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I am studying axiomatic systems and I have a hard time understanding how one is supposed to come up with an "abstract" model for an axiomatic system.

I will use the following example taken from here.

An example of axiomatic system

Primitive Terms

  • woozle (noun)
  • dorple (noun)
  • snarf (verb)

Axioms

  1. There exist at least three distinct woozles.
  2. A woozle snarfs a dorple if and only if the dorple snarfs the woozle.
  3. Each pair of distinct woozles snarfs exactly one dorple in common.
  4. There is at least one trio of distinct woozles that snarf no dorple in common.
  5. Each dorple is snarfed by at least two distinct woozles.

Definition of a model

A system obtained by replacing the primitive terms in an axiomatic system with more “concrete” terms in such a way that all the axioms are true statements about the new terms.

An example of model for the axiomatic system

Let the three distinct woozles be the points (0,0), (1,1), and (2,0) in the Cartesian plane. Let dorple now mean line in the plane, and let snarf now mean lies on. Convince yourself that the axioms of the system are all true with this interpretation of the primitive terms.

Question

If our model is based on another axiomatic system such as Euclidean geometry in this example, aren't the the terms still undefined/primitive/uninterpreted whatever you want to call them? I mean if we interpret a an undefined term with another undefined term, then we just end with an undefined term.

Can some explain why and how we can construct a model for axiomatic system A based on axiomatic system B?

Here is an excerpt taken from here that is related to my question:

But we are also able to talk about axiomatic systems, using other axiomatic systems. The language we use to talk about a system is called a metalanguage, and the language we are talking abut is called the object language. In any case, we can use metasyntactic symbols as variables that talk generally about the syntax of the object language. The object language would be an abstract model of the metalanguage.

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  • $\begingroup$ Point and line are objects of the model: cartesian plane. Wwe know them (their coordinate and equations)and we know to rules to manage (compute) them. $\endgroup$
    – Mauro ALLEGRANZA
    Commented Jul 6 at 17:29
  • $\begingroup$ @MauroALLEGRANZA Aren't point and line primitive terms in Euclid's geometry? I have added a small excerpt that might be beneficial for the discussion and possible answers. $\endgroup$
    – ado sar
    Commented Jul 6 at 17:50
  • $\begingroup$ Is there a mathematical question here? If not, you should probably take this to the philosophy site. $\endgroup$
    – Naïm Favier
    Commented Jul 6 at 20:06
  • $\begingroup$ You must produce a collection of woozles and darples and an operation "snarf" such that the woozles and darples satisfy the conditions on the axioms. If your formal education ends at calculus, you may not be ready for this kind of abstraction. This will become easier when you get to your junior level math courses. $\endgroup$
    – John Douma
    Commented Jul 6 at 22:52
  • $\begingroup$ @JohnDouma I am comfortable to find a collection of "woozles" etc. to satisfy the axioms from real world. E.g. "woozles" can be "apples", "organges" or whatever objects we can think of (and satisfy the axioms, same for the relation "snarf"). From Wikipedia: "A model is called concrete if the meanings assigned are objects and relations from the real world, as opposed to an abstract model which is based on other axiomatic systems." The bold part is what I am trying to understand. $\endgroup$
    – ado sar
    Commented Jul 6 at 23:22

1 Answer 1

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If our model is based on another axiomatic system such as Euclidean geometry in this example, aren't the the terms still undefined/primitive/uninterpreted whatever you want to call them?

Yes, taken literally, that is turtles all the way up: but, to make a long story short, the point is we would rather and eventually land at a meta-theory that is not formal but yet it is one we are pretty confident about despite not being formalized, typically because we are confident about it based on our intuition, and/or because it is simpler/clearer, ontologically to begin with, than our object theory. Viceversa, from such an informal meta-theory we get, to begin with, (the intended meaning of) the primitives of a formal theory.

For a slightly broader perspective (and for what it's worth), here is a post of mine where I elaborate on that problem: On the logic of "it"

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