Why does $\operatorname{Con}(\operatorname{Con}(\Gamma)) = \Gamma$? I'll understand if it was > $\Gamma \subseteq \operatorname{Con}(\operatorname{Con}(\Gamma))$ but why equals? There should be other formulas in $\operatorname{Con}(\operatorname{Con}(\Gamma$)) which aren't part of $\Gamma$. Edit (lecture notes): ![enter image description here](

I am certain that this is a typo, and should read "$Con(Con(\Gamma))=Con(\Gamma)$". This is because a closure operator $c$ on a family of sets $\mathcal{F}$ is one satisfying the following three properties:

* $X\subseteq c(X)$,

* $X\subseteq Y$ implies $c(X)\subseteq c(Y)$, and

* $c(c(X))=c(X)$




for all sets $X, Y\in\mathcal{F}$.

In particular, note the "that is" between the statement of the lemma and the three properties: the three properties listed are meant to _define_ what a closure operator is.

* * *

As further evidence for this being a typo, note that as written the properties imply that $Con(X)=X$ for all $X$:

* We have $X\subseteq Con(X)$ and $Con(X)\subseteq Con(Con(X))$ by property $1$.

* But $Con(Con(X))=X$, by property $3$ as your professor wrote it.

* So $X\subseteq Con(X)\subseteq X$, that is, $X=Con(X)$.




So clearly your professor didn't mean to write that, since otherwise $Con$ is trivial!