TITLE: What is the difference between being unique, unique up to isomorphism and unique up to unique isomorphism?
QUESTION [44 upvotes]: Could anyone explain the difference between the above mentioned terms "unique", "unique up to isomorphism" and "unique up to unique isomorphism", preferably with an example? Are there other characterisations of being unique "up to something"?

REPLY [2 votes]: For an arbitrary equivalence relation $\sim$, lets write $(\sim x)$ to mean 'if $x$ exists, it is unique up to $\sim$.' More precisely...
$$(\sim x) Px : \forall xy(Px \wedge Py \rightarrow x \sim y).$$
Then the statement that there exists a unique entity satisfying $P$ can be written
$$\exists xPx \wedge (= x)Px$$

Now write $f \in \mathrm{Iso}(X,Y)$ to mean that $f$ is an isomorphism with domain $X$ codomain $Y$. Further, let $X \sim Y$ denote the statement that there exists an isomorphism $f : X \rightarrow Y.$ More precisely
$$X \sim Y : \exists f(f \in \mathrm{Iso}(X,Y))$$
Since this is an equivalence relation, the statement that there exists a unique structure satisfying $P$, up to isomorphism, can be written 
$$\exists XP(X) \wedge (\sim X)P(X).$$

Now let $X \cong Y$ denote the statement that there exists a unique isomorphism $f : X \rightarrow Y.$ More precisely
$$X \cong Y : \exists f(f \in \mathrm{Iso}(X,Y)) \wedge (= f)(f \in \mathrm{Iso}(X,Y))$$
Since this is also an equivalence relation, thus the statement that there exist unique structures $X$ and $Y$ satisfying $P$, up to unique isomorphism, can be written
$$\exists XP(X) \wedge (\cong X)P(X).$$