Defining an equality

Question

When we define an equality ($=$) of things, for example of vectors in $\mathbb{R}^n$ or of sets in ZF by the Axiom of Extensionality, are there properties that we need to check in order for $=$ to be a "consistent equality notion"? For example, if I defined $v=u$ by $1=-8$ then this would be inconsistent... The Reflexivity axiom for first-order logic with equality would not be verified. So, for example, what do I need to check in order to be sure that the definition $$ (a,b)=(c,d)\iff a=c\text{ and }b=d $$ of equality in $\mathbb{R}^2$ is correct, that is, that it can be taken as an equality in the way we want equality to behave?

Answer 1

The basic laws of equality (or identity) are :

I.1 $∀x \ (x = x)$

I.2 $∀x∀y \ (x = y → y = x)$

I.3 $∀x∀y∀z \ (x = y ∧ y = z → x = z)$

I.4 $∀x_1 \ldots x_n y_1 \ldots y_n \ (x_1 = y_1 \land \ldots \land x_n = y_n → t(x_1, \ldots ,x_n) = t(y_1,\ldots, y_n))$,

$ \ \ \ \ \ ∀x_1 \ldots x_n y_1 \ldots y_n \ (x_1 = y_1 \land \ldots \land x_n = y_n → \varphi(x_1, \ldots ,x_n) \to \varphi(y_1,\ldots, y_n))$.

The first three formulae express : reflexivity, symmetry and transitivity, respectively, of equality.

The last one is a schema that must be "applied" to e.g. any functional symbol of the language.

Considering the language of arithmetic with the function symbol $+$ for sum, in this case I.4 will be :

$∀x_1 x_2 y_1 y_2 \ (x_1 = y_1 \land x_2 = y_2 → (x_1 + x_2) = (y_1 + y_2))$.

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Tho prove the above properties, the following axioms are sufficient [see Herbert Enderton, A Mathematical Introduction to Logic (2nd ed - 2001), page 112] :

1) $x = x$

2) $x = y → (\alpha → \alpha')$, where $\alpha$ is atomic and $\alpha'$ is obtained from $\alpha$ by replacing some (but not necessarily all) occurrences of $x$ by $y$.

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Regarding set theory, if we want to read the Axiom of Extensionality as a definition for $=$ :

$$A=B \leftrightarrow \forall x \ (x \in A \leftrightarrow x \in B)$$

this is not enough.

It is straightforward to verify that the first three identity laws above are satisfied :

I.1. $A=A$ : $x \in A \leftrightarrow x \in A$ : tautology; thus : $\forall x (x \in A \leftrightarrow x \in A)$, by generalization.

I.2. $A=B \to B=A$ : $(x \in A \leftrightarrow x \in B)$; thus : $(x \in B \leftrightarrow x \in A)$, by tautological consequence.

I.3. Exercise.

For the fourth one, we have that in f-o set theory, there is only one (binary) predicate symbol : $\in$ (and no function symbols).

Thus, we have to verify that :

$∀x_1 y_1 z \ (x_1 = y_1 → (z \in x_1) \to (z \in y_1))$

and

$∀x_1 y_1 z \ (x_1 = y_1 → (x_1 \in z) \to (y_1 \in z))$

For the first part, we have that $x_1 = y_1 \leftrightarrow \forall z (z \in x_1 \leftrightarrow z \in y_1)$.

But for the second one, we have to postulate it.