1 Introduction
Abstract

A recently proposed axiom system for André’s central translation structures is improved upon. First, one of its axioms turns out to be dependent (derivable from the other axioms). Without this axiom, the axiom system is indeed independent. Second, whereas most of the original independence models were infinite, finite independence models are available. Moreover, the independence proof for one of the axioms employed proof-theoretic techniques rather than independence models; for this axiom, too, a finite independence model exists. For every axiom, then, there is a finite independence model. Finally, the axiom system (without its single dependent axiom) is not only independent, but completely independent.

1 Introduction

Pambuccian has offered two axiom systems and  [8] for André’s central translation structures [2]. ( is together with the Fano principle that diagonals of parallelograms are not parallel.) In this note we further develop Pambuccian’s work by showing that:

  • has a dependent axiom. Without this axiom, the axiom system is indeed independent.

  • Without the dependent axiom, finite independence models exist for all axioms, whereas most of the independence proofs offered in [8] used infinite models. In particular, one of the independence proofs was accomplished by proof-theoretic methods rather than by a independence model; but for this axiom, too, there is a finite independence model;

  • Without its dependent axiom, is not just independent, but completely independent.

For the sake of completeness, we repeat here the definitions of the axiom systems and . Both are based on classical one-sorted first-order logic with identity. Variables are intended to range over points. A single language is used for both axiom systems; the language has three constants, , , and , a ternary relation for collinearity, and a single ternary function symbol for central translations: is the image of under the translation that shifts to . With the the binary operation symbol understood as

(essentially a point reflection), the axioms of are as follows:

A3

B1

B2

B3

B4

B5

B6

B7

(The appearance of without and is not an error. In the the official definition of from [8], rather than duplicating an axiom from André’s axiom system, the names of whose axioms all have the prefix “A”, under a new name, it is simply reused and the “B” axioms are offered.) The axiom system is together with

B8

captures the Fano principle that diagonals of parallelograms are not parallel.

Universal quantifiers will usually be suppressed.

2 A dependent axiom

It was claimed in [8] that is independent. This requires qualification: is indeed an independent axiom of (Proposition 1), but can be proved from with the help of the Fano principle (Proposition 2).

Proposition 1.

.

Proof.

Consider the domain and interpret and according to Table 1. A counterexample to

is provided by .

Triple
Table 1: A model of

For each pair , fails to be a transitive action because is never a value of . ∎

Lemma 1.

Proof.

The desired conclusion follows from and ( is not needed).∎

Lemma 2.

Proof.

This is equation (3) of [8]. It is derived without the help of . (Indeed, suffices).∎

Proposition 2.

.

Proof.

Suppose . From Lemmas 1 and 2, we have

\hb@xt@.01(0)

as well as

\hb@xt@.01(0)

for all , , , and . Thus, by (2), , whence (by Lemma 1 and (2),

for all and . Lemma 2 then gives us the desired conclusion . ∎

In light of Proposition 2, we define:

Definition 1.

.

In the following, the theory in focus is rather than .

3 Small finite independence models

The next several propositions show that every axiom of has a finite independence model. (Incidentally, the cardinalities of the independence models are minimal: when it is claimed that there is a independence model for of cardinality , it is also claimed that is true in every model of of cardinality less than .)

In the independence models that follow we give only the interpretation of the predicate and the function . Strictly speaking, this is not enough, because we need to interpret the constants , , and so that holds. But in the independence models there is always at least one triangle; from any one, an interpretation of the constants , , and can be chosen so that is satisfied.

Proposition 3.

There exists an independence model for of cardinality .

Proof.

Without one cannot prove

the failure of which opens the door to geometrically counterintuitive models. Consider the domain , interpret and as in Table 2. Note that the model is “collinear” in the sense that for any permutation of ; nonetheless, for many pairs of distinct points, the various collinearity statements one can make about and are false. A counterexample to

is .

Triple Triple Triple
Table 2: A model of

Proposition 4.

There exists an independence model for of cardinality .

Proof.

The difficulty here is that

fails without . Consider the domain and the interpretations of and as in Table 3. From the standpoint of , the model is nearly trivialized; the only triples where fails are where and . An example where

fails is .

Triple Triple Triple
Table 3: A model of

Proposition 5.

There exists an independence model for of cardinality .

Proof.

Consider the domain and the interpretations of and are as in Table 4. A counterexample to

in this structure is .

Triple Triple Triple
Table 4: A model of

Proposition 6.

There exists an independence model for of cardinality .

Proof.

is the only axiom that outright asserts that some points are collinear. Thus, if every triple of points constitutes a triangle (that is, holds for all , , and ), then clearly would be falsified. So take a -element structure and interpret so that for the unique triple of the structure is false. The interpretations of , , , and are forced. One can check that all axioms, except , are satisfied. ∎

Pambuccian was unable to find an independence model of . To show that is independent of the other axioms, methods of structural proof analysis [7] were employed. Specifically, an analysis of all possible formal derivations starting from was made, and by a syntactic-combinatorial argument it was found that is underivable from . By the completeness theorem, then, there must exist an independence model. Here is one:

Proposition 7.

There exists an independence model for of cardinality .

Proof.

For lack of space, we omit an explicit description of the table. The model was found by the finite model-finding program Mace4 [5]. ∎

Consideration of is skipped because it is a dependent axiom of (and in any case is not officially an axiom of ).

