# If every object in math can be reduced to sets, how are tuples explained via sets?

It seems to me that tuples are just as elementary as sets. Can tuples be reduced to sets?

Yes, they can! They are done inductively. First, ordered pairs are defined as $$(a, b) = \{\{a\}, \{a, b\}\}$$ (exercise: show that $$(a, b) = (c, d)$$ iff $$a = c$$ and $$b = d$$). Then, an $$n + 1$$ tuple $$(x_1, \dots, x_n, x_{n + 1})$$ is defined as $$((x_1, \dots, x_n), x_{n + 1})$$. Lastly, a (countably) infinite sequence, that is, an "infinite"-tuple in some sense, is defined as a function $$f: \mathbb{N} \rightarrow X$$, where $$X$$ is some set which contains every coordinate we want. The intuition for this is that I want $$f(n)$$ to define my $$n$$th coordinate.
• @csp2018 No. You can see why it won't be this if you consider $(a, b, a)$ and $(a, a, b)$. Under that definition, $(a, b, a) = \{\{a\}, \{a, b\}, \{a, b, a\}\} = \{\{a\}, \{a, b\}\} = \{\{a\}, \{a, a\}, \{a, a, b\}\} = (a, a, b)$ which is a big deal if $a \neq b$. Sep 5, 2019 at 3:02
• Instead, $(a, b, c) = ((a, b), c) = \{\{(a, b)\}, \{(a, b), c\}\} = \{\{\{\{a\}, \{a, b\}\}\}, \{\{\{a\}, \{a, b\}\}, c\}\}$. Sep 5, 2019 at 3:03