The Tortoise always beats Zeno at chess.
Zeno always beats Egbert at chess.
Egbert always beats the Tortoise at chess.The statements are not incompatible, although they describe a rather
bizarre circle of chess players. Hence, under this interpretation, the formal
system in which those three strings are theorems is internally consistent,
although, in point of fact, none of the three statements is true! Internal
consistency does not require all theorems to come out true, but merely that
they come out compatible with one another.
Now suppose instead that x by is to be interpreted as "x was invented
by y". Then we would have:The Tortoise was invented by Zeno.
Zeno was invented by Egbert.
Egbert was invented by the Tortoise.In this case, it doesn't matter whether the individual statements are true or
false-and perhaps there is no way to know which ones are true, and which
are not. What is nevertheless certain is that not all three can be true at once.
Thus, the interpretation makes the system internally inconsistent. This
internal inconsistency depends not on the interpretations of the three
capital letters, but only on that of b, and on the fact that the three capitals
are cyclically permuted around the occurrences of b. Thus, one can have
internal inconsistency without having interpreted all of the symbols of the
formal system. (In this case it sufficed to interpret a single symbol.) By the
time sufficiently many symbols have been given interpretations, it may be
clear that there is no way that the rest of them can be interpreted so that all
theorems will come out true. But it is not just a question of truth-it is a
question of possibility. All three theorems would come out false if the
capitals were interpreted as the names of real people-but that is not why
we would call the system internally inconsistent; our grounds for doing so
would be the circularity, combined with the interpretation of the letter b.
(By the way, you'll find more on this "authorship triangle" in Chapter XX.)Hypothetical Worlds and Consistency
We have given two ways of looking at consistency: the first says that a
system-plus-interpretation is consistent with the external world if every theo-
rem comes out true when interpreted; the second says that a system-plus-
interpretation is internally consistent if all theorems come out mutually compat-
ible when interpreted. Now there is a close relationship between these two
types of consistency. In order to determine whether several statements are
mutually compatible, you try to imagine a world in which all of them could
be simultaneously true. Therefore, internal consistency depends upon
consistency with the external world-only now, "the external world" is
allowed to be any imaginable world, instead of the one we live in. But this is
Consistency, Completeness, and Geometry^95