Do inaccessible-sized sets have any properties that smaller sets don't?

Question

When going from finite to infinite sets, properties arise that are impossible for finite sets. For example, an infinite set has bijections to proper subsets, and to the Cartesian product of the set with itself.

Now I wonder if for sets whose cardinality is $\ge$ an inaccessible cardinal, there are also new properties arising that are true for all such sets, but are not possible for smaller sets.

Now obviously there are new properties like “has a cardinality larger than an inaccessible cardinal” but I'm interested in things that can be formulated without reference to large cardinals (just as my two examples of infinite set properties nowhere reference infinity).

Clarification: The properties I'm after are set properties that under the assumption of ZFC + large cardinal, are equivalent to the property “the set's cardinality is greater or equal to an inaccessible cardinal”.

Answer 1

In general all the properties will somehow end up being somehow equivalent to "There is an inaccessible of size $\leq|X|$".

Sometimes, this can be boring. For example, suppose that there is only a single inaccessible cardinal $\kappa$, and there are no transitive models of $\sf ZFC$ which contain $V_\kappa$ (i.e. all transitive models of $\sf ZFC$ have height $\leq\kappa$), then a set has size $\geq\kappa$ if and only if it is not equipotent to a set which lies in a transitive model of set theory.

Yes, that's not a particularly attractive property.

When you go up in the food chain of large cardinals these can be nicer. For example, if $\kappa$ is a measurable cardinal, then $|X|\geq\kappa$ if and only if there is a $\kappa$-complete free ultrafilter on $X$. And if $\kappa$ is strongly compact, then you can even require that this ultrafilter is uniform.

But this is ultimately just a fancy restatement of $|X|\geq\kappa$ and $\kappa$ is a large cardinal of type $\varphi$. Maybe that's what you're looking for.

Answer 2

There are a few examples that you can find in these slides that somewhat hide the underlying large cardinal property.

Here are two:

1. $\vert X\vert$ is larger than a measurable iff for every $\omega$-model $(M, E)$ with $X\subseteq M$ there is a proper elementary extension which is still an $\omega$-model.

Here $(M, E)$ is an $\omega$-model means that $\omega$ is in the transitive collapse of the wellfounded-part of $(M, E)$.

2. $\vert X\vert$ is larger than an extendible cardinal iff $\vert X\vert$ is a strong reflection cardinal for all invariant $\Sigma_3$ properties of structures.

Here we call $\varphi(x)$ with only $x$ free an invariant property of structures iff whenever $\mathcal A$ and $\mathcal B$ are isomorphic structures then $\varphi(\mathcal A)\Leftrightarrow\varphi(\mathcal B)$. Furthermore, "$\lambda$ is a strong reflection cardinal for $\varphi$" means that whenever $\mathcal A$ is a structure in a countable language with $\varphi(\mathcal A)$ there is a substructure $\mathcal B$ of $\mathcal A$ of cardinality less than $\lambda$ with $\varphi(\mathcal B)$.

Unfortunatley I do not know any interesting example for just an inaccessible cardinal.