1.2SomeBRTSet062210

# 1.2SomeBRTSet062210 - 1 1.2 Some BRT settings The BRT...

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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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2 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), c|x| 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), c|x| 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 more conditions, each of which is either a basic lower bound condition or a basic upper bound condition. Here any of these basic lower bound and upper bound conditions an be modified as in the previous paragraph.
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