User blog:Ubersketch/Large Cardinals vs. Non-recursive Ordinals vs. Stages

In OCFs, there can be a lot of problems relating to the symbols used. This is especially evident in UNOCF, where the stage cardinal T, may not even be a cardinal. I'll lay out the 3 interpretations used.

Large cardinals

Using these has been the norm for a looooong time. However lots of issues can arise from using large cardinals. For example, at the consistency strengths required to proof the existence of large cardinals, their sizes may vary drastically. Take the Sigma_2 correct cardinals (not necessarily large cardinals but their size is dubious), they can either be larger than the first inaccessible cardinals, or much smaller. This is an issue if you were to use a Sigma_2 correct cardinal in an OCF.

Non-recursive ordinals

This one is much newer but has a lot of support. If you haven't noticed regular cardinals have admissible equivalents. Like recursively inaccessibles and recursively mahlos. One may be able to draw a connection between these and large cardinals. At first glance they may seem identical, but their sizes are absolute, and they come with less consistency problems and irrelevant properties. There also seems to be more diversity here, with stables and a possible stage "cardinal" equivalent.

Stages

This one was formulated by the googology community. Instead of regarding symbols used in OCFs as ordinals, we regard them as symbols which act like an ordinal and aren't. If you can't tell already, this solution came about when people realized that the stage cardinal probably wasn't a real cardinal at all.