# [FOM] Trivial bugs

Harvey Friedman friedman at math.ohio-state.edu
Thu Oct 7 21:59:00 EDT 2004

```The Pi01 statements in posting #221 earlier today have a trivial bug. E.g.,
look at

PROPOSITION 2. Let k >= 8 and R containedin Nk x Nk be strictly dominating
and order invariant. There exists A containedin Nk such that every element
of A x A x RA x A' is order/factorial equivalent to an element of A x A x A'
x RA in which k!!-1 does not appear.

Well, if we take A to be empty, then the conclusion is vacuously true. Also,
if we take A to be Nk, then the conclusion is vacuously true.

I am doing some serous checking for the obvious fix, which feels right:

PROPOSITION 2*. Let k >= 8 and R containedin Nk x Nk be strictly dominating
and order invariant. There exists nontrivial A containedin Nk such that
every element of A x A x RA x A' is order/factorial equivalent to an element
of A x A x A' x RA in which k!!-1 does not appear.

I.e., it seems like I just have to get the reversal started.

Of course, now I have to check that Proposition 2* does follow from the
Mahlo cardinals of finite order. This looks fine.

Also, I am checking a simplification of Proposition 2*.

Hopefully, I can make a numbered posting shortly.

Bear with me...

Harvey Friedman

```