# Tag Info

### A weak (?) form of Shelah cardinals

I think any measurable Woodin cardinal is a limit of weakly Shelah cardinals. To see this, note that, if $\kappa$ is a Woodin cardinal, for any $f : \kappa \to \kappa$, $\kappa$ is a limit of ...

### Reinhardt's ultimate classes

You can find Reinhardt's philosophy of set theory in "Set existence principles of Shoenfield, Ackermann, and Powell", Fundamenta Mathematica, vol 84, pp 5-34 and "Remarks on reflection ...
• 5,744
Accepted

### How large is the supremum of minimal $V$-heights of all first-order set theories formulated in a particular language of FOST?

Your ordinal $\beta_\mathcal{L}$ is perfectly well-defined: in my opinion it's more easily thought of as $$\sup\{\alpha: \forall \beta<\alpha(V_\beta\not\equiv V_\alpha)\},$$ and this definition ...
• 21.7k
Accepted

### What's the consistency status/strength of this limitation principle?

This principle is inconsistent: consider the formula $\theta(x)$ = "$x^+$ is the smallest infinite cardinal at which $\mathsf{CH}$ fails." The formula $\theta$ cannot hold on more than one ...
• 21.7k
1 vote

### Are there interesting examples of theorems proved using ‘height’ extensions?

Here is another example. The maximality principle in forcing is the scheme asserting of every statement $\varphi$ in the language of set theory that if there is forcing extension $V[G]$ of the set-...
### Equivalences of $\mathcal{F}$-Mahloness
The paper "Small Definably-large Cardinals" by Roger Bosch proves that an inaccessible cardinal is $\Sigma_{n+1}$-Mahlo if and only if it is $\Pi_n$-Mahlo except for $n=1$ (I'm referring to ...