Non-Standard Axioms for Various Math Structures 7

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

Note that the reduction of $Locl(C,c \rightarrow C)$ and $Locl(C,C \leftrightarrow c)$ to individual model equivalence depended entirely on the logic being closed under negation. So we can yield this reduction for any logic that is closed under negation.

With the intuition of negations of statements allowing this kind of reduction, I conjecture the same reduction for $Locl(C,C \rightarrow c)$ , for a logic closed under negation:

Conjecture. We have that $Locl(C,C \rightarrow c)$ is empty if $C$ consists of models not equivalent to each other, and otherwise is the class of all models equivalent to each element of $C$ .

Attempt. In the case that $C$ consists of models equivalent to each other, the desired classification of $Locl(C,C \rightarrow c)$ is immediate. In fact, we showed this earlier too. But for the sake of completeness: assume $C$ consisted of models all equivalent to each other. Clearly, any model equivalent to each element of $C$ would be in $Locl(C,C \rightarrow c)$ . For the converse, if $m \in Locl(C,C \rightarrow c)$ , then let $m^{'}$ be any element of $C$ . If a statement is true of $m^{'}$ , then it is true of every member of $C$ , and thus of $m$ . If a statement is true of $m$ , then the statement must be true of $m^{'}$ , for if the statement were not true of $m^{'}$ then the negation of the statement would be true of $m^{'}$ and thus of $m$ , contradiction. Thus, $m,m^{'}$ are equivalent.

So the “hard part” of this is showing that $Locl(C,C \rightarrow c)$ is empty if $C$ has two models not equivalent to each other. Assume for sake of contradiction that there existed $m \in Locl(C,C \rightarrow c)$ . There must exist $m_{1},m_{2} \in C$ and a statement $\lambda$ true of $m_{1}$ but not true of $m_{2}$ . Thus, by closure of the logic under negation, $\lambda$ is true of $m_{1}$ and $\neg\lambda$ is true of $m_{2}$ . But I’m not sure how to yield a contradiction here for $m$ , since presumably from a simple argument $m$ could satisfy either $\lambda$ or $\neg\lambda$ without issues.

We can show that if a statement is true of $m \in Locl(C,C \rightarrow c)$ , then it must be true of some member of $C$ . For if the statement weren’t true of some member of $C$ , then its negation would be true of all members of $C$ (using the closure of logic under negation), but then the negation would be true of $m$ too, contradiction.

If we can show the converse, that if a statement is true of some member of $C$ then it must be true of $m$ , then we’d be able to yield our desired. This is because if we have $m_{1},m_{2} \in C$ with $\lambda$ true of $m_{1}$ and $\neg\lambda$ true of $m_{2}$ , then both $\lambda$ and $\neg\lambda$ are true of some member of $C$ , thus they’d be both true for $m$ , contradiction.

I’m not sure how to show this, however. It remains to look into this further. $\blacksquare$

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So this is what we’ve shown so far. For any logic, $Locl(C,C \leftrightarrow c)$ reduces to individual model equivalence. Thus, we can dispense with that notation. For any logic closed under negation, $Locl(C,c \rightarrow C)$ reduces as well to individual model equivalence, in the same way. Thus, as far as we know now, the main nontrivial concepts are $Locl(C,C \rightarrow c)$ in any logic and $Locl(C,c \rightarrow C)$ in equational logic.

Can we show that the latter reduces to individual model equivalence in the case that $C$ consists of models equivalent to each other? We have that if $m$ is equivalent to each member of $C$ then if a statement is true of $m$ it must be true of each member of $C$ , hence $m \in Locl(C,c \rightarrow C)$ . Conversely, if $m \in Locl(C,c \rightarrow C)$ , then if a statement is true of $m$ it must be true of each member of $C$ ; thus, if $m^{'} \in C$ is arbitrary, every statement true of $m$ must be true of $m^{'}$ . But the reverse doesn’t seem easily extractable: if a statement is true of $m^{'}$ then it is true of every member of $C$ , but that doesn’t necessarily mean it is true of $m$ . Thus, the most we can show is that $Locl(C,c \rightarrow C)$ contains all models equivalent to each member of $C$ , even in this special case.

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We know that $Locl(C,C \rightarrow c)$ can be nontrivial in equational logic even if $C$ is a singleton (take for example the class of all abstract Boolean algebras), but this is not true for logics closed under negation. In other words, if the logic is closed under negation, then if $C$ is a singleton, $Locl(C,C \rightarrow c)$ is exactly the class of all models equivalent to the one element of $C$ . Actually, more generally: if $C$ consists of models equivalent to each other, then $Locl(C,C \rightarrow c)$ is exactly the class of all models equivalent to each member of $C$ . Thus, in this special case, $Locl(C,C \rightarrow c)$ reduces to individual model equivalence.

