Landau Foundations of Analysis Axiom 4: Is it necessary?

Question

Landau gives 5 axioms as the foundations for deriving the theorems in the first chapter:

• Axiom 1: 1 is a natural number.
• Axiom 2: If $x = y$ then $x' = y'$.
• Axiom 3: 1 is not a successor to any number.
• Axiom 4: If $x' = y'$ then $x = y$.
• Axiom 5 is the axiom of induction.

I do not understand why Axiom 4 is necessary. It seems to me that it can be derived from Axiom 2:

Suppose $x' = y' \Rightarrow x \neq y$. Then the contrapositive of this statement is $x = y \Rightarrow x' \neq y'$, which contradicts Axiom 2. Hence we obtain a contradiction, and it must be the case that $x' = y' \Rightarrow x = y$.

Am I missing something here? I mean there must be something wrong in the above reasoning, otherwise why would Landau list it as an axiom rather than a theorem?

Answer 1

Yes, you missed something: you incorrectly negated the statement "$x' = y'$ IMPLIES $x=y$".

Let's consider a general implication "P IMPLIES Q".

The method you used to negate that implication would be "P IMPLIES (NOT Q)". But that is incorrect.

Instead, its negation is "P AND (NOT Q)".

So the negation of (4) is "$x'=y'$ and $x \ne y$".

Answer 2

Are you sure the negation of the axiom is $x' = y' \implies x \ne y$ , and not, $x' = y'$ does not imply $x = y$ ?

Answer 3

To see why axiom 4 is not implied by the other axioms, consider this model of the theory with only axioms $1,2,3,5$: $$ \text{set of all natural numbers} = \{1,2\}\\ 1' = 2,\\ 2'=2. $$ This clearly satisfies axioms $1,2,3$ and $5$ [check this]. However, it does not satisfy axiom $4$, since $1'=2'$, but $1\neq 2$. Therefore, there can be no way to prove axiom $4$ from the others.