User blog comment:Deedlit11/Ordinal Notations V: Up to a weakly Mahlo cardinal/@comment-180.159.72.141-20160322163141/@comment-1605058-20160322174820

No, it doesn't mean that. There are \(\kappa+1\)-weakly inaccessible cardinals, but that's not important here. The important thing here is that you can't get to a weakly Mahlo "from below" using various inaccessible cardinals.

For comparison, let's look at weakly inaccessible cardinals first. I don't want to explain the formal definition of inaccessibility now, but the idea is that if \(\kappa\) is a weakly inaccessible cardinal, then we can't get \(\kappa\) or any larger cardinal by starting with a cardinal smaller than \(\kappa\) and applying the following operations: 1. Taking the smallest cardinal larger than a cardinal we have already reached, and 2. Summing a number of cardinals we have already reached, provided we have also already reached that number of summands.

Similarly, when we define 1-weakly inaccessible, we add one more operation, namely 3. Taking the \(\alpha\)th weakly inaccessible ordinal, where \(\alpha\) is an ordinal size of which we have already reached. Then, for 2-weakly inaccessible, we add yet another operation: 4. Taking the \(\alpha\)th 1-weakly inaccessible ordinal, where \(\alpha\) is an ordinal size of which we have already reached, and so on. This can be continued to arbitrary \(\beta\)-weakly inaccessible cardinal.

But the fun doesn't end here - for hyper-weakly inaccessible \(\kappa\) we state that we can't reach \(\kappa\) or above by starting with a cardinal smaller than \(\kappa\) and applying the following operation: Take \(\alpha\)th \(\beta\)-inaccessible cardinal, where \(\alpha.\beta\) have sizes we have already reached.

One can imagine that this can be continued further, to 1-hyper-weakly inaccessibles, hyper-hyper-weakly inaccessibles and what not. Now the weakly Mahlo cardinals are even larger - in certain formalizable sense, we can define weakly Mahlo cardinals as cardinals which cannot be exceeded by starting with a smaller cardinal and applying operations of any sort we can define in the manner above, so that this is a condition stronger than weak inaccessibility, hyper-hyper-weak inaccessibility and what not.