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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?

Nethesis
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  • I'm not familiar with this notation ($\iota$ seems to be used as the imaginary unit by some to avoid confusion with index $i$). Here $\iota!$ seems to introduce an universal property. Maybe you'll find some notation related to that term. – AlexR Mar 04 '15 at 21:06
  • It's meant to mean "the unique" – Nethesis Mar 04 '15 at 21:15
  • Universal property is the same in fancier ("the object uniquely defined by $\ldots$ up to some isomorphy") – AlexR Mar 04 '15 at 21:21
  • Possibly related? http://math.stackexchange.com/questions/852771/element-of-a-singleton-set-with-one-element-notation – DanielV Mar 10 '15 at 10:11
  • It would be quite handy if it were more standard, because it sometimes replaces verbiage with tidy calculation. For example, if $f: A \to B$ is a surjection, and $g: A \to C$ is any function such that $f(x) = f(x')$ implies $g(x) = g(x')$, one can write $h(y) = \iota(g(f^{-1}(y))$ as an explicit formula for the unique function $h: B \to C$ such that $g = h \circ f$, and prove the latter equation by a tidy calculation involving the identity $\iota{g(x)} = g(x)$. – Calum Gilhooley Mar 10 '15 at 10:14

1 Answers1

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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."

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".

Mauro ALLEGRANZA
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