Measurable cardinals as sets

Question

A philosopher said that measurable cardinals are the largest possible sets. Is this true? Are those sets at all? I mean, cardinals measure size of sets and for example $2=\{\{\},\{\{\}\}\}$ but can we represent measurable cardinals similarly? And is it true that those can not exist because defining them requires measurable cardinality amount of text, as he says?

Answer 1

No, they are not the largest possible sets. There is no largest possible set (Cantor theorem): If $A$ is a set, then $\mathcal P(A)$, the collection of all subsets of $A$, is a set of size strictly larger than the size of $A$. And neither there is a set $B$ that contains sets of all sizes, or that any set is smaller than one of the sets in $B$: Take the union $C$ of all sets in $B$. Then the power set of $C$ is strictly larger than all the sets in $B$.

And yes, they are ordinals, so they are sets. Special sets, at that, in the sense that they are transitive (so if $\kappa$ is measurable, and $a\in\kappa$, then also $a\subset\kappa$) and well-ordered by $\in$ (so $\in$ gives as a linear order on the elements of $\kappa$, and any non-empty subset of $\kappa$ has an $\in$-first element).

Some set theorists in fact expect/assume/work under the assumption that there is not just one measurable cardinal, but in fact a proper class of them, so given any set, there is a measurable cardinal that is larger. Others may on occasion assume that there are no measurables. But again, they can never be the largest size.

One possible explanation for what was meant is that if $\kappa$ is measurable, then $H(\kappa)$ satisfies all the axioms of set theory (is a model), and this model has size precisely $\kappa$. Here, $H(\kappa)$ is the collection of all sets $A$ such that $A$, and every $T\in A$, and every $S$ in such a $T$, and every $R$ in such and $S$, etc, all of them, have size strictly smaller than $\kappa$. Informally one sometimes describes this situation by saying that the (proper) class $\mathsf{ORD}$ of all ordinals is measurable. For example, this is how the early descriptions of Steel's work on the core model were presented.

Measurability is an example of a large cardinal property. As far as large cardinals go, it is on the small side of the spectrum. Nowadays, we routinely consider properties that give us much much larger cardinals. You may want to take a look at Cantor's attic for some examples.