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Joel David Hamkins
Joel David Hamkins
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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.

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.

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.

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.

Source Link
Joel David Hamkins
Joel David Hamkins
  • 236.2k
  • 44
  • 777
  • 1.4k

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.

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.