Should "together with" be taken as slang for an n-tuple?

Question

When an algebraic structure is defined, it is often defined as a set $S$ "along with"/"together with"/"having" operations $\circ_1, \circ_2, \ldots, \circ_n$, and "denoted" by $(S, \circ_1, \circ_2, \ldots, \circ_n)$.

This is true of metric spaces as well. We often say that a metric space is a set $X$ with some distance function $f: X \times X \to \mathbb{R}$.

But none of this seems very formal to me. Are we -- the reader -- supposed to infer that it in each of these cases, we're really talking about an $n$-tuple (in ZFC)?

So for instance, when a theorem says something like this:

Thm Let $X$ be a metric space. If $X$ is a foo, then $X$ is a noo.

It is formally:

Thm Let $(X, d)$ be a metric space. If (the underlying set) $X$ is a foo, then $X$ is a noo.

I guess I'm just confused as to why tuples are glossed over and replaced with "together with" so often, when it detracts from the "axiomaticity" of the material.

Answer 1

It's often okay to define a mathematical object without an explicit set theoretic model. If we need a model we can always take a step back and make sure there is one.

Take for example the following definition:

A map $f$ from a set $A$ to a set $B$ assigns to every $a\in A$ a unique value $f(a)\in B$. We write $f\colon A\to B$. Two maps $f,g\colon A\to B$ are equal if and only if $f(a)=g(a)$ for all $a\in A$.

We can use this definition of a map to define surjectivity, injectivity and any other property of maps and work with those without ever thinking about how we would model this in set theory.

Or do you always think of a map $f\colon A\to B$ as a tuple $(F, A, B)$ where $F\subseteq A\times B$ is a right-unique relation, and the $3$-tuple $(F, A, B)$ is really build from Kuratowski pairs $(F, (A, B))$ with $(A, B) = \{\{A\}, \{A, B\}\}$?

Answer 2

One can, but I do not think one should.

Also, $n$-tuples are not basic objects of standard set theories, they have to be defined. Take a probability space $(\Omega,\Sigma,\mu)$. Should we take this as $(\Omega,(\Sigma,\nu))$ with some definition of an ordered pair? How about $((\Omega,\Sigma),\mu)$? Or maybe it is a function $f$ with domain $\{0,1,2\}$ such that $f(0)=\Omega$, $f(1)=\Sigma$, $f(2)=\nu$. ? Many other choices are possible, but it doesn't really matter. If a result in, say, geometry essentially depends on how you define $n$-tuples it is probably of no interest to any geometer.