# Is analysis a “complete theory”, i.e., does it end somewhere?

As in, up to structure types...

For example, in algebra, we start with groups (most basic), then rings, then vector spaces, then inner product spaces, then Normed Linear Spaces, and so on, with the main motivation behind these structures is that the interactions in different structures/frameworks are different and also sometimes useful in other areas of interest.

Up to isomorphisms or structural equivalence ideas, are there infinite or finite number of structures?

It would either seem that either: you are now not sure about the question and it is a bit unclear or alternatively the answer is "obviously there are infinite structures".

I guess that ultimately the question I am getting at is: is it possible for a human to know all the possible structure types(and have some intuition for each one?) Does it end? Is there going to ever be a point where someone finally finishes a book on analysis (for example) and says "This is the end. The theory is complete. There isn't anything else which is useful to consider?"

We are in the realm of either algebra or analysis here (you can think of the two realms separately).

It is also possible that this question makes no sense because my intuition of maths is wrong. However, to me it seems intuitively like a very human and very curious question to ask. If you look at complicated advanced analysis book with loads of symbols... and then an even more complicated, more advanced analysis book, I can't help but wonder if the whole theory ends somewhere...

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You should probably clarify what you mean by 'structure types', because that has an interpretation in the context of species theory. I'm inclined to think that one can't really achieve the sort of closure you allude to -- there are scads of functions and some of them are special cases of each other, so I'm not sure something like the Enormous theorem (of group theory) would be useful for them. – deoxygerbe Apr 27 '12 at 0:32
Surely some analog to Euclid's proof is available here - some sort of "multiply all the structure types, and add 1". – Gerry Myerson Apr 27 '12 at 1:11
Is "group, ring, vector space, ..." supposed to be a list of examples of structure types? This could be made much clearer ("structure type" has a precise meaning in mathematics as deoxygerbe says). In that case the answer is that there are infinitely many natural "structure types" (take $R$-modules for every ring $R$) but I don't really know what this has to do with analysis being a "complete theory" (if you want some kind of "analytic structures" then take continuous representations of topological groups $G$ on Banach spaces for every $G$). – Qiaochu Yuan Apr 27 '12 at 1:19
One specific question can be completely answered. But the collection of all questions can never be completely answered. Thus I don't understand what you're asking- what do you mean by a "structure type" and what is your collection of questions? – Daniel Moskovich Apr 27 '12 at 5:09
I had not even heard of species theory before. Maybe this is what I am after... I have no idea. Qiaochu - Ah, I see there are lots of different types of mathematical structures, each grouped on its own. E.g. there are algebraic structures (groups, rings,...), categories, topologies, etc. I thought these were all linked but maybe they are not (are they??) Is there a complete theory in category theory for example, i.e. the theory has some sort of closure (in terms of being able to learn all the concepts)? To Gerry - I know what you mean but eventually it might give us a familiar structure... – Adam Rubinson Apr 27 '12 at 10:34