5 grammar

It is very near the bottom of Kanamori's chart. The very bottom of the chart is the level of a (strongly) inaccessible cardinals, which is the smallest large cardinal axiom. Right above the inaccessibles in the chart are the α-inaccessible cardinals. It turns out that the Universe Axiom (UA) is strictly weaker than the existence of a 2-inaccessible cardinal. In fact, κ is 2-inaccessible if and only if κ is regular and Vκ ⊧ ZFC + UA.

Specifically, a cardinal κ is:

• 0-inaccessible iff κ is regular,
• 1-inaccessible iff κ is a regular strong limit of 0-inaccessibles,
• 2-inaccessible iff κ is a regular strong limit of 1-inaccessibles,
• etc.

So an inaccessible cardinal is exactly the same as a 1-inaccessible cardinal, which are also precisely the regular cardinals κ such that Vκ ⊧ ZFC. If κ is 2-inaccessible, then there are unboundedly many inaccessibles λ < κ. These are inaccessible in Vκ too, which is why Vκ satisfies UA.

Note that the existence of a 2-inaccessible cardinal κ does not directly imply the Universe Axiom. Indeed, κ may well be the last inaccessible cardinal, which means that there may be no universe that contains κ itself. However, if κ is 2-inaccessible then the universe Vκ does satisfy UA, which means that the existence of a 2-inaccessible proves the consistency of ZFC + UA.

Although UA is indeed a large cardinal axiom, there is no way to formulate UA as the existence of a single large cardinal. However, morally speaking, you can think of UA as saying "the class of all ordinals (viewed as a cardinal number) is 2-inaccessible." Of course, this doesn't make sense since the class of all ordinals is not a set, but this is exactly what κ looks like when viewed from inside Vκ.

It is very near the bottom of Kanamori's chart. The very bottom of the chart is the level of a (strongly) inaccessible cardinals, which is the smallest large cardinal axiom. Right above the inaccessibles in the chart are the α-inaccessible cardinals. It turns out that the Universe Axiom (UA) is strictly weaker than the existence of a 2-inaccessible cardinal. In fact, κ is 2-inaccessible if and only if κ is regular and Vκ ⊧ ZFC + UA.

Specifically, a cardinal κ is:

• 0-inaccessible iff κ is regular,
• 1-inaccessible iff κ is a regular strong limit of 0-inaccessibles,
• 2-inaccessible iff κ is a regular strong limit of 1-inaccessibles,
• etc.

So an inaccessible cardinal is exactly the same as a 1-inaccessible cardinal, which are also precisely the regular cardinals κ such that Vκ ⊧ ZFC. If κ is 2-inaccessible, then there are unboundedly many inaccessibles λ < κ. These are inaccessible in Vκ too, which is why Vκ satisfies UA.

Note that the existence of a 2-inaccessible cardinal κ does not directly imply the Universe Axiom. Indeed, κ may well be the last inaccessible cardinal, which means that there may be no universe that contains κ itself. However, if κ is 2-inaccessible then the universe Vκ does satisfy UA, which means that the existence of a 2-inaccessible proves the consistency of ZFC + UA.

3 correction

It is very near the bottom of Kanamori's chart. The very bottom of the chart is the level of a (strongly) inaccessible cardinals, which is the smallest large cardinal axiom. Right above the inaccessibles in the chart are the α-inaccessible cardinals. It turns out that the Universe Axiom (UA) is strictly weaker than the existence of a 2-inaccessible cardinal. In fact, κ is 2-inaccessible if and only if κ is regular and Vκ ⊧ ZFC + UA.

Specifically, a cardinal κ is:

• 0-inaccessible iff κ is regular,
• 1-inaccessible iff κ is a regular strong limit of 0-inaccessibles,
• 2-inaccessible iff κ is a regular strong limit of 1-inaccessibles,
• etc.

So an inaccessible cardinal is exactly the same as a 1-inaccessible cardinal, which are also precisely the regular cardinals κ such that Vκ ⊧ ZFC. If κ is 2-inaccessible, then there are unboundedly many inaccessibles λ < κ. These are inaccessible in Vκ too, which is why Vκ satisfies UA.

2 grammar
1