Non-Standard Axioms for Various Math Structures 4

We continue the discussion from the previous post in the series.

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03/25/2023: Some additions and modifications.

Upon a new point of view, we will change the terminology and introduce some notation.

Right now, we can equivalently define our current terminology as follows. Let $\mathcal{L}$ be a logic (the definition could probably make sense for a higher-order logic too) and $L$ be a language in the logic of $\mathcal{L}$ . Let $C$ be a class of models of $L$ .

Definition. The narrow closure of $C$ is the class of all models $c$ such that, for any statement $\lambda$ in the language of $L$ (and in the logic of $\mathcal{L}$ ), $\lambda$ is true of all members of $C$ if and only if it is true of $c$ .

Definition. The wide closure of $C$ is the class of all models $c$ such that, for any statement $\lambda$ in the language of $L$ (and in the logic of $\mathcal{L}$ ), if $\lambda$ is true of all members of $C$ , then it is true of $c$ .

We could consider a new concept inspired by the form of these: the class of all models $c$ such that, for any suitable statement $\lambda$ , if $\lambda$ is true of $c$ , then it is true of all members of $C$ .

We introduce some notation to accommodate this view.

Definition. We denote the previously-called narrow closure of $C$ by $Clo(C,c \leftrightarrow C)$ .

Definition. We denote the previously-called wide closure of $C$ by $Clo(C,C \rightarrow c)$ .

Our new concept can then be denoted by $Clo(C,c \rightarrow C)$ .

This is very readable and the notation is automatically suggestive of the meaning. (We use $Clo$ since the just-shorter $Cl$ could possibly conflict with “class.”)

This is the notation that we will use henceforth, so we will ditch the terminology of “wide” and “narrow” closure, and simply talk about “model closures,” or more precisely closures of classes of models, with closure in the specific context investigated here. This could be qualified as “logical closure,” since it is fundamentally about logical implication of logical statements. Thus, we could say we are studying logical closures of classes of models. If the term isn’t already used or subsumed, we could call this topic “logical closure theory,” which would then be a part of model theory.

Thus, we will henceforth say we are discussing logical closure theory, with the fundamental notation given above. Clearly, we have $Clo(C,c \leftrightarrow C) = Clo(C,c \rightarrow C) \cap Clo(C,C \rightarrow c)$ . As examples, we have that:

$Clo\left( \left\{ \mathbb{R} \right\},c \leftrightarrow C \right)$ is the class of real-closed fields, with $\mathcal{L}$ first-order logic and $L$ the language of fields.
In general, given a model $M$ of the language $L$ and assuming that $L$ is first-order, $Clo\left( \left\{ M \right\},c \leftrightarrow C \right)$ is the class of all models of $L$ that are elementarily equivalent to $M$ .
If $S$ is the class of symmetric groups, then $Clo(S,C \rightarrow c)$ is the class of all groups, with $\mathcal{L}$ equational logic and $L = \left\{ \times ,e,a \rightarrow a^{- 1} \right\}$ .

We also ask:

If $M$ is the set $\left\{ M_{n \times n}\left( \mathbb{R} \right):n \in \mathbb{Z}_{+} \right\}$ , where $M_{m \times n}\left( \mathbb{R} \right)$ is the set of real $m \times n$ matrices, then is $Clo(M,C \rightarrow c)$ the class of all rings, for $\mathcal{L}$ either equational or first-order logic and $L$ the language of rings (including identities and additive inverse)?

It remains to investigate this question.

We can reframe the previous post’s discussion with this new notation: if $\mathcal{L}$ is a logic, $L_{1}$ and $L_{2}$ are languages in the logic of $\mathcal{L}$ , and $C$ is a class of models of $L_{1}$ and also a class of models of $L_{2}$ , then under suitable “re-axiomatizability” conditions of the closure are the closures by $L_{1}$ and $L_{2}$ necessarily equal? Is equality of closure equivalent to such “re-axiomatizability”? (We can presumably ask this for each type of closure, not just wide as before.)

We define this “re-axiomatizability” as follows: we say that, given sets of axioms $A_{1}$ in $L_{1}$ and $A_{2}$ in $L_{2}$ , we require that there exists an expression of the symbols of $L_{2}$ in terms of the symbols of $L_{1}$ , such that for any interpretation of $L_{1}$ satisfying $A_{1}$ , the corresponding interpretation of $L_{2}$ satisfies $A_{2}$ — and vice versa. This satisfaction can be proven in some logic (maybe $\mathcal{L}$ or even something stronger than and containing $\mathcal{L}$ ) or can be just known — for some $\mathcal{L}$ , Godel’s completeness theorem guarantees that proof must be possible in the case of “just knowing,” but this is a side curiosity in terms of stating the conditions of the question.

