Show that well-ordering is not a first-order property.

Question

Problem description: Show that well-ordering is not a first-order notion. Suppose that $\Gamma$ axiomatizes the class of well-orderings. Add countably many constants $c_i$ and show that $\Gamma \cup \{c_{i+1} < c_i \mid i \in \mathcal{N}\}$ has a model.

So, I don’t quite get how to go about this. This post seems to address the same question, but I don’t understand the answer. I assume I’ll start something like this:

Suppose that $\Gamma$ axiomatizes the class of well-orderings (do I need to define this class?). Let $\{c_i \mid i \in \mathcal{N}\}$ be a set of new constants. Consider the set $A = \Gamma \cup \{c_{i+1} < c_i \mid i \in \mathcal{N}\}$.

And here I can’t quite see how to proceed. How do I show that every finite subset $A_0 \subseteq A$ is satisfiable?

Anyway, suppose I manage to do the above. Then I use compactness to show that $A$ is satisfiable, i.e. $A$ has a model.

Then what? How do I finish it? Or if it is finished, how to I explain that it is?

I should add: hints only, no direct answers — stop me if I’m asking questions that aren’t possible to answer without revealing the solution. Thanks!

Answer 1

Re: your second question: let $\mathcal{M}$ be a model of $A$. $\mathcal{M}$ is a linear order with a bunch of named elements (the $c_i$s); we can consider the "underlying linear order" $\mathcal{L}$ of $\mathcal{M}$.

• Is $\mathcal{L}$ a well-order?

• Why is that a problem?

Re: your first question, it might be easier to consider a concrete example: can you find a well-ordering with two elements $c_1>c_2$? How about three elements $c_1>c_2>c_3$? How about an arbitrary finite number?

Answer 2

HINT for your first question: For $k\in\Bbb N$ let $\varphi_k$ be $c_{k+1}<c_k$. If $A_0$ is a finite subset of $A$, $A_0'=A_0\setminus\Gamma$. If $A_0'$ is empty, $A_0$ certainly has a model: pick any well-ordering. If $A_0'\ne\varnothing$, let

$$m=\max\{k\in\Bbb N:\varphi_k\in A_0'\}\;,$$

and construct a model of $A$ whose underlying set is $\{0,\ldots,m\}$.