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‘I do know the obvious isomorphism between them.’ And presumably you also know the isomorphism between $\operatorname{quot}_{\Bbb R}$ and $\Bbb Q$, whatever your particular definition of $\Bbb Q$ may be. Such isomorphisms are the answer to all of your question. Spivak simply identifies $\Bbb N$ with its image in $\Bbb R$ and $\Bbb Q$ with $\operatorname{quot}_{\Bbb R}$ in order to reduce notational clutter. If you really wish to do so, you can introduce names for the various isomorphisms and rewrite all of the statements that are bothering you in formally correct fashion. It is perhaps worthwhile to carry out such an exercise once, but in practice the more careful version simply obscures the real idea.

‘I do know the obvious isomorphism between them.’ And presumably you also know the isomorphism between $\operatorname{quot}_{\Bbb R}$ and $\Bbb Q$, whatever your particular definition of $\Bbb Q$ may be. Such isomorphisms are the answer to all of your question. Spivak simply identifies $\Bbb N$ with its image in $\Bbb R$ and $\Bbb Q$ with $\operatorname{quot}_{\Bbb R}$ in order to reduce notational clutter. If you really wish to do so, you can introduce names for the various isomorphisms and rewrite all of the statements that are bothering you in formally correct fashion. It is perhaps worthwhile to carry out such an exercise once, but in practice the more careful version simply obscures the real idea.

Added: ‘Specifically, I believe he is treating numbers as rational numbers which are at best isomorphic to them.’

Although I haven’t a copy of the book to hand, I’m pretty sure that at this point Spivak takes it for granted that $\Bbb N\subseteq\Bbb Q\subseteq\Bbb R$. No, this isn’t the $\Bbb N$ from which you built $\Bbb Z$ as a set of equivalence classes of ordered pairs; but that $\Bbb N$ has an isomorphic copy in $\Bbb Z$, and isomorphic means that as far as algebraic and order properties are concerned the two are interchangeable. That $\Bbb Z$ and it’s isomorphic copy of $\Bbb N$ aren’t subsets of the $\Bbb Q$ that you constructed from it, but they have isomorphic copies in that $\Bbb Q$, which again are interchangeable with their originals as far as algebraic and order properties are concerned. And that $\Bbb Q$ and its $\Bbb Z$ and $\Bbb N$ aren’t subsets of the $\Bbb R$ that you construct via Dedekind cuts (which isn’t the one that you construct via equivalence classes of Cauchy sequences, or the one that can be constructed by Conway’s surreal number procedure), but they have isomorphic copies in that $\Bbb R$, to which the same boring refrain applies. And neither of these versions of $\Bbb Q$ is literally identical to the field of quotients of the copy of $\Bbb N$ that lives in this $\Bbb R$, but both are isomorphic to it. And at this point, when the goal is to prove that up to isomorphism $\Bbb R$ is the unique complete ordered field, none of these formal details of construction matter: you’re entitled to use all of the standard properties of $\Bbb R$, and to think of real numbers as just real numbers, not as special sets of rational numbers.

‘I do know the obvious isomorphism between them.’ And presumably you also know the isomorphism between $\operatorname{quot}_{\Bbb R}$ and $\Bbb Q$, whatever your particular definition of $\Bbb Q$ may be. Such isomorphisms are the answer to all of your question. Spivak simply identifies $\Bbb N$ with its image in $\Bbb R$ and $\Bbb Q$ with $\operatorname{quot}_{\Bbb R}$ in order to reduce notational clutter. If you really wish to do so, you can introduce names for the various isomorphisms and rewrite all of the statements that are bothering you in formally correct fashion. It is perhaps worthwhile to carry out such an exercise once, but in practice the more careful version simply obscures the real idea.

Added: For example, let $h:\Bbb Q\to\operatorname{quot}_{\Bbb R}$ be the canonical isomorphism. Then the statement

Given $x,y\in\Bbb R$, it is clear that $x<y\implies\Theta_x\subset\Theta_y$.

is to be understood as shorthand for

Given $x,y\in\Bbb R$, it is clear that $x<y\implies\Theta_{h(x)}\subset\Theta_{h(y)}$.

‘I do know the obvious isomorphism between them.’ And presumably you also know the isomorphism between $\operatorname{quot}_{\Bbb R}$ and $\Bbb Q$, whatever your particular definition of $\Bbb Q$ may be. Such isomorphisms are the answer to all of your question. Spivak simply identifies $\Bbb N$ with its image in $\Bbb R$ and $\Bbb Q$ with $\operatorname{quot}_{\Bbb R}$ in order to reduce notational clutter. If you really wish to do so, you can introduce names for the various isomorphisms and rewrite all of the statements that are bothering you in formally correct fashion. It is perhaps worthwhile to carry out such an exercise once, but in practice the more careful version simply obscures the real idea.

‘I do know the obvious isomorphism between them.’ And presumably you also know the isomorphism between $\operatorname{quot}_{\Bbb R}$ and $\Bbb Q$, whatever your particular definition of $\Bbb Q$ may be. Such isomorphisms are the answer to all of your question. Spivak simply identifies $\Bbb N$ with its image in $\Bbb R$ and $\Bbb Q$ with $\operatorname{quot}_{\Bbb R}$ in order to reduce notational clutter. If you really wish to do so, you can introduce names for the various isomorphisms and rewrite all of the statements that are bothering you in formally correct fashion. It is perhaps worthwhile to carry out such an exercise once, but in practice the more careful version simply obscures the real idea.

Added: For example, let $h:\operatorname{quot}_{\Bbb R}\to\Bbb Q$ be the canonical isomorphism. Then the statement

Given $x,y\in\Bbb R$, it is clear that $x<y\implies\Theta_x\subset\Theta_y$.

is to be understood as shorthand for

Given $x,y\in\Bbb R$, it is clear that $x<y\implies\Theta_{h(x)}\subset\Theta_{h(y)}$.

‘I do know the obvious isomorphism between them.’ And presumably you also know the isomorphism between $\operatorname{quot}_{\Bbb R}$ and $\Bbb Q$, whatever your particular definition of $\Bbb Q$ may be. Such isomorphisms are the answer to all of your question. Spivak simply identifies $\Bbb N$ with its image in $\Bbb R$ and $\Bbb Q$ with $\operatorname{quot}_{\Bbb R}$ in order to reduce notational clutter. If you really wish to do so, you can introduce names for the various isomorphisms and rewrite all of the statements that are bothering you in formally correct fashion. It is perhaps worthwhile to carry out such an exercise once, but in practice the more careful version simply obscures the real idea.

Added: For example, let $h:\Bbb Q\to\operatorname{quot}_{\Bbb R}$ be the canonical isomorphism. Then the statement

Given $x,y\in\Bbb R$, it is clear that $x<y\implies\Theta_x\subset\Theta_y$.

is to be understood as shorthand for

Given $x,y\in\Bbb R$, it is clear that $x<y\implies\Theta_{h(x)}\subset\Theta_{h(y)}$.