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An axiomatic system is:

  1. consistent: if no logical contradiction can be derived from the axioms. (Don't see how you can prove there is no logical contradiction possible.)
  2. independent: if an axiom cannot be proved or disproved from the other axioms. (Not sure how this itself can be proved).
  3. complete: "if every statement expressible in the terms of the system is either provable or has a provable negation". (Don't see how you can prove either way this is true or not).
  4. categorical: if any two models of the system are isomorphic (essentially, there is only one model for the system). (Don't see how you can prove this either).

For all of these, I don't see how you can prove that there are no negative cases. By that I mean, like for (4) there are no alternative models, or for (1), there are no logical contradictions.

It seems for all of this you would need to iterate through every possible (infinite) combination of things to see if it was a negative case. So I don't see how.

Please show me how I could prove something fits these 4 properties (just at a high level, don't need to go into a formal proof unless it's helpful). I am looking at Foundations of Geometry, and wondering how they know the geometry axioms in any of the systems (Hilbert, Euclid, Birkhoff, etc.) for sure are consistent, independent, complete, and categorical, or that they are not. Basically I would like to know a methodology for:

  • How to look at some axiom in a math book and determine, with a proof, the logical values the axioms take for these 4 properties.

That is:

| consistent | independent | complete | categorical |
|------------|-------------|----------|-------------|
| true       | true        | true     | true        |
| true       | true        | true     | false       |
| true       | true        | false    | true        |
| true       | false       | true     | true        |
| false      | true        | true     | true        |
| true       | true        | false    | false       |
...

Basically how to determine which row it is in this set of combinations.

Don't need to know specifically, as that would require knowing which axiom I am considering, I just would like to know the general methodology to apply to any axiom.

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