# 856: Finite Increasing reducers/1

Harvey Friedman hmflogic at gmail.com
Tue Jun 16 17:15:59 EDT 2020

```We say that f s a finite increasing reducer if and only if
1) f is a finite partial function from some Q^k into Q.
2) for all x in dom(f), f(x) <= min(x)
3) for all x,y in dom(f), if x <= y coordinatewise then f(x) <= f(y).

The rank of p (in f) is the least number of f's needed to write p as a
term with only f's and the constant 0. If p cannot be so written then
its rank is infinity.

THEOREM 1. Let n >> k. There is no finite increasing reducer f such
that 1,2,...,n have ranks at most 1,2,...,n, respectively.

THEOREM 2. Theorem 1 is equivalent to the 1-consistency of arithmetic
transfinite induction on each cutoff of the proof theoretic ordinal
"theta sub capital omega to the little omega at 0", also known as the
ordinal of Kruskal's theorem. In fact, Theorem 1 is equivalent to my
finite form of Kruskal's theorem over EFA.

*****************

THEOREM 3. . Let K be a finite set of functions of the form c1 x1 +
... + ck xk + d, where c1,...,ck are rationals >= 1 and d is a
rational >= 0. Then starting with 0, K generates a well ordered set of
nonnegative rationals.

CONJECTURE. Theorem 3 is equivalent to "theta sub capital omega to the
little omega at 0", is well ordered.

CONJECTURE. One can read off a nice Pi02 sentence and a nice algorithm
associated with the sets generated by the K in Theorem 3, that are
equivalent to my finite form of Kruskal's theorem.

Already what happens with just a single c1 x1 + c2 x2 + d in Theorem
3, as in fusables?

Harvey Friedman

#######################################

My website is at https://u.osu.edu/friedman.8/ and my youtube site is at
This is the 855th in a series of self contained numbered

postings to FOM covering a wide range of topics in f.o.m. The list of
previous numbered postings #1-799 can be found at

800: Beyond Perfectly Natural/6  4/3/18  8:37PM
801: Big Foundational Issues/1  4/4/18  12:15AM
802: Systematic f.o.m./1  4/4/18  1:06AM
803: Perfectly Natural/7  4/11/18  1:02AM
804: Beyond Perfectly Natural/8  4/12/18  11:23PM
805: Beyond Perfectly Natural/9  4/20/18  10:47PM
806: Beyond Perfectly Natural/10  4/22/18  9:06PM
807: Beyond Perfectly Natural/11  4/29/18  9:19PM
808: Big Foundational Issues/2  5/1/18  12:24AM
809: Goedel's Second Reworked/1  5/20/18  3:47PM
810: Goedel's Second Reworked/2  5/23/18  10:59AM
811: Big Foundational Issues/3  5/23/18  10:06PM
812: Goedel's Second Reworked/3  5/24/18  9:57AM
813: Beyond Perfectly Natural/12  05/29/18  6:22AM
814: Beyond Perfectly Natural/13  6/3/18  2:05PM
815: Beyond Perfectly Natural/14  6/5/18  9:41PM
816: Beyond Perfectly Natural/15  6/8/18  1:20AM
817: Beyond Perfectly Natural/16  Jun 13 01:08:40
818: Beyond Perfectly Natural/17  6/13/18  4:16PM
819: Sugared ZFC Formalization/1  6/13/18  6:42PM
820: Sugared ZFC Formalization/2  6/14/18  6:45PM
821: Beyond Perfectly Natural/18  6/17/18  1:11AM
822: Tangible Incompleteness/1  7/14/18  10:56PM
823: Tangible Incompleteness/2  7/17/18  10:54PM
824: Tangible Incompleteness/3  7/18/18  11:13PM
825: Tangible Incompleteness/4  7/20/18  12:37AM
826: Tangible Incompleteness/5  7/26/18  11:37PM
827: Tangible Incompleteness Restarted/1  9/23/19  11:19PM
828: Tangible Incompleteness Restarted/2  9/23/19  11:19PM
829: Tangible Incompleteness Restarted/3  9/23/19  11:20PM
830: Tangible Incompleteness Restarted/4  9/26/19  1:17 PM
831: Tangible Incompleteness Restarted/5  9/29/19  2:54AM
832: Tangible Incompleteness Restarted/6  10/2/19  1:15PM
833: Tangible Incompleteness Restarted/7  10/5/19  2:34PM
834: Tangible Incompleteness Restarted/8  10/10/19  5:02PM
835: Tangible Incompleteness Restarted/9  10/13/19  4:50AM
836: Tangible Incompleteness Restarted/10  10/14/19  12:34PM
837: Tangible Incompleteness Restarted/11 10/18/20  02:58AM
838: New Tangible Incompleteness/1 1/11/20 1:04PM
839: New Tangible Incompleteness/2 1/13/20 1:10 PM
840: New Tangible Incompleteness/3 1/14/20 4:50PM
841: New Tangible Incompleteness/4 1/15/20 1:58PM
842: Gromov's "most powerful language" and set theory  2/8/20  2:53AM
843: Brand New Tangible Incompleteness/1 3/22/20 10:50PM
844: Brand New Tangible Incompleteness/2 3/24/20  12:37AM
845: Brand New Tangible Incompleteness/3 3/28/20 7:25AM
846: Brand New Tangible Incompleteness/4 4/1/20 12:32 AM
847: Brand New Tangible Incompleteness/5 4/9/20 1 34AM
848. Set Equation Theory/1 4/15 11:45PM
849. Set Equation Theory/2 4/16/20 4:50PM
850: Set Equation Theory/3 4/26/20 12:06AM
851: Product Inequality Theory/1 4/29/20 12:08AM
852: Order Theoretic Maximality/1 4/30/20 7:17PM
853: Embedded Maximality (revisited)/1 5/3/20 10:19PM
854: Lower R Invariant Maximal Sets/1:  5/14/20 11:32PM
855: Lower Equivalent and Stable Maximal Sets/1  5/17/20 4:25PM

Harvey Friedman
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