Proposition 8.

There exists a independence model for of cardinality .

Proof.

Consider the domain and the interpretations of and as in Table 5. A counterexample to

is given by .

Triple Triple Triple
Table 5: A model of

We are unable to improve upon the independence proof for given in [8]: the independence model given there has cardinality .

Proposition 9.

There exists an independence model for of cardinality 2.

Proof.

Indeed, a suitable structure is already available: the countermodel for (over ) also falsifies ; since is not an axiom of , works. A counterexample to

is given by . In this structure, the value of , and hence , is always . ∎

4 Complete independence

The notion of completely independent set was proposed by E. H. Moore [6]. It is a considerably stronger property of an axiom system than the familiar notion of independence.

Definition 2.

An axiom system is said to be completely independent if, for all subsets of , the set is satisfiable.

If an axiom system is completely independent then it is also dependent: for every sentence of , we have that is satisfiable, or (by the completeness theorem), that . When an axiom system is completely independent, no Boolean combination of its axioms can be proved from the other axioms.

Theorem 1.

is completely independent.

Proof.

Since has axioms, by following the definition of complete independence one sees that there are sets of formulas to check for satisfiability. (Such an enumeration of cases is best executed mechanically rather than by hand; we were assisted by the Tipi program [1].) But for all cases, very small finite models can be found with the help of a finite model-finder for first-order classical logic (e.g., Paradox [3]). For lack of space we do not present all the models here. ∎

Theorem 2.

is completely independent.

Proof.

As with , has axioms, so again one has sets of formulas to check for satisfiability. Except for two cases, very small finite models can be found almost immediately. The only cases—both involving —that cannot be immediately dispensed with are:

  1. , and

  2. .

Case (1): The satisfiability of is, by the completeness theorem, the same thing as the independence of . The proof in [8] works. Recall that the smallest independence model for this axiom (27) is much larger than the other independence models (which are all size 3 or less).

Case (2): is satisfiable. Take a model of in which is false. Since fails, there exists points , , , and in such that but . An appropriate model is obtained from by changing ’s interpretation of , , and to, respectively, , , and . ∎

The treatment of case (2) in the preceding proof might be regarded as somewhat odd. The model at work there did contain triangles (that is, it was “nonlinear”), yet it falsified . But if is false, shouldn’t the model be “linear”?

It is worth noting that is just barely completely independent; a seemingly innocent change to one it is axioms destroys complete independence. Consider the existential generalization of

B7

and let be .

Intuitively, says the same thing as . Every model of is, by ignoring the interpretations of , , and , a model of . And every model of can be extended to a model of by choosing, for the interpretation of , , and , any witness to the truth of . Nonetheless, the two theories are subtly different:

Proposition 10.

is independent but not completely independent.

Proof.

The independence proofs for are easily adapted to . As for complete independence, note that is true in every model of . Thus, is unsatisfiable. ∎

In other words, if is false, then there is a triangle, i.e., holds. This illustrates a relationship among the axioms of that the notion of complete independence is intended to rule out.

is quite far from being completely independent. Although cannot be proved from the other axioms, with seven exceptions (see Table 6), if any of ’s axiom is negated, becomes provable; that is, the other Boolean combinations are incompatible with . In a rough sense, then, is “almost” a theorem of .

Table 6: Boolean combinations of axioms of compatible with .

The difference between and is now clear. makes an assertion about three specific (though undetermined) points which are not mentioned anywhere else in the axioms and are thus “semantically inert”. By contrast, is a purely existential sentence that can “interact” with the other axioms (specifically, ).

Similarly, in the foundations of logic, a result similar to Proposition 10 was discovered by Dines [4]: among several axioms, only one (also having the flavor of a minimal-cardinality principle) was an obstacle to complete independence.

Interestingly, the axiom system from which is derived has many dependent axioms. is, moreover, very far from being completely independent. Thus is, from a certain methodological standpoint, to be preferred to .

References

  • [1] J. Alama, Tipi: A TPTP-based theory development environment emphasizing proof analysis, arXiv preprint arXiv:1204.0901 (2012).
  • [2] Johannes André, Über Parallelstrukturen, II: Translationsstrukturen, Mathematische Zeitschrift 76 (1961), no. 1, 155–163.
  • [3] Koen Claessen and Niklas Sörensson, New techniques that improve MACE-style finite model finding, Proceedings of the CADE-19 Workshop: Model Computation-Principles, Algorithms, Applications, 2003, pp. 11–27.
  • [4] L. L. Dines, Complete existential theory of Sheffer’s postulates for Boolean algebras, Bulletin of the American Mathematical Society 21 (1915), no. 4, 183–188.
  • [5] W. McCune, Prover9 and Mace4, http://www.cs.unm.edu/~mccune/prover9/, 2005–2010.
  • [6] Eliakim Hastings Moore, Introduction to a form of general analysis, New Haven Mathematical Colloquium (1910), 1–150.
  • [7] Sara Negri and Jan Von Plato, Structural Proof Theory, Cambridge University Press, 2008.
  • [8] Victor Pambuccian, Two notes on the axiomatics of structures with parallelism, Note di Matematica 20 (2001), no. 2, 93–104.
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