Comments on  Models for Axiomatic Systems. 
Revised 29jan13.



\begin{document}
\maketitle
\textbf{ $\C$  2010, Prof. George K. Francis, Mathematics Department, University of Illinois} 

\section{Introduction}

Section 1.5 of Hvidsten is too \textit{soft} for our purposes. We need to be more precision.
Aside from greater precision, this lesson complements the textbook and also the printed notes 
\textit{Axiomatic Systems for Geometry}. But that document covers more material than we 
consider at this time. For present purposes, you should learn this lesson. It 
supplements lesson A3 and continues lesson A4.

\section{Consistency}

We need to be more precise. By  
\begin{itemize}
\item \textit{Primitives:} we mean the so called undefined terms. \\
\item \textit{Propositions:} we mean complete sentences about the primitives.  \\
\item \textit{Axioms:} we mean propositions we accept without proof. \\ 
\item \textit{Theorems:} we mean propositions that follow logically from the axioms and 
previously proved theorems. 
\item \textit{Interpretation:} we mean that the primitives are identified with objects 
 and relations in a familiar, and trusted mathematical context. \\
\item \textit{Model:} we mean an interpretation of the primitives in which the axioms, 
and therefore all theorems are true. \\ 
\item \textit{System of Axioms:} we mean a set of propositions about the primitives.\\
\item \textit{Axiomatic System:} we mean a system of axioms together with all of its theorems.\\ 
\end{itemize}

These concepts were treated in earlier lessons. Next, there are more concepts we need to understand:
\begin{itemize}
\item \textit{Categorical}: An axiom system is categorical if all models are isomorphic. But not a more practical definition below.
\item \textit{Isomorphic}: Two models are isomorphic if there is a bijection between the primitives, and corresponding objects have corresponding relations.  
\item \textit{Consistent}: An axiomatic system is consistent if no two of its theorems are contradictory. But note a more practical definition below. 
\item \textit{Independent}: A proposition (sentence about primitives) is independent from an axiom system if it cannot be proved using the axioms and other theorems in the system. But note a more practical definition below. 
\end{itemize}

Now, as stated above, some of these definitions are not very practical, because
they would seem to require that we test \textbf{all} possible cases. How can we test all possible pairs of models to see if they are isomorphic? And could be check every pair of possible theorems in the system for being non-contradictory?
Finally, how can we possibly check every possible proof in the system to see 
that some axiom is independent of the others? 


Fortunately, logicianas have devised methods using models, which yield more
practical criteria to check. We have already seen in an earlier lesson how
it can be shown that an axiom system is or is not categorical. Namely, if 
a particular interpretation of the primitives forces a model to be a certain 
way, leaving no alternatives, then it must be categorical. Independence 
requires its own discussion.

\section{... and Independence}

Let's reconsider the triangle system, written in good logical form. Note
that now we recognize that T4 is the \texbf{dual} of T2, because you get 
T4 from T2 by exchanging the role of Points and Lines with each other.

\begin{eqnarray*} 
T1 & \{P_1,P_2,P_3\} & \mbox{ There are exactly 3 points } \\
T2 & (\forall A \ne B)(\exists ! \ k)(Ak \and Bk) & \mbox{Euclid's 2nd postulate} \\
T3 & (\forall k)(\exists A)( \neg Ak)  & \mbox{ Dimension > 1 } \\
T4 & (\forall k \ne \ell)(\exists ! \ A)(Ak \and A\ell) & \mbox{Dual of T2} \\
\end{eqnarray*}

We now make a notational definition, and write $ (PQ) $ for that unique line through 
points $P$ and $Q$ mentioned in $T2$. Next consider the proposition which says that
every line has exactly two points on it.

\begin{eqnarray*} 
T5 & (\forall h)(\exists ! \ A \ne B)(h=(AB)) \\
\end{eqnarray*}

This says that every Line has exactly 2 Points on it.
This is certainly true for the model we discovered for T1-4 earlier. Recall that is 
consists in a triangle, whose vertices at the Points, an whose sides are the 
Lines in the interpretation. 

Here is a logical deduction written out in the customary style. See Hvidsten,
for example.

\textit{Proof ot T5:} By T2 we may label 
\begin{eqnarray*} 
 \ell_1 = (P_2 P_3) & \ell_2 = (P_3 P_1) &  \ell_3 = (P_1 P_2) \\
\end{eqnarray*}
By T3 the third point is not on a line, hence T5 holds. 

It is certainly easier to just look at the model to conclude that T5 holds than to understand
the two line argument above.  You you may have to amplify amplify it before 
it makes sense to you.

