Gödel, Escher, Bach An Eternal Golden Braid by Douglas R. Hofstadter

subject", such as "so-and-so". A string with free variables is like a predicate

with "so-and-so" as its subject. For instance,

(SO+SO)=b

is like saying "I plus 1 equals so-and-so". This is a predicate in the variable

b. It expresses a property which the number b might have. If one were to

substitute various numerals for b, one would get a succession of formulas,

most of which would express falsehoods. Here is another example of the

difference between open formulas and sentences:

Vb:Vc:(b+c)=(c+b)

The above formula is a sentence representing, of course, the commutativity

of addition. On the other hand,

Vc:(b+c)=(c+b)

is an open formula, since b is free. It expresses a property which the

unspecified number b might or might not have-namely of commuting

with all numbers c.

Translating Our Sample Sentences

This completes the vocabulary with which we will express all number-

theoretical statements! It takes considerable practice to get the hang of

expressing complicated statements of N in this notation, and conversely of

figuring out the meaning of well-formed formulas. For this reason we

return to the six sample sentences given at the beginning, and work out

their translations into TNT. By the way, don't think that the translations

given below are unique-far from it. There are many-infinitely many-

ways to express each one.

Let us begin with the last one: "6 is even". This we rephrased in terms

of more primitive notions as "There exists a number e such that 2 times e

equals 6". This one is easy:

3e:(SSO' e) =SSSSSSO

Note the necessity of the quantifier; it simply would not do to write

(SSO· e) =SSSSSSO

alone. This string's interpretation is of course neither true nor false; it just

expresses a property which the number e might have.

It is curious that, since we know multiplication is commutative, we

might easily have written

3e:(e 'SSO)=SSSSSSO

instead. Or, knowing that equality is a symmetrical relation, we might have

chosen to write the sides of the equation in the opposite order:

Typographical Number Theory 209