User blog comment:Ubersketch/Why haven't people been formalizing and fixing things?/@comment-35470197-20181209220623/@comment-35470197-20181210011115

> Anyways we can easily solve these intuition-based OCF problems by simply treating the terms as general symbols rather than cardinals.

Well, depending on what "we" means. Googologists here tried such a natural stratesy many times by googologists here, but found that they had no idea to formalise a recursive (arithmetic) interpretation of the system of fundamental sequences of UNOCF. Namely, their formalisations became ill-defined, not well-founded, or too much weaker than they expected to call the resulting stuffs "formalisation".

So "you" might be able to do that, but others could not at this point. Interpreting an OCF-like function system into a (completely arithmetic) notation system is very difficult in general. It is much harder than to define a notation system associated to an actual (not intuition-based) OCF, because arithmetic is much weaker than set theory on presentability.

Well, if you do not mind the well-foundedness or the strength of the resulting notation system, then I agree with you. Then there are many trials. Take a look at the following examples:
 * Trial of formalising UNOCF by Syst3ms
 * Trial of formalising UNOCF by Syst3ms
 * Trial of formalising UNOCF by Syst3ms
 * Formalisation of PSS and its termination by me (72 page long in pdf in Japanese)
 * Formalisation of BMS by me

Then (even though you do not read them carefully) you will understand that the point is not the laziness, but the difficulty.

> An alternative would be to get these problems into the wider math world, even though that's way harder than formalizing them yourselves.

At least, any naive trials will give us not well-founded notation system, which is useless to analyse large functions. Guaranteeing the well-foundedness heavily depends on arguments with actual cardinals, because arithmetic is also much weaker than set theory on provability. That is why we need an actual OCF in order to ensure that a notation system works.