Kinds of uniqueness
Definitions
Say we define a value xx by giving a condition ϕ\phi that must hold on it. Then we say:
xx is unique if the condition ϕ\phi determines at most one value. Mathematically, we consider xx to be unique if for all a,ba, b satisfying ϕ\phi we have that a=ba = b
xx is unique up to isomorphism if all values satisfying ϕ\phi are isomorphic to each other Likewise, this is expressed as ϕ(a)ϕ(b)    ab\phi(a) \land \phi(b) \implies a \cong b
Generally for any equivalence relation \sim we say that xx is unique up to \sim if all values satisfying ϕ\phi are related by \sim Expressed as ϕ(a)ϕ(b)    ab\phi(a) \land \phi(b) \implies a \sim b
Examples
Define xx to be the value xRx \in \mathbb R such that x2=2x^2 = 2. Then xx is not unique, since both 2\sqrt 2 and 2-\sqrt 2 satisfy this condition.
Define xx to be the value xRx \in \mathbb R such that x>0x > 0 and x2=2x^2 = 2. Then xx is unique and is given by x=2x = \sqrt 2
Define xx to be the 0-element set. Then xx is unique and is given by x=x = \varnothing.
Define xx to be the 1-element set. Then xx is not unique but is unique up to isomorphism, since all 1-element sets are isomorphic (wrt the category Set\textbf{Set})



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