# Relations of infinite arity

Infinitary logic considers languages being infinite by infinite conjunctions and disjunctions.

I wonder why it not considers languages being infinite by relations and functions of infinite arity.

Relations of finite arity $n$ over a base set $A$ can be seen as unary predicates of functions $f:[n] \rightarrow A$. Nothing prohibits us to consider more general functions $f:\mathbb{N} \rightarrow A$ or even $f:\mathbb{R}^+_0 \rightarrow A$.

Is there a model theory assuming a language that allows for relations and functions of infinite and even uncountable arity?

I asked this question at MSE but did get no feedback.

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I assume that not considering relations of infinite arity has to do with the fact, that in practical situations most arities > 2 are considered "many" and ignored. (See: en.wikipedia.org/wiki/Counting#Education_and_development) There are some relations of arity 3 that are taken serious - e.g. betweenness -, very few of arity 4, and I don't know one of arity 5. So why bother about arities like 2011 or even $\omega$? – Hans Stricker Feb 22 '11 at 19:46
@Hans: there is another MO question about operations of higher arity. I mentioned there that a natural example is compact Hausdorff spaces, which can be viewed as sets equipped with J-ary operations for every set J corresponding to ultrafilters on J. – Qiaochu Yuan Feb 22 '11 at 21:18
@Hans: operations of arbitrary arity also naturally occur in the theory of operads and related areas of higher category theory. Arguably many of the binary operations we care about are more naturally thought of as n-ary operations for all finite n that satisfy enough compatibility relations; see, for example, Leinster's Higher operads, higher categories at arxiv.org/abs/math.CT/0305049 . – Qiaochu Yuan Feb 22 '11 at 21:20
@Qiaochu: Thank you very much for your valuable hints. Since I never heard of operads before, and got answers/comments concerning operads on two seemingly (at least to me) unrelated questions - mathoverflow.net/questions/55641/… and this one - and since you pointed me to higher category theory concerning a third question - math.stackexchange.com/questions/23120/… - I dare to ask, if YOU can see the interconnection between my three questions, which is hidden to me? – Hans Stricker Feb 22 '11 at 21:51
Generalized operads and multicategories can be applied to theories of arbitrary arity just as well as to ones of finite arity: nlab.mathforge.org/nlab/show/generalized+multicategory . See also nlab.mathforge.org/nlab/show/infinitary+Lawvere+theory . – Mike Shulman Feb 23 '11 at 5:16