I am wondering if there is a foundation of mathematics where not sets or "set-like objects" (such as objects of a suitable topos as in ETCS) are the primitive notion, but rather **lists**. These lists (aka sequences, tuples, ...) should be mathematical objects denoted as $[a_1,a_2,\dotsc]$ with an information in which position an element $a_i$ occurs, and of course the elements may appear multiple times. The length of a list does not have to be finite, but if it helps to answer the question feel free to make this assumption, thus answering the question in finitism. Generally, they should be indexed by (an equivalent of) ordinal numbers.

The question is partially motivated by the observation that in many programming languages lists are actually more common and also the building blocks of more complex objects. Here, sets are often seen as *derived from lists* or *special lists*, namely lists in which the order does not matter and in which no element appears twice. In the set-theoretic foundations I am aware of, it is vice versa: lists are special sets. So my question is basically if we can **turn this paradigm around** and see lists as the primary notion and build mathematics around it?

Maybe I did not use the right search terms, and surely I am not an expert in foundation of mathematics, but I cannot find any such approach. And I find it hard to come up with something on my own, because every axiom for lists I can think of involves set-like objects in the end. It feels like set theory (and category theory of course, but here we also have *sets of objects* etc. so this does not get us out of the paradigm) permeates every thought.

For instance, what are the indices of the list elements, or how do they behave, when we are not allowed to talk about sets and hence of ordinal numbers? How can we even formulate that each list and every position yields an element without talking about maps? The "cheap" way out is to define sets as special lists as above and write down the ZFC axioms in terms of these special lists - but of course it would be much more satisfactory to develop something which does not just use ZFC as an intermediate step.

I am aware that the whole idea might not work at all. In this case, an answer is also appreciated explaining why it does not work.

Just to give Andreas Blass' comment extra visibility: Oliver Deiser has developed such a theory, called **Axiomatic List Theory**, in his Habilitationsschrift (in German), an excerpt of which has been published as well (in English).

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