The Possibility of Multiple Interpretations
A full formalization of geometry would take the drastic step of making every
term undefined-that is, turning every term into a "meaningless" symbol of
a formal system. I put quotes around "meaningless" because, as you know,
the symbols automatically pick up passive meanings in accordance with the
theorems they occur in. It is another question, though, whether people
discover those meanings, for to do so requires finding a set of concepts
which can be linked by an isomorphism to the symbols in the formal system.
If one begins with the aim of formalizing geometry, presumably one has an
intended interpretation for each symbol, so that the passive meanings are
built into the system. That is what I did for p and q when I first created the
pq-system.
But thete may be other passive meanings which are potentially percep-
tible, which no one has yet noticed. For instance, there were the surprise
interpretations of p as "equals" and q as "taken from", in the original
pq-system. Although this is rather a trivial example, it contains the essence
of the idea that symbols may have many meaningful interpretations-it is
up to the observer to look for them.
We can summarize our observations so far in terms of the word
"consistency". We began our discussion by manufacturing what appeared
to be an inconsistent formal system-one which was internally inconsistent,
as well as inconsistent with the external world. But a moment later we took
it all back, when we realized our error: that we had chosen unfortunate
interpretations for the symbols. By changing the interpretations, we re-
gained consistency! It now becomes clear that consistency is not a property of a
formal system per se, but depends on the Interpretation which is proposed for it. By
the same token, inconsistency is not an intrinsic property of any formal
system.
Varieties of Consistency
We have been speaking of "consistency" and "inconsistency" all along,
without defining them. We have just relied on good old everyday notions.
But now let us say exactly what is meant by consistency of a formal system
(together with an interpretation): that every theorem, when interpreted,
becomes a true statement. And we will say that inconsistency occurs when
there is at least one false statement among the interpreted theorems.
This definition appears to be talking about inconsistency with the
external world-what about internal inconsistencies? Presumably, a system
would be internally inconsistent if it contained two or more theorems
whose interpretations were incompatible with one another, and internally
consistent if all interpreted theorems were compatible with one another.
Consider, for example, a formal system which has only the following three
theorems: TbZ, ZbE, and EbT. If T is interpreted as "the Tortoise", Z as
"Zeno", E as "Egbert", and x by as "x beats y in chess always", then we have
the following interpreted theorems:94 Consistency, Completeness, and Geometry