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1
1.2. Some BRT settings.
The BRT settings were defined in Definition 1.11.
Most areas of mathematics have a naturally associated
family of (multivariate) functions and sets. This usually
forms a natural and interesting BRT setting on which to do
BRT.
This book focuses on three principal BRT settings, as noted
below in subsection I. We have only been able to scratch
the surface of BRT on these principal settings.
As discussed in section 1.1, we have not completed EBRT in
the signature A,B,fA,fB on any of these principal settings,
and have only barely scratched the surface of EBRT in the
signature A,B,C,fA,fB,fC,gA,gB,gC on the setting (ELG,INF).
However, in the latter case, we know that particularly
simple
α
statements have exotic logical properties
(provability with large cardinals but not without).
In this section, we survey a huge range of mathematically
interesting BRT settings. This will give the reader a sense
of the unusual scope of BRT and a glimpse of what can be
expected in the future development of BRT.
We provide a plausible estimate that at least 1,000,000 of
these mathematically interesting BRT settings represent
significantly different BRT phenomena. Any substantial
probing of BRT on these settings is beyond the scope of
this book.
In sections 1.3 and 1.4, we will investigate the status of
the Complementation Theorem and the Thin Set Theorem in
some very modest sampling of the BRT settings presented in
this section. This will give a modest indication as to the
depth of BRT and its sensitivity to the choice of BRT
setting.
I. On N.
We now consider a number of natural conditions on functions
in MF. These conditions are of three kinds.
1. Bounding conditions.
2. Regularity conditions.
3. Choice of norm.
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The basic lower bound conditions that we propose are as
follows.
i. There exists c op i, d op’ j, such that for all x
∈
dom(f), cx
d
op’’ f(x). Here op,op’
∈
{<,>,
≤
,
≥
,=}, op’’
∈
{<,
≤
}, i,j
∈
{0,1/2,1,3/2,2}, and   is the l
∞
norm, the l
1
norm, or the l
2
norm.
The basic upper bound conditions that we propose are as
follows.
ii. There exists c op i, d op’ j, such that for all x
∈
dom(f), cx
d
op’’ f(x). Here op,op’
∈
{<,>,
≤
,
≥
,=}, op’’
∈
{>,
≥
}, i,j
∈
{0,1/2,1,3/2,2}, and   is the l
∞
norm, the l
1
norm, or the l
2
norm.
Each of these conditions in i,ii above results from the
choice of 6 parameters. Note that some of the choices of
parameters result in degenerate conditions.
Each of these basic lower bound conditions and basic upper
bound conditions can be modified by using “for all but
finitely many x” instead of “for all x”. This doubles the
number of lower and upper conditions, and the resulting
conditions are called the lower bound conditions and the
upper bound conditions.
The bounding conditions consist of a conjunction of zero or
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 Fall '08
 JOSHUA
 Math, Sets

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