FOM: 33:A Variant
Harvey Friedman
friedman at math.ohio-state.edu
Thu Mar 4 07:52:14 EST 1999
This is the 33rd in a series of self contained postings to fom covering a
wide range of topics in f.o.m. Previous ones are:
1:Foundational Completeness 11/3/97, 10:13AM, 10:26AM.
2:Axioms 11/6/97.
3:Simplicity 11/14/97 10:10AM.
4:Simplicity 11/14/97 4:25PM
5:Constructions 11/15/97 5:24PM
6:Undefinability/Nonstandard Models 11/16/97 12:04AM
7.Undefinability/Nonstandard Models 11/17/97 12:31AM
8.Schemes 11/17/97 12:30AM
9:Nonstandard Arithmetic 11/18/97 11:53AM
10:Pathology 12/8/97 12:37AM
11:F.O.M. & Math Logic 12/14/97 5:47AM
12:Finite trees/large cardinals 3/11/98 11:36AM
13:Min recursion/Provably recursive functions 3/20/98 4:45AM
14:New characterizations of the provable ordinals 4/8/98 2:09AM
14':Errata 4/8/98 9:48AM
15:Structural Independence results and provable ordinals 4/16/98
10:53PM
16:Logical Equations, etc. 4/17/98 1:25PM
16':Errata 4/28/98 10:28AM
17:Very Strong Borel statements 4/26/98 8:06PM
18:Binary Functions and Large Cardinals 4/30/98 12:03PM
19:Long Sequences 7/31/98 9:42AM
20:Proof Theoretic Degrees 8/2/98 9:37PM
21:Long Sequences/Update 10/13/98 3:18AM
22:Finite Trees/Impredicativity 10/20/98 10:13AM
23:Q-Systems and Proof Theoretic Ordinals 11/6/98 3:01AM
24:Predicatively Unfeasible Integers 11/10/98 10:44PM
25:Long Walks 11/16/98 7:05AM
26:Optimized functions/Large Cardinals 1/13/99 12:53PM
27:Finite Trees/Impredicativity:Sketches 1/13/99 12:54PM
28:Optimized Functions/Large Cardinals:more 1/27/99 4:37AM
28':Restatement 1/28/99 5:49AM
29:Large Cardinals/where are we? I 2/22/99 6:11AM
30:Large Cardinals/where are we? II 2/23/99 6:15AM
31:First Free Sets/Large Cardinals 2/27/99 1:43AM
32:Greedy Constructions/Large Cardinals 3/2/99 11:21PM
A complete archiving of fom, message by message, is available at
http://www.math.psu.edu/simpson/fom/
Also, my series of self contained postings (only) is archived at
http://www.math.ohio-state.edu/foundations/manuscripts.html
FAVORITE SELF CONTAINED POSTINGS: 21, 25, 27, 31, 32, 33.
This short note presents a variant of the independent Proposition 2 from
posting 31. The current plan is to lead off with this alternative (see
Proposition 2*) below, and then immediately follow with the previous
Proposition 2 and further technical variants that don't mention F-free
sets.
Recall Proposition 2 from posting 31:
PROPOSITION 2. Let k,n >= 1 and F:N^k into N. There exists sets A[1]
containedin A[2] ... containedin A[n] containedin N such that
i) A[1] is an infinite set of odd integers;
ii) for all 1 <= i <= n-1, A[i+1] is the first F-free subset of A[i+1]
union A[i]+A[i].
Here is the variant:
PROPOSITION 2*. Let k,n >= 1 and F:N^k into N. There exists sets A[1]
containedin A[2] ... containedin A[n] containedin N such that for all 1 <=
i <= n-1, A[i+1] is the first F-free subset of A[i+1] union A[i]+A[i] with
infinitely many odd elements.
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