Solution Axiomatic Systems Exercise 1.1.
"For example," is not a proof.
—Jewish Proverb

Exercise 1.1. Consider the following axiom set.

Postulate 1. There are at least two buildings on campus.
Postulate 2. There is exactly one sidewalk between any two buildings.
Postulate 3. Not all the buildings have the same sidewalk between them.

1. What are the primitive terms in this axiom set?
   The undefined terms are building, campus, sidewalk, and between. Note that building, campus and sidewalk are elements and between is a relation, since it indicates some relationship between sidewalk and building.

2. Deduce the following theorems:
   Theorem 1. There are at least three buildings on campus.
   Proof.  By Postulate 1, there are two buildings, call them b₁ and b₂. Then by Postulate 2, there is exactly one sidewalk between b₁ and b₂, call it S₁₂. Since by Postulate 3 not all buildings have the same sidewalk between them, there must be another building b₃ that does not have S₁₂ between it and b₁ or b₂. Hence, there are at least three buildings on campus.//
   
   Theorem 2. There are at least two sidewalks on campus.
   Proof .  By Postulate 1, there are two buildings, call them b₁ and b₂. Then by Postulate 2, there is exactly one sidewalk between b₁ and b₂, call it S₁₂. Since by Postulate 3 not all buildings have the same sidewalk between them, there must be another building b₃ which does not have S₁₂ between it and b₁ or b₂. This implies with Postulate 2 that there must be a sidewalk between either b₁ and b₃ or b₂ and b₃ which is not S₁₂.//

3. Show by the use of models that it is possible to have
   exactly two sidewalks and three buildings; at least two sidewalks and four buildings; and, exactly three sidewalks and three buildings.
   In each model, the buildings are dots and sidewalks are segments/curves. Between is defined differently in the third model from the first two. In the third model between means the segment connecting the dots; whereas, in the first two models between means the separation of the dots.

              
   Note that all three of the postulates are satisfied for each model. Postulate 1 is satisfied, since each model has three or more points. Postulate 2 is satisfied, since there is exactly one segment/curve between each pair of dots. Postulate 3 is satisfied, since each has two pairs of dots that have a different segment/curve between them.

4. Is the system complete? Explain.
   The system is not complete, since the models in part (c) are not isomorphic.

5. Find two isomorphic models.
      We expand on the third model from part (c).
           A, B, C
          {A, C}, {A, B}, {B, C}
       For the two models, make the following pairings:
           U ↔ A, V ↔ B, T ↔ C, 1 ↔ {A, C}, m ↔ {A, B}, n ↔ {B, C}.

6. Demonstrate the independence of the axioms.
           A model where Postulate 2 and Postulate 3 are true, but Postulate 1 is not true. Consider a model with one building and no sidewalks. The model satisfies both Postulate 2 and Postulate 3 vacuously. But Postulate 1 is not satisfied, since there is only one building and not two as required.
           A model where Postulate 1 and Postulate 3 are true, but Postulate 2 is not true.
   The dots are buildings and curves are sidewalks, and between is defined as a curve connecting two dots. Postulate 1 is satisfied, since there are three dots. Postulate 2 is not satisfied, since the left and right-hand dots have two curves connecting them. Postulate 3 is satisfied, since between was defined as a curve connecting two dots.
           A model where Postulate 1 and Postulate 2 are true, but Postulate 3 is not true.
   Postulate 1 is satisfied, since there are two buildings. Postulate 2 is satisfied, since the two buildings have exactly one sidewalk between them. Postulate 3 is not satisfied, since all the buildings have the same sidewalk between them.

   buildings	sidewalk
   A, B	AB