math logic and axiom
Consistency Models
Name:
Create a consistency model for the following axiom systems.
1. Axiom System II
The Primitive Terms: element, set, and in
Axiom 1: There exists at least one set.
Axiom 2: For each element, there exists exactly one other element such that no set contains both.
Axiom 3: Each set contains exactly two elements.
Axiom 4: Each element is contained in exactly four sets.
Solution: Let elements by points and sets be lines.
e1
e2
e3
e4
s1
s2
s3s4
s5
s6
s7s8
OR
e1
e2e3
e4e5
e6
s1s2
s3s4
s6s7 s8
s9
s10s12
s5 s11
2. Axiom System III The Primitive Terms: point, line, and on
Consider the following definitions:
Definition 1. Two lines are parallel if they are not on any common point.
Definition 2. Three or more points are called noncollinear if they are not all on one common line.
Consider the axioms:
Axiom 1: Every line is on at least one point.
Axiom 2: There exist at least two points.
Axiom 3: Any two distinct points are on exactly one common line.
Axiom 4: For each line there is at least one point not on it.
Axiom 5: If ` is a line and P is a point not on `, then there exists exactly one line on P parallel to `.
Solution:
p1
p2
p3
p4
l1
l2
l3
l4l5 l6
3. Axiom System IV
The Primitive Terms: player, team, and recruit
Axiom 1: There exists at least one team.
Axiom 2: Each team recruits exactly two players.
Axiom 3: For each team, there is exactly one other team such that no player is recruited by both.
Axiom 4: Each player is recruited by exactly two teams.
Solution: Let players be points and teams be lines
p1
p2
p3
p4
t1
t2
t3
t4
4. Axiom System V
The Primitive Terms: citizen, superhero, protect
Axiom 1: There exists at least one superhero.
Axiom 2: Each superhero protects exactly three citizens.
Axiom 3: Each pair of superheroes protects exactly two citizens in common.
Axiom 4: Each pair of citizens is protected by exactly two common superheroes.
Solution: Let citizens be points and superheroes be lines
c1
c2
c3
c4
s1
s2
s3s1
s3
s4
s4s2
5. Primitive terms: rider, tdragon, fly
Axiom 1: There exists at least one rider.
Axiom 2: Each rider flies on at least two dragons.
Axiom 3: Any pair of riders flies on exactly one common dragon.
Axiom 4: For every dragon, there are exactly three riders that do not fly it.
Solution:
r1
r2r3
r4r5
d1d2
d3d4
d5
d6d7 d8
d9
d10
6. Primitive terms: spaceship, master builder, build.
Axiom 1: There exists at least one master builder.
Axiom 2: Each spaceship is built by exactly three master builders.
Axiom 3: Every master builders builds exactly two spaceships.
Axiom 4: For each pair of spaceships is built by at most one common master builder.
Solution: Let spaceships be points and master builders be lines
s1
s2
s3
s4
m1
m2
m3
m4m5 m6