[FOM] 578: Provably Falsifiable Proposiitons
Arnon Avron
aa at tau.ac.il
Mon Mar 9 10:06:47 EDT 2015
As is well-known, Hilbert has divided mathematical
propositions into two types: the `real' ones and the
`ideal" ones, taking only the real ones as really
meaningful.
What is the *essential* difference, if any, between
Hilbert's real propositions and the "provably falsifiable" ones?
(Except of course that the latter is rigorously defined,
while the concept of real proposition was not, and that
"provably falsifiable" is relative to a theory, while
"real proposition" was not supposed to be).
Was not "that in order for a mathematical question to be regarded
as truly significant, it must be first seen to be provably falsifiable"
exactly Hilbert's view, on which his program was based?
Arnon
On Sat, Mar 07, 2015 at 02:54:40PM -0500, Harvey Friedman wrote:
> 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.
>
> ************************************************************
> 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
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