[FOM] 578: Provably Falsifiable Proposiitons

Sat Mar 7 14:54:40 EST 2015

There is a well recognized key property of certain mathematical
propositions. Informally,

*if phi is false then it is automatically refutable*

We say that phi is "provably falsifiable". E.g., before FLT was
proved, it was well recognized that FLT is provably falsifiable. After
FLT was proved, FLT is of course seen to be provably falsifiable by
default.

DEFINITION 1. A sentence phi in the language of set theory is provably
falsifiable over ZFC if and only if the sentence "if phi is false then
phi is refutable in ZFC" is itself provable in ZFC.

Here is a stronger form.

DEFINITION 2. A sentence phi in the language of set theory is provably
falsifiable over ACA_0 if and only if the statement "if phi is false
then phi is refutable in ACA_0" is itself provable in ACA_0.

THEOREM 1. Let phi be a sentence in the language of ZFC. Suppose phi
is implicitly Pi01 in the sense that there is an algorithm alpha such
that "phi iff alpha goes on forever" is provable in ZFC (ACA_0). Then
phi is provably falsifiable over ZFC (ACA_0).

Theorem 1, even with ACA_0, applies to all of the propositions we have
presented recently in Concrete Mathematical Incompleteness, such as in
FOM posting #577. Thus they are all provably falsifiable over ACA_0.

The condition "provably falsifiable" is related to falsifiability of
physical theories. Generally speaking, physical theories that are not
falsifiable by observations have rather controversial reputations.
Such physical theories are often outright rejected as not being
meaningful by many physical scientists.

Will this kind of attitude be adopted by mathematicians? I.e., that in
order for a mathematical question to be regarded as truly significant,
must it be first seen to be provably falsifiable? This attitude has
already been perhaps arguably adopted by a significant segment of
applied mathematicians.

We can use the phrases

Provably Falsifiable Mathematical Incompleteness
Provably Falsifiable Concrete Mathematical Incompleteness

as alternatives to

Pi01 Mathematical Incompleteness.

Implicitly Pi01 sentences (over ZFC, ACA_0 respectively) are
automatically Provably Falsifiable (over ZFC, ACA_0 respectively). We
do not have to use Explicitly Pi01 sentences for Provable
Falsifiability.

The work on Concrete Mathematical Incompleteness provides the only
current examples of Provably Falsifiable Mathematical Incompleteness.
The first Provably Falsifiable Incompleteness is of course due to
Goedel.

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My website is at https://u.osu.edu/friedman.8/ and my youtube site is at
https://www.youtube.com/channel/UCdRdeExwKiWndBl4YOxBTEQ
This is the 578th 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  1/27/15 10:39AM
577: New Pi01 Incompleteness

Harvey Friedman