All of elementary arithmetic can be expressed in this language, but sometimes a bit of ingenuity is required. We can, for example, compensate for the lack of subtraction and division signs. \({ x = z - y }\) can be expressed as \({ [ ( x + y ) = z ] }\). The second statement implies \({ ( y ≤ z ) }\), without which the first wouldn’t make sense. Similarly, \({ x = z / y }\) can be expressed as \({ [ ( x · y ) = z ] }\).
Knowing that statements about division can be expressed via statements about multiplication we can see how to express divisibility. The condition that \({ z }\) is divisible by \({ x }\), i.e. that \({ x }\) is a divisor of \({ z }\), for example, can be expressed as \({ \{ ∃ y . [ ( x · y ) = z ] \} }\).
We can also express primality. The following sentence is one way of saying that \({ z }\) is prime: \[ \{ [ ∀ x . ( ∀ y . \{ [ ( x · y ) = z ] ⊃ [ ( x = z ) ∨ ( y = z ) ] \} ) ] ∧ ( 0 ' < z ) \} . \] As with all statements, this one is best understood by breaking it into smaller phrases. Starting with \[ [ ∀ x . ( ∀ y . \{ [ ( x · y ) = z ] ⊃ [ ( x = z ) ∨ ( y = z ) ] \} ) ] \] we can peel off the universal quantifiers and and ask when \[ \{ [ ( x · y ) = z ] ⊃ [ ( x = z ) ∨ ( y = z ) ] \} \] is true. This means if \({ [ ( x · y ) = z ] }\), i.e. if \({ z }\) is the product of \({ x }\) and \({ y }\), then \({ [ ( x = z ) ∨ ( y = z ) ] }\), i.e. at least one of \({ x }\) or \({ y }\) is equal to \({ z }\). Since the only way to write a prime as a product of natural numbers is 1 times itself, in either order, the statement \[ [ ∀ x . ( ∀ y . \{ [ ( x · y ) = z ] ⊃ [ ( x = z ) ∨ ( y = z ) ] \} ) ] \] is true whenever \({ z }\) is prime. There are two non-prime values of \({ z }\) for which the statement above is true though, \({ 0 }\) and \({ 1 }\). To exclude these we add the additional condition \[ ( 0 '' ≤ z ) , \] which ensures that \({ z }\) is greater than 1.
We can express even more complicated thoughts. We can say, for example, that there are infinitely many primes. It’s not immediately obvious how to do this. We’ve just seen how to express the fact that any particular number is prime but how can we make a statement about infinitely many numbers in language which doesn’t have a notation for sets or infinity? There is a standard trick for this. To say that there are infinitely many primes we say that for every number \({ w }\) there is a prime number \({ z }\) greater than \({ w }\). In our language this is \[ \{ ∀ w . [ ∃ z . ( \{ w < z \} ∨ \{ [ ∀ x . ( ∀ y . \{ [ ( x · y ) = z ] ⊃ [ ( x = z ) ∨ ( y = z ) ] \} ) ] ∧ ( 0 ' < z ) \} ) ] \} . \]
As you can see, our expressions are starting to get unwieldy. It’s helpful to introduce a notational convention. I’ll use capital letters to refer to substitution instances of particular expressions and follow those letters with the particular expressions to be substituted, e.g. I’ll write \[ D ( x , z ) ≡ \{ ∃ y . [ ( x · y ) = z ] \} \] to mean that \({ D }\) followed by an open parenthesis, then a numerical expression, then a comma, then another numerical expression, represents the expression obtain by substituting the first numerical expression for the \({ x }\) and the second numerical expression for \({ z }\) in the Boolean expression \({ \{ ∃ y . [ ( x · y ) = z ] \} }\). Although this is the literal meaning of the notation, once we understand the expression defining \({ D }\) we think of \({ D ( x , z ) }\) simply as “\({ x }\) divides \({ z }\)”. Note that neither \({ D }\) nor \({ ≡ }\) is part of our language for elementary arithmetic. These are part of a notation for talking about the language of elementary arithmetic. Similarly, we could introduce the shorthand \[ P ( z ) ≡ \{ [ ∀ x . ( ∀ y . \{ [ ( x · y ) = z ] ⊃ [ ( x = z ) ∨ ( y = z ) ] \} ) ] ∧ ( 0 ' < z ) \} \] to express primality and then write the statement that there are infinitely many primes as \[ \{ ∀ w . [ ∃ z . ( \{ w < z \} ∨ P ( z ) ) ] \} . \] This isn’t part of the language of elementary arithmetic, since \({ P }\) isn’t part of that language, but it uniquely identifies a statement which is part of the language, namely the one given previously to express the same meaning.