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There is an informal notion of "identifying" two mathematical objects that I have run into several times, and I'm am wondering how to formally express this idea. A case of this I ran into long ago was "identifying" a finite dimensional vector space with its double dual space. It is well known that these spaces are canonically isomorphic, but what does it mean exactly (preferably in set theoretic language) to "identify" the two? Or is this inherently an informal idea, applied to shorten notation?

Another case that I recently ran into comes from non-standard analysis. Here, after constructing a non-standard extension $^*X$ of some set $X$ (e.g. the reals $\mathbb{R}$), we then "identify" the original set $X$ with a subset of $^*X$. As in the above case, the idea is to consider two objects that are a priori different to be the same.

Asaf Karagila
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Yly
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  • In both of your examples it simply means that instead of futzing about with an isomorphism, we’ll pretend that the set $X$ is its isomorphic copy. In general it simplifies the notation and does no harm. If you like, you can think of it as replacing the isomorphic copy by the original and making all of the necessary corresponding changes to the larger structure. – Brian M. Scott Jun 16 '16 at 05:51

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In this context, identifying two objects $A$ and $B$ is not an operation within the base formal system, but a meta-mathematical convention. It says: Whatever you do or formulate with or about either $A$ or $B$ can be mechanically converted into analogous and equivalent statements about $B$ resp. $A$ by inserting the formal objects through which you have seen $A$ and $B$ to be 'essentially the same' (isomorphisms).

One can consider the need of doing this as a deficiency of the base formal system, as the notion of equality that it provides is apparently too fine-grained. You might be interested in looking at Homotopy Type Theory, which through the Univalence Axiom enforces the equivalence of logical equality and structural isomorphism.

Hanno
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