivalent-to-but-conservative-over-the-Vopenka-scheme]

Abstract.The Vopěnka principle, which asserts that every proper class of first-order structures in a common language admits an elementary embedding between two of its members, is not equivalent over GBC to the first-order Vopěnka scheme, which makes the Vopěnka assertion only for the first-order definable classes of structures. Nevertheless, the two Vopěnka axioms are equiconsistent and they have exactly the same first-order consequences in the language of set theory. Specifically, GBC plus the Vopěnka principle is conservative over ZFC plus the Vopěnka scheme for first-order assertions in the language of set theory.

The *Vopěnka principle* is the assertion that for every proper class $\mathcal{M}$ of first-order $\mathcal{L}$-structures, for a set-sized language $\mathcal{L}$, there are distinct members of the class $M,N\in\mathcal{M}$ with an elementary embedding $j:M\to N$ between them. In quantifying over classes, this principle is a single assertion in the language of second-order set theory, and it makes sense to consider the Vopěnka principle in the context of a second-order set theory, such as Godel-Bernays set theory GBC, whose language allows one to quantify over classes. In this article, GBC includes the global axiom of choice.

In contrast, the first-order *Vopěnka scheme* makes the Vopěnka assertion only for the first-order definable classes $\mathcal{M}$ (allowing parameters). This theory can be expressed as a scheme of first-order statements, one for each possible definition of a class, and it makes sense to consider the Vopěnka scheme in Zermelo-Frankael ZFC set theory with the axiom of choice.

Because the Vopěnka principle is a second-order assertion, it does not make sense to refer to it in the context of ZFC set theory, whose first-order language does not allow quantification over classes; one typically retreats to the Vopěnka scheme in that context. The theme of this article is to investigate the precise meta-mathematical interactions between these two treatments of Vopěnka’s idea.

**Main Theorems.**

- If ZFC and the Vopěnka scheme holds, then there is a class forcing extension, adding classes but no sets, in which GBC and the Vopěnka scheme holds, but the Vopěnka principle fails.
- If ZFC and the Vopěnka scheme holds, then there is a class forcing extension, adding classes but no sets, in which GBC and the Vopěnka principle holds.

It follows that the Vopěnka principle VP and the Vopěnka scheme VS are not equivalent, but they are equiconsistent and indeed, they have the same first-order consequences.

**Corollaries.**

- Over GBC, the Vopěnka principle and the Vopěnka scheme, if consistent, are not equivalent.
- Nevertheless, the two Vopěnka axioms are equiconsistent over GBC.
- Indeed, the two Vopěnka axioms have exactly the same first-order consequences in the language of set theory. Specifically, GBC plus the Vopěnka principle is conservative over ZFC plus the Vopěnka scheme for assertions in the first-order language of set theory. $$\text{GBC}+\text{VP}\vdash\phi\qquad\text{if and only if}\qquad\text{ZFC}+\text{VS}\vdash\phi$$

These results grew out of my my answer to a MathOverflow question of Mike Shulman, Can Vopěnka’s principle be violated definably?, inquiring whether there would always be a definable counterexample to the Vopěnka principle, whenever it should happen to fail. I interpret the question as asking whether the Vopěnka scheme is necessarily equivalent to the Vopěnka principle, and the answer is negative.

The proof of the main theorem involves the concept of a *stretchable* set $g\subset\kappa$ for an $A$-extendible cardinal, which has the property that for every cardinal $\lambda>\kappa$ and every extension $h\subset\lambda$ with $h\cap\kappa=g$, there is an elementary embedding $j:\langle V_\lambda,\in,A\cap V_\lambda\rangle\to\langle V_\theta,\in,A\cap V_\theta\rangle$ such that $j(g)\cap\lambda=h$. Thus, the set $g$ can be stretched by an $A$-extendibility embedding so as to agree with any given $h$.

Some alternative ideas for the terminology of

Vopenka schemecardinal might be:weakly Vopenkacardinal, oralmost-Vopenkacardinal. In the original MathOverflow post I had used the almost-Vopenka cardinal terminology. Please reply with comments on this or alternative suggestions.While “weakly” can be used for many things, I think it’s a good idea to keep in the scheme of compactness. In this aspect, “almost-Vopenka” sounds better.

Perhaps something like definably-Vopenka; or something like the indescribability hierarchy, suggesting how far you can push the scheme. Then you can get a “Totally Vopenka cardinal”, and bring a photo of Keanu Reeves as Ted Logan, stating that “This cardinal is totally Vopenka, dude!”

Regarding the large cardinal naming query and the very well-chosen photo of this post from the notion of “Indra’s Net” in Hinduism and Buddhism which combines the concepts of reflection, infinity and embedding a world into another at once, I would like to add that maybe it is not a bad idea to name one of such large cardinal axioms that grow out of various forms of reflection and embedding after Eastern gods and philosophers whose deep thoughts led them to consider such notions thousands of years before the modern treatment of infinity. Anyway I think names like Indra or Buddha cardinal sound really nice and beautiful!

https://en.wikipedia.org/wiki/Indra%27s_net

https://en.wikipedia.org/wiki/Indra

https://en.wikipedia.org/wiki/Ananta_(infinite)

http://www.hindupedia.com/en/Mathematics_of_the_Vedas

I had chosen the image because I found it to suggest infinity and infinite reflection, such as arises in Vopenka’s axiom. I imagined the spherical jewels in the image as representing the structures in a class instance of Vopenka’s axiom, stretching to infinity. The structures are not identical, and yet they reflect each other. One flaw in the image, as far as I can tell, is that a close inspection reveals the edge of the lattice—it is not actually infinite, unfortunately.

In some sense the subject of naming new large cardinal axioms (and in general naming mathematical objects, concepts, methods and theorems) reminds me the standards that exist in the other disciplines like astronomy and chemistry for naming astronomical objects and chemical compounds. For example see the followings:

https://en.wikipedia.org/wiki/Astronomical_naming_conventions

https://en.wikipedia.org/wiki/IUPAC_nomenclature_of_organic_chemistry

In both cases an international organization defined precise rules for clarifying the ways that any existing and new found object is supposed to be named. For astronomy the International Astronomical Union (IAU) and for chemistry the International Union of Pure and Applied Chemistry (IUPAC) did this task.

https://en.wikipedia.org/wiki/International_Astronomical_Union

https://en.wikipedia.org/wiki/International_Union_of_Pure_and_Applied_Chemistry

In both cases defining such a unique naming system are absolutely necessary because there are too many objects and some referring confusions might arise if the naming system isn’t uniquely determined in these fields.

I don’t know if it is a perfect description to consider the current situation of naming many existing large cardinal axioms (and other mathematical objects) as something similar to the traditional system of naming astronomical and chemical object based on people’s names and aesthetically descriptions.

I am also not sure if having such a naming system in mathematics is necessary but maybe it could be an option for the International Mathematical Union (IMU) to design such a system for mathematical objects at least for the classification purposes in the future.

https://en.wikipedia.org/wiki/International_Mathematical_Union

Anyway such ideas should be discussed publicly with math community members (e.g. in Mathoverflow, social networks, personal blogs, conferences, etc.) to get a better idea of the community’s general opinion.

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