Kinds of uniqueness

Definitions

Say we define a value $x$ by giving a condition $\phi$ that must hold on it. Then we say:
$x$ is unique if the condition $\phi$ determines at most one value.
Mathematically, we consider $x$ to be unique if for all $a, b$ satisfying $\phi$ we have that $a = b$

$x$ is unique up to isomorphism if all values satisfying $\phi$ are isomorphic to each other
Likewise, this is expressed as $\phi(a) \land \phi(b) \implies a \cong b$

Generally for any equivalence relation $\sim$ we say that $x$ is unique up to $\sim$ if all values satisfying $\phi$ are related by $\sim$
Expressed as $\phi(a) \land \phi(b) \implies a \sim b$

Examples

Define $x$ to be the value $x \in \mathbb R$ such that $x^2 = 2$. Then $x$ is

*not*unique, since both $\sqrt 2$ and $-\sqrt 2$ satisfy this condition.Define $x$ to be the value $x \in \mathbb R$ such that $x > 0$ and $x^2 = 2$. Then $x$

*is*unique and is given by $x = \sqrt 2$Define $x$ to be the 1-element set. Then $x$ is

*not*unique but is unique up to isomorphism, since all 1-element sets are isomorphic (wrt the category $\textbf{Set}$)Referenced by:

- 59418182846446194661r
- 76251275596394291247r
- F2
- cardinality
- cauchys-functional-equation
- closures-and-completions
- cycle-decomposition
- functional-track
- gcd
- group
- hausdorff
- induction-recursion
- injective-surjective-bijective
- multiplicative-groups-of-integers
- partial-ordering
- propositional-logic
- structural-strength
- terminal-object