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I asked a poorly received question on physics stack exchange, about the universe.

Today I left the comment:

can we imagine the universe "expanding into nothingness" if we're clear that "nothingness" has no physical properties? i'd have thought one could easily re-plot the universe on another set of axis, with the extra "space" being an empty set

to which the user replied:

no we cannot. The universe is not expanding into anything, not even an empty set

I wanted to check whether a model of something (here the universe, but I'm not asking about physics but the general properties of mathematical space) can have an extra dimension or axis (sorry, I don't know the proper terminology) added to it without changing the content of the model?

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The problem is that you didn't specify what "mathematical space" or "dimension" or "axis" means at all, so there is no answer to your question. For example, I could just map every object $x$ in some model to the pair $(x,0)$, and modify all functions and predicates, and then indeed the resulting collection of objects is isomorphic to the original. So what? That's clearly nothing to do with dimension or axis in the sense of linear algebra. That's the point. If you want a vector space to remain the same under some map, then you reasonably have to preserve its dimension too. If you don't, then what would you mean by "without changing the contents"?

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