Abandoned LCAs on Cantor's Attic : Grand Reflection cardinals, universe cardinals, weak universe cardinals

Question

Cantor's Attic is a really great website for the various descriptions of large finite numbers, large countable ordinals, and large cardinal axioms.

However, after looking through the archives of the website, I have found that originally, the following cardinals were included and never given a definition:

• Grand reflection cardinals
• Universe cardinals
• Weak universe cardinals

The universe cardinals and weak universe cardinals were replaced by the worldly cardinals in the same spot, so it makes sense that the term "universe" was renamed to worldly. However, that doesn't explain what "weak universe" cardinals are.

The grand reflection cardinals were created and never replaced. They still remain on the upper attic today, although hidden by code. You can see the link there, but it links to nothing.

So what are these cardinals? Does anybody know? The best person I could think of to answer this would be @JoelDavidHamkins himself, who was the one to put these on Cantor's Attic.

Answer 1

Thanks for the question. I had briefly used the universe/weak-universe terminology in 2010 in my answer to a question on MathOverflow, What interesting/nontrivial results in Algebraic geometry require the existence of universes?

My thinking at that time was that $\kappa$ should be a universe cardinal, if $V_\kappa$ was an (uncountable) Grothendieck universe, which means that this is the same as an inaccessible cardinal.

What we now call the worldly cardinals can be seen as a weak version of this, where one drops the requirement that $\kappa$ is regular. So $\kappa$ is worldly (or a weak universe cardinal in that MO answer), if $V_\kappa\models\text{ZFC}$.

That answer also mentions a very weak universe concept, which is simply a transitive model of $\text{ZFC}$.

I think I had created the Cantor's Attic entries at first at about the same time I had posted that answer, which was just a little before I had started using the worldly cardinal terminology, which I think is better and which I am happy to see has become comparatively established. So I think I may have created the early entries, and then switched over to the worldly terminology later.

The Grand Reflection cardinals, in contrast, were introduced by Philip Welch. This is a cardinal $\kappa$ relating truth in $V_\kappa$ for subsets of $V_\kappa$ to truth in $V$ for proper classes.

Incidentally, Cantor's Attic is a community-run affair, and knowledgeable people are welcome to help with adding to or improving entries.