My Axiom System

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My Axiom System

• By,
• Steven Dunn

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Primitive Terms Point, Line, On

• Axiom 1 There exists exactly three points
• Axiom 2 There exists at most three lines
• Axiom 3 For every point there exists at least
  one line not containing it.
• Axiom 4 There exists at least two points that
  are on exactly two lines in common
• Theorem 1 There exists exactly three lines
• Theorem 2 There exists at least one line that
  contains at most one point.

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Consistency Models
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Independence Models

• A1 Fails

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Independence Models

• A2 Fails

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Independence Models

• A3 Fails

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Independence Models

• A4 Fails

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Theorem 2

• T2 There exists at least one line that contains
  at most one point.
• Proof Suppose there is no line that contains at
  most one point. Then each line contains at least
  two points. By A4 there exists two points, say P1
  and P2, that are on exactly two lines in common,
  say L1 and L2. By T1 there exists another line
  say L3. Now be A3 there must be a line so P1 is
  not on it and since, by T1, there exists three
  lines that line must be L3. Similarly P2 is not
  on L3. But by assumption L3 must have two points
  on it, say P3 and P4. Which contradicts A1.
  Therefore there exists at least one line that
  contains at most one point.

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Theorem 1

• T1 There exists exactly three lines
• Proof By A2 there exists at most three lines.
  Now we will prove there exists at least three
  lines. By A4 there exists two points, say P1 and
  P2 on two common lines, say L1 and L2. By A3
  there exists a line, say L3 not containing it,
  say P1. Clearly the three lines are distinct.
  Therefore there exist exactly three lines.