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Let’s talk about the cardinality of logical bases. From my exploration on an abstract approach to generating sets, without involving more specifics from mathematical logic it is very possible that bases could have different cardinalities. Let’s delve into this further, using an example with groups.

Lemma. Let $p_{i}$ be the $i$ th prime number. Then, for any $k$ , the set of additive groups $\left\{ \mathbb{Z}_{p_{1}},\ldots,\mathbb{Z}_{p_{k}} \right\}$ is (logically) independent.

Proof. We want to show that for any $i$ , $\left\{ \mathbb{Z}_{p_{1}},\ldots,\mathbb{Z}_{p_{k}} \right\} - \left\{ \mathbb{Z}_{p_{i}} \right\}$ doesn’t generate $\mathbb{Z}_{p_{i}}$ . To do this, we’ll exhibit a statement true for $\mathbb{Z}_{p_{1}},\ldots,\mathbb{Z}_{p_{i - 1}},\mathbb{Z}_{p_{i + 1}},\ldots,\mathbb{Z}_{p_{k}}$ but not true for $\mathbb{Z}_{p_{i}}$ . In first-order logic (with existential quantifiers), this seems particularly easy: in general, in $\mathbb{Z}_{n}$ we have $\exists x\left( x^{n} = e \right)$ , which is an equation with a term of length $n$ . This means that for any $\mathbb{Z}_{a_{1}},\ldots,\mathbb{Z}_{a_{l}}$ , this first-order statement will hold: $\exists x\left( x^{lcm\left( a_{1},\ldots,a_{l} \right)} = e \right)$ . In particular, the statement $\exists x\left( x^{p_{1}\ldots p_{i - 1}p_{i + 1}\ldots p_{k}} = e \right)$ will hold in $\mathbb{Z}_{p_{1}},\ldots,\mathbb{Z}_{p_{i - 1}},\mathbb{Z}_{p_{i + 1}},\ldots,\mathbb{Z}_{p_{k}}$ but not in $\mathbb{Z}_{p_{i}}$ , since no factor of $p_{1}\ldots p_{i - 1}p_{i + 1}\ldots p_{k}$ can divide $p_{i}$ and in a group the order of an element must divide the order of the group. This shows the desired independence. $\blacksquare$

In particular, we have showcased arbitrarily large finite sets of independent groups. Thus, if a proposed “logical dimension” conjecture holds true (that for the class of models of a first-order theory any two bases that are sets must have the same cardinality), then this dimension would probably need to be infinite. In fact, I suspect that in general (for abstract generations) the uniqueness of logical dimension would imply that we can’t have a spanning set of smaller cardinality — I will investigate this in that topic’s thread. I had another conjecture, the “singleton” conjecture, that the class of models of any first-order theory is the span of some singleton. If my suspicions turn out to be true, then this conjecture would be incompatible with the logical dimension conjecture — the truth of one of these would imply the falsity of the other.

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Let’s go back to the point that the only nontrivial concepts are $Locl(C,C \rightarrow c)$ and equational logic $Locl(C,c \rightarrow C)$ . Topically, these two concepts seem rather different; I’m not sure what “meaning” or intuitively what information $Locl(C,c \rightarrow C)$ in equational logic would carry. Thus, while that would be interesting to investigate on its own, we will split that up from the ensuing discussion here. Henceforth, we will only discuss $Locl(C,C \rightarrow c)$ here. Any discussion of $Locl(C,c \rightarrow C)$ in equational logic would be spun off from this point to a separate thread.

Going back to the roots of where the investigation of this concept started from (with the motivating examples of abstract Boolean algebras and symmetry groups), a good name for this study would be logical generation theory. This captures both trying to find just generating subclasses as well as potentially independent generating subclasses (bases), so it avoids the implications of narrower scope that our previous term of “logical basis theory” would have. This in general summarizes well the idea of what this subject would be about: finding generating subclasses of classes of structures/models.

Thus, we henceforth say that we are studying logical generation theory. I would probably now invoke heavier-duty mathematical logic to see what further results I can produce here in general, and for specific cases like rings I can investigate those possibly with methods specific to those theories. It remains to undertake this further.

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We continue this discussion in the next post of the series.

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