A uniform phase transition spanning from ISigma_d to ID_1

[Andreas Weiermann]

Dear FOM'ers,

maybe the following is of interest for some of you.


Let f denote a binary numbertheoretic function.

Let G(f) be the following statement:

For all K there exists M such that for all a_0,...,a_M :

if for all i<=M we have that a_i<= f(K,i)

then there exists an i<M such that a_i[i+2:=i+3]<=a_{i+1}.


Here a_i[i+2:=i+3] refers to the base change replacing all

occurrences of base i+2 by i+3 in the standard complete base i+2 
exponential

representation of a_i.

Let 2_l(x) denotes a tower of two's with an x on top.

Let f_d(K,i):=2_{d-1}((i+2)^K).

G(f_d) is then equivalent with the one consistency of ISigma_d for d>0.



Let us now assume a base i+2 representation for number as in Arai, 
Wainer and Weiermann BSL 2021

For f=F_\omega^K(i) the statement G(f) is independent of ATR_0.

For f=F_(epsilon_}^K(i) the statement G(f) is independent of ID_1, etc.

So one obtains a uniform statement and the normal form used will be the 
same in all statements.



The strength of the assertion G(f) only depends on the growth of the 
bounding function for the a_i.

So we have a phase transition phenomenon where the thresholds for one 
singular function correspond to logical systems.


All the best,

Andreas Weiermann