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In the book Understanding analysis, by Abbot, when discussing the Archimedean property, the author states that there are ordered field extensions of $mathbb{Q}$ that include "numbers" bigger than every natural number.

Could someone provide examples and and explanation why this could be the case?

A concrete example

Consider the ring $mathbb{Q}[x]$ of polynomials over $mathbb{Q}$. These can be linearly ordered as in this answer, in a way that preserves the ordering of the rationals. Now we have the "regular" naturals in this field: 1, 2,3, and so on. But the polynomial $x$ is larger than all of these, and $x^2$ is larger than $x$, and so on. So in this ring there are elements that are greater than all the ordinary natural numbers.

By standard abstract algebra, the ordering on $mathbb{Q}[x]$ can be extended to an ordering on the field of fractions of the ring. That will give an ordered field that contains $mathbb{Q}[x]$ as a subring, and as such contains elements larger than all the ordinary natural numbers.

There are other algebraic constructions that give other non-Archimedean ordered fields, as well.

An example from logic

In logic, the comapctness theorem says that if we have a set of axioms so that every finite subset of the axioms is consistent on its own, then the whole set is consistent. Take our axioms to include the axioms of an ordered ring, and also the axiom $z > n$ for each $n$. For any finite subset of these axioms, we can choose a value of $z$ large enough to make that finite subset true in $mathbb{Q}$. So there is a model in which all the axioms are true at once. In that model, whatever $z$ is must be an element larger than every natural number.

In nonstandard analysis

In nonstandard analysis, we work with a field that extends of the reals in which there are infinitesimals. For this purpose, an infinitesimal is an element $delta$ so that $0 < delta$ but $delta < 1/n$ for each ordinary natural number $n$. Then if $delta$ is infinitesimal, $1/delta$ exists (because $delta not = 0$, and $1/delta$ is larger than $n$ for all $n$.

Consider the field of rational functions over $mathbb{Q}$. I don't know what the standard notation for this is, but let's denote it by $mathbb{Q}(x)$.

$$mathbb{Q}(x)=left{frac{p(x)}{q(x)}: p(x),q(x)inmathbb{Q}[x] text{ and } q(x)neq 0 text{ is a monic polynomial.}right}$$

To define an order on it, define $f>g$ if and only if $f-g>0$ where a rational function is positive if and only if its leading coefficient of the numerator is positive. I leave the process of checking the axioms to you.

Now compare $frac{x}{1}$ and $frac{n}{1}$. We see that $frac{x}{1}-frac{n}{1}=frac{x-n}{1}>0$. So, $x$ is bigger than any natural number $n$.

Edit: After Carl Mummert's wonderful comment, it is worthy to add that in order to have a well-defined order, we should first set the leading coefficient in the denominator to be $1$ by multiplying the whole fraction by the inverse of this coefficient if necessary. So, we can assume that $q(x)$ is monic. Now, we can see that the order we have defined is also an extension of the order on $mathbb{Q}$ as well.

Here's a well known construction of an ordered field extension of $mathbb Q$ that comes from logic and from nonstandard analysis, as mentioned in the answer of @CarlMummert.

Choose a nonprincipal ultrafilter on the natural numbers $mathbb N$: an ultrafilter is a finitely additive, $0,1$-valued measure defined on all subsets $A subset mathbb N$, such that $mathbb N$ has measure $1$; and $mu$ is nonprincipal if ${i}$ has measure zero for every $i in mathbb N$. The existence of a nonprincipal ultrafilter is an exercise in applying the axiom of choice.

Consider the set $mathbb Q^{mathbb N}$ which is the set of all sequences of rational numbers. Define an equivalence relation on this set: given $underline x = (x_i)$ and $underline y = (y_i) in mathbb Q^{mathbb N}$, define $underline x approx underline y$ if the set of indices $i$ for which $x_i = y_i$ has measure $1$. Let $[underline x] = [x_1,x_2,x_3,...]$ denote the equivalence class of $underline x = (x_1,x_2,x_3,...]$. Define addition and multiplication in the obvious way: $[underline x] + [underline y] = [underline z]$ means that the set of indices for which $x_i + y_i = z_i$ has measure $1$, and similarly for multiplication. Define inequality similarly: $[underline x] < [underline y]$ if the set of indices for which $x_i < y_i$ has measure $1$. Check that everything is well-defined, and that you get an ordered field.

To embed $mathbb Q$ into this field, map the rational number $q$ to $[q,q,q,q,q,q,q,q,q,q,...]$. Check that this is an embedding of ordered fields.

To identify a number greater than any natural number, take $[1,2,3,4,5,6,7,8,9,...]$.