[FOM] 581: Impossible Counting/more

[Harvey Friedman]

In http://www.cs.nyu.edu/pipermail/fom/2015-May/018752.html there is a
silly typo. gmail wrote

I said that PA is interpretable in RCA_0 + BWQ. This shows very
fantastically that Con(RCA_0 + BWQ) implies Con(PA). However, RCA_0 +
BWQ does not prove Con(PA).

when I meant

fI said that PA is interpretable in RCA_0 + BWQ. This shows very
FINITISTICALLY that Con(RCA_0 + BWQ) implies Con(PA). However, RCA_0 +
BWQ does not prove Con(PA).

In fact, one also gets that

RCA_0 + BWQ proves the consistency of every finite fragment of PA

and the above is also proved very finitistically.

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I updated the manuscript to

https://u.osu.edu/friedman.8/foundational-adventures/downloadable-manuscripts/
#88. Impossible Counting, June 2, 2015, 10 pages, draft.

I added that Theta'(k) is recursively unsolvable if we use only finite
bops. Also this at the end;

We can measure how well a standard extension does in computing
Theta(12). Specifically, let Theta(12,T) be the least n such that T
proves that Theta(12) <= n.

THEOREM 3.4. Let T,T' be standard extensions as in Theorem 3.3,
neither proving the inconsistency of itself. If T' proves the
consistency of T then Theta(12,T') < Theta(12,T).

We have not been precise about the exact finite list of standard
extensions (of ZFC and ZF by large cardinal hypotheses) that we are
using. This will be presented at a later time. The list will include

ZFC + "there is a nontrivial elementary embedding from some
V(lambda+1) into V(lambda+1)".

NBG + "there is a nontrivial elementary embedding from V into V".

************************************************************
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This is the 581st 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 #1-527 can be found at the FOM posting
http://www.cs.nyu.edu/pipermail/fom/2014-August/018092.html

528: More Perfect Pi01  8/16/14  5:19AM
529: Yet more Perfect Pi01 8/18/14  5:50AM
530: Friendlier Perfect Pi01
531: General Theory/Perfect Pi01  8/22/14  5:16PM
532: More General Theory/Perfect Pi01  8/23/14  7:32AM
533: Progress - General Theory/Perfect Pi01 8/25/14  1:17AM
534: Perfect Explicitly Pi01  8/27/14  10:40AM
535: Updated Perfect Explicitly Pi01  8/30/14  2:39PM
536: Pi01 Progress  9/1/14 11:31AM
537: Pi01/Flat Pics/Testing  9/6/14  12:49AM
538: Progress Pi01 9/6/14  11:31PM
539: Absolute Perfect Naturalness 9/7/14  9:00PM
540: SRM/Comparability  9/8/14  12:03AM
541: Master Templates  9/9/14  12:41AM
542: Templates/LC shadow  9/10/14  12:44AM
543: New Explicitly Pi01  9/10/14  11:17PM
544: Initial Maximality/HUGE  9/12/14  8:07PM
545: Set Theoretic Consistency/SRM/SRP  9/14/14  10:06PM
546: New Pi01/solving CH  9/26/14  12:05AM
547: Conservative Growth - Triples  9/29/14  11:34PM
548: New Explicitly Pi01  10/4/14  8:45PM
549: Conservative Growth - beyond triples  10/6/14  1:31AM
550: Foundational Methodology 1/Maximality  10/17/14  5:43AM
551: Foundational Methodology 2/Maximality  10/19/14 3:06AM
552: Foundational Methodology 3/Maximality  10/21/14 9:59AM
553: Foundational Methodology 4/Maximality  10/21/14 11:57AM
554: Foundational Methodology 5/Maximality  10/26/14 3:17AM
555: Foundational Methodology 6/Maximality  10/29/14 12:32PM
556: Flat Foundations 1  10/29/14  4:07PM
557: New Pi01  10/30/14  2:05PM
558: New Pi01/more  10/31/14 10:01PM
559: Foundational Methodology 7/Maximality  11/214  10:35PM
560: New Pi01/better  11/314  7:45PM
561: New Pi01/HUGE  11/5/14  3:34PM
562: Perfectly Natural Review #1  11/19/14  7:40PM
563: Perfectly Natural Review #2  11/22/14  4:56PM
564: Perfectly Natural Review #3  11/24/14  1:19AM
565: Perfectly Natural Review #4  12/25/14  6:29PM
566: Bridge/Chess/Ultrafinitism 12/25/14  10:46AM
567: Counting Equivalence Classes  1/2/15  10:38AM
568: Counting Equivalence Classes #2  1/5/15  5:06AM
569: Finite Integer Sums and Incompleteness  1/515  8:04PM
570: Philosophy of Incompleteness 1  1/8/15 2:58AM
571: Philosophy of Incompleteness 2  1/8/15  11:30AM
572: Philosophy of Incompleteness 3  1/12/15  6:29PM
573: Philosophy of Incompleteness 4  1/17/15  1:44PM
574: Characterization Theory 1  1/17/15  1:44AM
575: Finite Games and Incompleteness  1/23/15  10:42AM
576: Game Correction/Simplicity Theory  1/27/15  10:39 AM
577: New Pi01 Incompleteness  3/7/15  2:54PM
578: Provably Falsifiable Propositions  3/7/15  2:54PM
579: Impossible Counting  5/26/15  8:58PM
580: Goedel's Second Revisited  5/29/15  5:52 AM

Harvey Friedman