How do we compare the set of natural numbers in different models of ZFC

Question

In a model of ZFC, an inductive set is a set $A$ satisfying $\emptyset\in A$ and $n\cup\{n\}\in A$ for every $n\in A$. Suppose that $X$ is the inductive set given by the axiom of infinity. In my last question, I stated that the set of natural numbers is defined as $$\omega := \bigcap_{X'\subset X, X'\text{ is an inductive set}} X'.$$ Now, we say that different models of ZFC do not necessarily give the same $\omega$. But in what sense? We cannot compare the $\omega$'s in different models, can we? If we say that one is embeddable into another, an embedding $\omega\to\omega'$ between the two sets of natural numbers in different models is involved, but it should an element of $\omega\times\omega'$, which has no sense at all. If we cannot compare the $\omega$'s in different models, then it's meaningless to say that one is different from the other. So what do we actually mean here?

Sorry if I have asked a foolish question. Any help appreciated.

Answer 1

We cannot compare the $\omega$s in different models, can we? If we say that one is embeddable into another, an embedding $\omega\rightarrow\omega'$ between the two sets of natural numbers in different models is involved, but it should an element of $\omega\times\omega'$, which has no sense at all.

In fact we absolutely can compare the $\omega$s of different models; it's no more tricky than comparing (say) the centers of different groups. The point is that the putative isomorphism we're looking for lives, not in either of the two models in question, but in "reality."

---

First, suppose $\mathcal{M}_1,\mathcal{M}_2$ are models of $\mathsf{ZFC}$. This means that each is a set equipped with a binary relation symbol, say $$\mathcal{M}_1=(M_1;\varepsilon_1)\quad\mbox{and}\quad\mathcal{M}_2=(M_2; \varepsilon_2).$$ Note that these models themselves "live" in the "true mathematical universe;" it's within this larger universe that the relevant comparison-making objects will exist.

Now there are elements $m_1\in M_1$ and $m_2\in M_2$ such that $\mathcal{M}_1$ thinks $m_1$ is $\omega$ and $\mathcal{M}_2$ thinks $m_2$ is $\omega$ (note my use of "$\in$" here rather than "$\varepsilon_i$" - this is because I'm talking about the literal underlying sets of the models in question, so I'm using the "true" elementhood relation). Connected to these are a pair of sets, namely $$[m_1]=\{a\in M_1: (a,m_1)\in\varepsilon_1\}\quad\mbox{and}\quad [m_2]=\{a\in M_2: (a,m_2)\in\varepsilon_2\}.$$ Note that these are probably different things from $m_1$ and $m_2$ themselves; that's OK. We can now "compare the $\omega$s" of our two different models:

Is there a function $f:[m_1]\rightarrow [m_2]$ such that, for every $a,b \in [m_1]$, we have $$(a,b)\in\varepsilon_1\leftrightarrow (f(a),f(b))\in\varepsilon_2?$$

Note that for simplicity here I'm just thinking of $\omega$ as a linear order; given that nonisomorphic models of $\mathsf{PA}$ can be order-isomorphic, it's a good exercise to write out carefully what the definition of "$\mathcal{M}_1$ and $\mathcal{M}_2$ have isomorphic semirings of natural numbers" should be.

Answer 2

Let $(M,\varepsilon)$ and $(M',\varepsilon')$ be models of ZFC. Remember that $M$ and $M'$ are sets (in our ambient set-theoretic universe), and $\varepsilon$ and $\varepsilon'$ are binary relations on $M$ and $M'$, respectively.

Now $M$ has an element $\omega^M$ that satisfies the definition of $\omega$ in $M$. Let $\Omega = \{m\in M\mid (m,\omega^M)\in \varepsilon\}\subseteq M$. So $\Omega$ is a set (in our ambient set-theoretic universe), consisting of those of elements of the model $M$ that $M$ thinks are natural numbers. Note that $\varepsilon$ restricts to a linear order on $\Omega$.

Similarly, $M'$ has an element $\omega^{M'}$ that satisfies the definition of $\omega$ in $M'$, we can define $\Omega' = \{m\in M'\mid (m,\omega^{M'})\in \varepsilon'\}\subseteq M'$, and $\varepsilon'$ restricts to a linear order on $\Omega'$.

Now what people mean when they say "the $\omega$ of $M$ can be different from the $\omega$ of $M'$" is that the ordered sets $(\Omega,\varepsilon)$ and $(\Omega',\varepsilon')$ might not be isomorphic. That is there may be no function $f\colon \Omega\to \Omega'$ (in our ambient set-theoretic universe) which is an isomorphism between these ordered sets.

Answer 3

Perhaps the most straightforward way of exhibiting a larger model $M_2$ once we have a model $M_1$ of ZFC is by the ultrapower construction. There is a natural "diagonal" embedding of the smaller into the larger model. In particular, the natural numbers of $M_1$ are included in the natural numbers of $M_2$ (as an initial cut).

In fact it is possible to develop a situation where each of the models is in fact the larger model obtained by ultrapower of the smaller one. In the multiverse of Hamkins, there is a related "well-foundedness mirage" axiom.