Consistency,Completeness and Geometry

Explicit and Implicit Meaning

What is clapping? Most people associate the act of hitting hands together with the sound that follow, implicitly. But this implicit association of clapping with sound ignores the role of the medium i.e. air. We attribute sound to the act, rather than to the link between the act and the real world.

In the last posts we saw how meaning in case of simple formal systems arises when there is a link between rule-governed symbols and the real world. This is true of language as well, words in themselves are empty sounds, but we often associate meaning implicitly to words, ignoring the isomorphism between them and the real world.

The Modified pq System and Inconsistency

Axiom Schema II : If x is a hyphen string ,then xp%qx is an axiom.

Remember the original interpretation, where the numbers of hyphen strings to the left of p and in between p and q added to the number of hyphen strings to the right of q. We tied the theorems of this original pq system to the process of addition of two digit numbers in the real world.

However, the addition of the second axiom schema allows for the derivation of theorems such as %%p%%q%%%, which could be interpreted as implying 2 + 2 = 3. Thus, our new system contains lots of false statements, thus our original isomorphism between addition and pq system is rendered inconsistent with the external world.

This new pq system is also internally inconsistent as it contains theorems like %p%q%% and %p%q% (1 + 1 = 2 and 1 + 1 = 1). Note that this inconsistency is only with respect to our original isomorphism. It would be silly to think that adding new axioms would not force us to change our interpretation.

Regaining Consistency

To regain consistency in our modified pq system is fairly easy. Just reinterpret the symbol “q” as “is greater than or equal to”.

Now, our "contradictory" theorems -p-q-and -p-q--come out harmlessly as: "1 plus 1 is greater than or equal to 1", and" 1 plus 1 is greater than or equal to 2". We have simultaneously gotten rid of (1) the inconsistency with the external world, and (2) the internal inconsistency. And our new interpretation is a meaningful interpretation; of course the original one is meaningless .

Building on this idea, Hofstader points out the similarity between the meaning change required in the definition of trivial things like points and straight line when we substitute new axiom in place of the fifth axiom of Euclid’s geometry. It turned out that the trivial definition of point is more like our interpretation of the original pq system. Adding new axiom in place of the fifth postulate( that famous one, which cannot be proved using other fours ) required new meaning/ definition. Thus, geometry is more closer to rule governed symbol shunting of formal systems than was previously thought.

Varieties of Consistency

Let’s look at the definition of consistency (together with an interpretation)

External consistency : Every theorem when interpreted becomes a true statement.

Internal consistency : All theorems of a formal system are compatible with each other, one theorem when interpreted doesn’t contradict the other theorem.

example of Internal Inconsistency: let’s consider a formal system which have only three theorems. AbC, CbD, DbA and b stands for beats in the race in our interpretation of the formal system.

Ist theorem : A beats C

IInd theorem : C beats D

IIIrd theorem : D beats A

Obviously, in the interpretation of beating in a race, it is impossible that all three statements are true at the same time.

Hence, the formal system is internally inconsistent under our current interpretation.