Let’s look again at groups as an example. We have a first-order axiomatization in $\left\{ \times \right\}$ , and separately an equational axiomatization in $\left\{ \times ,e,a \rightarrow a^{- 1} \right\}$ . We also have that equational logic is a subset of first-order logic, and there is an expression of $\left\{ \times ,e,a \rightarrow a^{- 1} \right\}$ in terms of $\left\{ \times \right\}$ (and vice versa) such that we have re-axiomatizability. But according to our formulation, which only supports talking about one logic, here the logic would be first-order. Then, what we can conclude is that the first-order closure of some subclass of groups is the same regardless of $\left\{ \times \right\}$ or $\left\{ \times ,e,a \rightarrow a^{- 1} \right\}$ . But for the equational closures to be the same, we’d instead need an equational axiomatization of groups in the language of just $\left\{ \times \right\}$ . This is exactly what we’d expect from the previous post’s discussion. Thus, our formulation above makes sense. And really, it’s just a formal rephrasing of an intuitively clear idea. But we’d want to go through and prove it for the sake of completeness (and practice.)

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03/28/2023: Some new thoughts.

First, the definition of re-axiomatizability given above actually seems a bit suspect when compared to common uses. For example, for groups, given the first-order axiomatization with language $\left\{ \times \right\}$ , we define a corresponding interpretation and equational axiomatization for language $\left\{ \times ,e,a \rightarrow a^{- 1} \right\}$ , but this interpretation isn’t just an expression in $\left\{ \times \right\}$ — rather, the first-order axioms for $\left\{ \times \right\}$ assert the existence and uniqueness of identity, and we let that be $e$ . So a better formulation would be:

Definition. Assume $\mathcal{L}$ is a logic, $L_{1}$ and $L_{2}$ are languages in the logic of $\mathcal{L}$ , and $A_{1}$ and $A_{2}$ are sets of statements in $L_{1}$ and $L_{2}$ respectively. We say that $A_{1}$ can be re-axiomatized to $A_{2}$ if for every model $M = \left( U,I_{1} \right)$ of $A_{1}$ , there exists an interpretation $I_{2}$ of $L_{2}$ on $U$ that satisfies $A_{2}$ .

But then this also seems suspect to me, since it seems like $I_{2}$ may not even need to be “attached to” $I_{1}$ . Given $U$ , if there is some interpretation $I^{'}$ of $L_{2}$ that satisfies $A_{2}$ , then it doesn’t matter what $I_{1}$ is: given $\left( U,I_{1} \right)$ , regardless of $I_{1}$ , we can always select $I_{2} = I^{'}$ . Constant dependence is still dependence, so the logical dependence of $I_{2}$ on $I_{1}$ (which is inherent in the definition) doesn’t change this. We could require that given $I_{1}$ that $I_{2}$ must be unique up to some level, but this seems restrictive and not generally possible — for any permutation $P$ of a set and any satisfactory interpretation $I^{'}$ , the new interpretation $I^{'} \circ P$ would seem to be satisfactory as well. (And of course, except for a singleton, there will exist more than one such permutation.) Furthermore, this uniqueness requirement seems to also be a condition on $A_{2}$ and $U$ that doesn’t actually involve $I_{1}$ at all.

So how can we formalize this notion of a “re-axiomatization” or an “equivalent axiomatization”? By this definition, there exist many alternate axiomatic systems for groups that aren’t what we would naturally consider “sufficiently equivalent.”

Also, “re-axiomatized” isn’t the proper word to use in the above definition, since that implies equivalence which would require both ways; a better word is “implies.”

Maybe it’s about logical proof. In other words, maybe we can define:

If we are introducing expressions again, then the non-logical proof-based definition is better; in fact, the condition that every model satisfies $A_{2}$ too is itself a logical proof (just maybe not in the first-order proof system.) Restricting to a particular proof system seems unsatisfactory.

Instead, maybe we should generalize our concept of “expression” to include common use cases like groups mentioned above. Maybe we want something like: if $A_{1}$ implies the existence of certain things, then we can “pick” such a thing and use that in the expression of $L_{2}$ in terms of $L_{1}$ . This sounds very “Choice”-y, and from a highly formal standpoint (like in the context of a computerized formalization), I’m not sure what would need to be required to state or formalize this. But more informally, we can state our definition as follows:

Definition. Assume $\mathcal{L}$ is a logic, $L_{1}$ and $L_{2}$ are languages in the logic of $\mathcal{L}$ , and $A_{1}$ and $A_{2}$ are sets of statements in $L_{1}$ and $L_{2}$ respectively. Let $S_{1}$ be the set of “concepts” whose existence are implied by $A_{1}$ . We say that $A_{1}$ implies $A_{2}$ if there exists an expression of $L_{2}$ in terms of $L_{1}$ and “instances” of “concepts” in $S_{1}$ such that for every model of $L_{1}$ satisfying $A_{1}$ , the corresponding interpretation of $L_{2}$ satisfies $A_{2}$ .

