Are there any proposed operations to actually construct an inaccessible set?

Question

When we postulate the smallest infinite set, we define it using an interative process involving iterations on the empty set.

When we postulate the existence of a set of continuum cardinality, we again have a process in mind to construct it, namely the Power set operation.

But what's the situation regarding the inacessible set? Is it:

1. We know that no such process can be defined.

Or :

2. There is research going on to define this set using a process

If the situation is the former, then are we sure that there exists some unknowable "collection of objects" that has this cardinality? Is it also possible that there's no such collection, and the notion of an inaccessible set only exists as an axiom for us to play with in our formal string manipulation game?

If the situation is the latter, then what are some of the proposed processes?

Answer 1

Maybe you will like this characterization.

The axiom of infinity says that there exists a set which contains 0 and is closed under the successor operation $S(x) = x \cup \{x\}$. You could, if you like, think of that as a "process" in the following way: start with 0, and then repeatedly take successors. Continue this process "forever" until it "stalls", i.e. until you are no longer adding any new elements. The set of everything you end up with is an infinite set.

Of course, the axiom of infinity is just the assertion that this process does in fact "stall". Without this axiom, it would be consistent that the process never stalls; that in every set, there is an element whose successor is not in the set, so that the process can always continue. This is the situation in the model HF of hereditarily finite sets, which satisfies all the ZFC axioms except Infinity.

So let's try to reformulate "a strong inaccessible exists" in this way. A strongly inaccessible cardinal $\kappa$ is one which is:

• uncountable, i.e. contains $\omega$

• has $2^\lambda \in \kappa$ for every $\lambda \in \kappa$; i.e. it is closed under the "power cardinality" operation $\lambda \mapsto |\mathcal{P}(\lambda)|$.

• is regular, i.e. its cofinality equals itself. That is, if $\lambda \in \kappa$ and $A \subset \kappa$ is a set of ordinals with order type $\lambda$, then $\bigcup A \in \kappa$. We can think of this as being closed under an operation $U(\lambda, A) = \bigcup A$ where $\operatorname{type} A = \lambda$.

So let's express this as follows. Start with $\omega$. Repeatedly do the following:

• apply the "power cardinality" operation to any ordinal $\lambda$ produced so far;

• apply the $U$ operation to any ordinal $\lambda$ produced so far, and any set $A$ of ordinals produced so far that has order type $\lambda$.

Continue this process until it stalls, i.e. until neither operation adds any new elements. When you get done, the set of everything you produced is a strongly inaccessible cardinal.

And the issue is parallel to the above: the proposed axiom "a strong inaccessible exists" is exactly the statement that the process does stall. In a model with no strong inaccessibles, e.g. $H(\kappa)$ for the least strong inaccessible $\kappa$ (which satisfies all the ZFC axioms), the process never stalls: given any set $B$ of ordinals, by applying either the power cardinality or $U$ operation to some elements of $B$, you can always produce an ordinal that is larger than any in $B$, and in particular is not in $B$.

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In fact any "closure" or "generation" operation can be imagined this way, if you wish. For instance, the topological closure of a set in a metric space: the operation is "given any convergent sequence of points generated so far, take its limit". The subgroup of a group generated by a set: take products and inverses. The transitive closure of a relation $R$: given any $a,b,c$ for which $(a,b), (b,c) \in R$, take the pair $(a,c)$. And so on.