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Is a definition either intensional or extensional?

  1. Can a definition be neither?
  2. Can a definition be both?

    How about this definition? when there is only one object that satisfies a definition, e.g.

    define $A:= 1$

    I think this definition of $A$ is both intensional and extensional.

    Intensional because being $1$ is a necessary and sufficient condition for something being $A$.

    an intensional definition gives the meaning of a term by specifying all the properties required to come to that definition, that is, the necessary and sufficient conditions for belonging to the set being defined.

    Here we have "the set" being singleton i.e. $\{1\}$.

    Extensional, because we list all objects that satisfies the definition, and there is only one object $1$.

    An extensional definition of a concept or term formulates its meaning by specifying its extension, that is, every object that falls under the definition of the concept or term in question.

Thanks.

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You want "intensional." –  Qiaochu Yuan Jul 19 at 0:39
    
That definition of $A$ is extensional, by definition. –  Git Gud Jul 19 at 0:42
    
Is it intensional? @Git –  Tim Jul 19 at 0:43
    
@Tim No, an intentional definition would be $A:=\{x\in \mathbb N\colon 1\leq x\leq 3\}$. –  Git Gud Jul 19 at 0:45
    
The issue of whether a definition is intensional or extensional is not cleanly tied to whether there is only one object that satisfies the definition. In many cases we would complain that a definition was incomplete if more than one object satisfied the formulation. Here many Readers would understand your definition of $A$ to be extensional since it depends on the relation of $A$ to other objects ($1,2,3$ as members, everything else as nonmembers). –  hardmath Jul 19 at 0:45

1 Answer 1

Extensional: $A:=\{1\}$

Intensional: $A:=\{x\in\mathbb{R}:x=1\}$

This is due to the agreement in naive (and probably also in ZF) set theory that sets can be specified in either listing its elements or specifying some rules (in the context of a universe) - so precisely by giving an extensional or intensional definition.

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