FOM: 120:Discrepancy Theory/4
Harvey Friedman
friedman at math.ohio-state.edu
Sat Jan 26 13:33:38 EST 2002
I have discovered that if one, at least, temporarily, abandons the quest
for perfect symmetry, then one gets some very good compensation. This is my
current favorite:
PROPOSITION 1. For all multivariate f1,f2,f3 from N into N of quadratic
growth, there exist infinite sets A1,A2,A3 containedin N obeying the system
of inclusions
A1 containedin Ai delta fjAk containedin Ai delta f1Ak+1.
Here is a more general form.
PROPOSITION 1. For all n,m >= 1 and multivariate f1,...,fn from N into N of
quadratic growth, there exist infinite sets A1,...,Am containedin N obeying
the system of inclusions
A1 containedin Ai delta fjAk containedin Ai delta f1Ak+1.
THEOREM 2. Propositions 1 and 2 are each provably equivalent to the
1-consistency of ZFC + {there exists an n-Mahlo cardinal}n over ACA.
Note that Propoisiton 1 is clearly part of the original Boolean relation
theory (for three functions and three sets). It does not need a tower
A1,...,Am, and it does not need a notion of largeness. Further, we are
considering only inclusions of the form
U delta gV containedin W delta hX
which has nice symmetry.
As usual we can use other notions of growth such as expansive linear
growth, or expansive linearly trapped.
**********************************************
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This is the 120th in a series of self contained postings to FOM covering
a wide range of topics in f.o.m. Previous ones counting from #100 are:
100:Boolean Relation Theory IV corrected 3/21/01 11:29AM
101:Turing Degrees/1 4/2/01 3:32AM
102: Turing Degrees/2 4/8/01 5:20PM
103:Hilbert's Program for Consistency Proofs/1 4/11/01 11:10AM
104:Turing Degrees/3 4/12/01 3:19PM
105:Turing Degrees/4 4/26/01 7:44PM
106.Degenerative Cloning 5/4/01 10:57AM
107:Automated Proof Checking 5/25/01 4:32AM
108:Finite Boolean Relation Theory 9/18/01 12:20PM
109:Natural Nonrecursive Sets 9/26/01 4:41PM
110:Communicating Minds I 12/19/01 1:27PM
111:Communicating Minds II 12/22/01 8:28AM
112:Communicating MInds III 12/23/01 8:11PM
113:Coloring Integers 12/31/01 12:42PM
114:Borel Functions on HC 1/1/02 1:38PM
115:Aspects of Coloring Integers 1/3/02 10:02PM
116:Communicating Minds IV 1/4/02 2:02AM
117:Discrepancy Theory 1/6/02 12:53AM
118:Discrepancy Theory/2 1/20/02 1:31PM
119:Discrepancy Theory/3 1/22.02 5:27PM
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