Is this notation standard?

Question

In a book I'm reading, "Sets: An introduction, by M.D.Potter", the symbol $\iota$ is used to mean "a definite", so that:

$\iota ! y(x\in y \iff \Phi(x))=\{x:\Phi(x)\}$

is read as $\{x:\Phi(x)\}$ is a definite unique $y$ such that $x\in y\iff \Phi(x)$, in other words "the unique $Y$ such that...".

Is this notation standard? or is there another symbol more often used?

Answer 1

Yes, it is the standard (but not commonly used in textbooks of mathematical logic) symbol for Definite descriptions.

It was introduced by Bertrand Russell in his fundamental analysis of definite descriptions.

The reading of :

$\iota x \ \phi(x)$

is simply :

"the (unique) object $x$ such that $\phi(x)$", or "the (unique) object such that $\phi$ holds of it."

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Thus : $\iota ! \ y \ (x \in y \Leftrightarrow \Phi(x))$ (but $!$ is redundant) means, as you say :

"the unique $y$ such that $x \in y$ iff $\Phi(x)$ holds".