It seems immediate then to define $A_{1}$ and $A_{2}$ to be equivalent if $A_{1}$ implies $A_{2}$ and $A_{2}$ implies $A_{1}$ , but is that enough? We can recall our discussion in the Topology as an Algebraic Structure series, where even when we had an interpretation of $L_{2}$ it still didn’t feel like enough, especially if the construction couldn’t be reversed uniquely.

A stronger definition of equivalence that could allow us to go back and forth between $A_{1}$ and $A_{2}$ would be:

Definition. Assume the setup in the previous definition. We say that $A_{1}$ and $A_{2}$ are equivalent if $A_{1}$ implies $A_{2}$ , $A_{2}$ implies $A_{1}$ , and the expressions are “inverses” of each other: if $f_{1 \rightarrow 2}$ is the expression of $L_{2}$ in terms of $L_{1}$ , and if $f_{2 \rightarrow 1}$ is the expression of $L_{1}$ in terms of $L_{2}$ , then $f_{1 \rightarrow 2}$ applied to the $L_{2}$ symbols substituted by $f_{2 \rightarrow 1}$ just yields expressions equal to the original $L_{1}$ , and similarly the other way.

I’m sure this idea of “equivalent axiomatizations” for different languages is something that has been thought about more generally and formalized. I just need to understand what it is and how I can identify the subject that discusses this. (It would clearly be a part of model theory and mathematical logic, but what is this called formally?)

Out of the concepts discussed here, it seems that “equivalent axiomatization” would very much be something already formalized, while “logical closure” could be new. I would probably need to study more model theory to learn the standard formalization of “equivalent axiomatization” as well as the general landscape of frameworks for these kinds of concepts and ideas.

This raises another point. I have been learning often that many concepts I first investigate have generalizations and “fulfillments” in existing frameworks and subjects, but discovering these generalized subjects is actually quite tricky. From a technical standpoint, our current technology understands “tell me about concept X” much better than “if we generalized concept X in Y way, does there exist a name for that yet?” This kind of technology, what I like to call “reverse search,” seems like it could revolutionize the way we study and discover math, and it is very much a to-be-addressed problem. This is actually part of one of my personal project ideas to drastically improve the technology available to mathematicians — I think a “search index” that isn’t just a generic Knowledge Graph but something that understands logical proofs and derivations specific to math (and theoretical science) could generate immense domain-specific value. This seems to be something that companies are starting to do today to various degrees — I wouldn’t be surprised for example to see that Wolfram Research has invested in a project like that — but I think there’s a lot more that is waiting to happen in this space. But in the meantime, I wonder if there are better approaches that people use to figure out / search questions like “if we generalized concept X in Y way, does there exist a name for that yet?” Manually adding to a database of information that just exposes these subjects to more students like me could be very valuable. This is part of what I want to do with my expository posts, but I’m thinking it could be a good idea to start a page that records more of these connections between topics so that people have a much quicker way of finding out about them, even at a high level before I learn the details fully enough to write expository presentations of them.

A couple more points to make before we end this post. (This post is starting to get long, so a continuation of this discussion should happen in a follow-up post, unless the continuation can be stated very succinctly.) First, logical closure theory can be investigated independent of this discussion on equivalent axiomatization. To that end, I have some questions:

For which first-order theories $T$ is the class of models of $T$ the (first-order) logical closure of a singleton? (In other words, a single specific model contains all first-order statements true for all models of $T$ .)
What about the analogous question for equational logic (equational theories and equational closures)? Clearly, abstract Boolean algebras are the logical closure of a singleton, namely $\left\{ \left\{ 0,1 \right\} \right\}$ . What about other logics? (Maybe just within first-order logic to tame things a bit.)

Also, this line of thought was motivated in the beginning by abstract Boolean algebras, which in our current terminology corresponds to $Clo(C,C \rightarrow c)$ . Thus, the interpretation (in an informal, not model-theoretic sense) or meaning of $Clo(C,C \rightarrow c)$ and also $Clo(C,C \leftrightarrow c)$ are well-understood. But what about $Clo(C,c \rightarrow C)$ ? What exactly does this mean, in terms of the information it yields (about either the closure or $C$ ?)

It remains to investigate this.

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We continue this discussion in the next post.

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