User blog comment:PsiCubed2/How to make Deedlit's Mahlo-level notation more intuitive/@comment-35470197-20180807000338/@comment-30754445-20180808095604

"Oh, so your statement 'Anybody with even a cursory understanding of Deedlit's notation could tell you that, without needing all the heavy set-theoretical stuff you've mentioned.' just means that they can state what I verified, right...? It is trivial."

No, what I'm saying is deeper than that.

I'm saying that without the assumption that M > I(1,0,0), Deedlit's notation makes absolutely no sense and this entire discussion would have been pointless. It's part of a common ground we all accept "on faith" (or more specificallly: because we know that it was proven in the past, even without seeing or understanding those proofs).

So there's no point in retreading the precise proof for this, any more than there's a point to cite the elementary set-theoretic proof that 2+2=4. At least, not if you trust that Deedlit knows what he is doing (which you previously stated that you do). "But in order to solve arguments like what we did, we need precise proofs, don't we?"Sure.

But what "argument" did your linked proofs resolve? You just gave proof to something we already knew. There was never any disagreement about the fact that M>I(1,0,0).

Besides, there's little point in providing a proof that the other person can't follow, is there? "Some mathematician says so, and here is their reasoning (which you won't be able to follow)" isn't a more persuasive argument than "Deedlit says so".

I would also like to remind both you and others (especially those that the word "proof" fills them with dread): A "proof" does not need to be written in fancy and mysterious mathematical language. Nor does is need to rely on obscure axioms that only a select few are able to understand.

Here is what a proof is:

A step-by-step logical argument, that begins with a core of assumptions that we all agree upon, and ends with the certain conclusion.

That's all it is. Nothing mysterious or difficult. And certainly nothing to be afraid of.

And as a person who values rigorousness, I think that exercising our logical ability to prove our own personal claims is far more useful then retreading complicated proofs of basic things we already know.

"Or if you have more elementary proofs, I would like to learn them in order to improve my ability to deal with Mahlo. Since I am not so good at set theory, I thought that I need to interpret (1,0)-weak inaccessibility into a statement in the cumulative universe in order to use the stationarity condition."

You're apparently better in set theory than me, because I haven't understood what you just said.

So unfortunately, I cannot help you here.

The good news is that the precise properties of Mahlos don't really matter for understanding their use in collapsing fuctions (they are necessary to justify what people like Rathjen and Deedlit are doing, but not to understanding how these systems work).

"To be more precise, "undecidable" should be "unprovable", because we do not know the negation is also unprovable. Remember that the consistency of Mahlo is formally unorovable under the consistency of ZFC. It might occur that the existence of a Mahlo cardinal causes contradiction."

Thanks for the correction.