Hvidsten points out that T5 is not independent of the previous four. He observes that
it it impossible to create a model of T1-4 in which T5 is false. 
If there were a Line with 3 Points on it in this geometry, which axiom would 
require you to violate which other axiom?

What about T4, is it independent of the axiom system $\{T1,T2,T3\}$ ? 
To make this a tractable problem we adopt the following two meta theorems from mathematical logic.

\section{Meta Theorems}

A meta-theorem concerns how to handle axioms systems in general. It is not a 
proposition inside an axiom system which follows from its axioms.

\textbf{Meta Theorem 1.} \textit{ An axiom system has a model then it is 
consistent.} 

Strictly speaking, it is as consistent as the mathematics it is modelled in.
But since we accept all of Euclidean geometry as you know it as true,  
we won't worry about such issues as "equiconsistency" in this course. 

\textbf{Meta Theorem 2.} \textit{ A proposition about the primitives which is
a theorem in an axiomatic system, is also true for every model of it.}

Recall that a model is an interpretation of the primitives of an axiomatic
system for which the axioms are true. Since the model resides in a geometry
we trust to be logical, a logical consequence of the axioms has to be true
of the model as well. The converse of this meta-theorem is our most powerful
tool for checking consistency of an axiom system. 

\textbf{Meta Theorem 3.} \textit{ A proposition about the primitives which 
is true in a model, is a theorem in an axiomatic system, provided the 
axiomatic system is categorical. 

If we did not accept this meta theorem we would have to prove them, 
and this course would 
turn into a logic course rather than a geometry course. 

If an axiom system is not categorical, then it would have two different models.
There would be a statement about one of the models which is not true for the
other. This statement cannot be a theorem in the axiomatic system, i.e. it
cannot be derived logically from the axioms. In particular, such an an 
axiomatic system permits independent propositions. So we have a better
and surely more practical definition of independence.

\textbf{Meta Theorem 4.} \textit{ A proposition is independent of 
a given axiom system if there 
are two different models for the axiom system, one in which the proposition is true, and 
another in which it is false.} 

\section{Application to the Triangle Geometry }
For example, here is different take on the triangle axiom system we studied
in the previous lesson. The triangle with its vertices the Points and edges the Lines, is obviously a model for T123, the system with only axioms, T1,T2 and T3. 
But it is not the only model,
since the first 3 axioms do \textbf{not} forbid adding any number of Lines 
that just don't happen to have any Points on them. 
If there is even one such Line, then $T4$ is false because $T4$ 
says that every pair of Lines intersect in a common Point. 
If there are no such extra Lines, then $T4$ is true. 
So $T4$ is independent of the other 3 triangle geometry axioms because
we have two different models, one in which T1, T2, T3 and T4 are true,
and another in which T1, T2, T3, and notT4 are true. We say that 
T4 is independent of T123. This means that in T123, T4 is neither true
not false!

\textbf{Preview:} This will be an important distinction between the three
geometries we study in the next section: Absolute Geometry, Euclidean Geometry
and non-Euclidean Geometry.

\section{Some Exercises to check that you understood.}

Here is a collection of exercises you might try to see that you have understood
this section. Put any solutions you can find into your Journal for future
reference. 

\subsection{Practice Problem}
To show that you understand this, convince yourself that each of the four triangle geometry
axioms is independent of the other three. There are four cases of a geometry
and an independent proposition:  

(T123, T4), (T234, T1),  (T341, T2),  (T412, T3). 

To save a little time, you can use the triangle model in all four cases. Why?

You'll need three three models different from the triangle to prove the 
3 remaining independencis.  For example, if you remove one of the Lines from
the triangle model, then which three axioms still hold? Which fails? Which
case have you just settled?

This leaves you with only two cases. Go for it!

Can you find a different model for T134 in which T2 is false. Hint: add
an second Line connection two of the Points.


\subsection{Challenge Problem} 
To see whether you are really reaching full steam
on this material, show that each axiom of Pythagoras' "geometry of five" (the
Icecream Geometry) is independent of the others (or maybe not?). Ditto for the 
tetrahedral geometry you've worked on in the the homework.
Since each of these has 3 axioms, you have a total of 6 theorems to prove, and 
8 models to compare.  With this many repetitive arguments, you might develop a 
language for your strategy of proving the two additional cases efficiently.

Put whatever parts of this challenge problem you solve into your Journal for 
future reference. There will be an opportunity to submit this work for 
substitute credit if you need it. 
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