According to BBFSK a relation with the properties of comparativity and reflexivity satisfies symmetry and transitivity. Comparativity is defined as $x\sim{z}\land{y\sim{z}}\implies{x\sim{y}}$. Apparently the proof of this is as simple as $x\sim{x}\land{y\sim{x}}\implies{x\sim{y}},$ which shows symmetry. From symmetry and comparativity, transitivity clearly follows.

Using the sets $$\mathcal{S}\equiv\left\{1,2,3\right\};\mathcal{P}\equiv\left\{\langle2,1\rangle,\langle3,1\rangle,\langle3,2\rangle\right\};\mathcal{Q}\equiv\left\{\langle1,3\rangle,\langle3,1\rangle\right\};$$

and relations

$$F(x,y)\equiv\text{False};L(a,b)\equiv\langle a,b\rangle\notin\mathcal{P};A(a,b)\equiv\langle a,b\rangle\notin\mathcal{Q};$$

it is possible to show that each relation satisfies exactly two of the axioms of an equivalence relation. Each pair of satisfied axioms is different from that of another of the relations. This demonstrates that the axioms are independent, that is orthogonal.

I know that "orthogonality" is something of a metaphor when applied to axioms. Nonetheless, I find it unsettling that a system of three axioms can be shown to be isomorphic to a system of two axioms. Is this an acceptable state of affairs?

  • 3
    $\begingroup$ I'm confused by why you find that "unsettling" or even surprising. (I'm also confused by what "acceptable" is supposed to mean. The equivalence just is, whether you "accept" it or not.) As long as there are multiple perspectives on and ways of analyzing some concept, there will be multiple distinct but equivalent axiomatizations. $\endgroup$ – Derek Elkins Jan 27 at 5:21
  • $\begingroup$ It's a question of the number of "objects" and relations between them. In programing it is often possible to simplify the first hack down the a bare minimum of code, but once that is reached, any further "simplification" means moving the complexity somewhere else. We might rely on the OS to do the work for us; or use a library call, overload an operator, etc. But there will still be some minimum number of independent variables and instructions needed to accomplish a task. That is based on experience, not on theory. But I'll bet there is theory to back up my assertion. $\endgroup$ – Steven Hatton Jan 27 at 19:27

In fact, it's equivalent to a system with one axiom, since we can always take the conjunction of any finite number of sentences.

Of course, you may object that doing so changes the form of the axioms in some way, and there are interesting questions to be asked here. For example, in the context of general algebras (= structures in a purely functional signature), we can ask $(i)$ whether a class of structures can be axiomatized by equations (this has a complete answer, essentially) and $(ii)$ if so, how many equations are needed (e.g. for the class of groups). But this is a side issue; the situation is completely acceptable, some axiomatizations are different from others.

Separately, it's worth saying that your statement

"orthogonality" is something of a metaphor when applied to axioms

is quite true, and is why the word "orthogonality" really shouldn't be used here: I suspect your worry comes from the sense that somehow we have an object which is both three-dimensional and two-dimensional, but this is only due to a misrepresentation of a logical phenomenon as a geometric one.

  • $\begingroup$ I think my misgiving derives from the same intuition that leads one to speak of orthogonality of axioms. Having a system completely determined by two independent rules which is equivalent to a system completely determined by three independent rules make me suspect that the ideas haven't been completely atomized. I know there are other places where this happens. In the theory of groups the axioms of identity and inverse are redundant, but they may be replaced by left and right division. It just bothers me. $\endgroup$ – Steven Hatton Jan 27 at 5:58
  • $\begingroup$ The more interesting question than how few "orthogonal" axioms can we use to characterize a system may be: what is the maximum number of independent axioms possible? $\endgroup$ – Steven Hatton Jan 27 at 9:30
  • $\begingroup$ @StevenHatton That also doesn't work properly: consider the single sentence $p$ versus the pair of sentences $p\vee q, p\vee\neg q$ for $q$ "sufficiently independent of" $p$. You need to do more work to get an interesting notion of "number of axioms." Finitely axiomatizable versus non-finitely axiomatizable works, as does smallest number of axioms after restricting to a sufficiently narrow syntactic class of sentences. $\endgroup$ – Noah Schweber Jan 27 at 18:50
  • $\begingroup$ I need to step away from the crack pipe. I love this stuff, but it is distracting me from other goals. Please see my comment in response to Derek Elkins for a sketch of how I'm thinking about this matter. $\endgroup$ – Steven Hatton Jan 27 at 19:35

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