[FOM] 782: Revolutionary Possibilities/1
Harvey Friedman
hmflogic at gmail.com
Fri Jan 12 11:26:05 EST 2018
Here I want to raise some EXTREME possibilities for the impact of f.o.m. on
mathematics. I don't expect that these possibilities are going to be
realized this century in the extreme form that I state them. HOWEVER, I do
expect that they will be realized in substantially less extreme forms, but
still in strong enough form as to represent a major upheaval in mathematics
and mathematical thinking.
In fact, we already have realized such possibilities in various highly
suggestive forms that can't at this point be characterized as "a major
upheaval in mathematics and mathematical thinking". In fact, it is not yet
clear just how we would want to characterize these developments.
Right now, I am focused on two phenomena where there has been great
developments over a long period of time, and where the future
possibilities, of various levels of force, are extremely rich. These two
are, of course,
1. Algorithmic Unsolvability.
2. Incompleteness.
I wrote quite a bit about this in unnumbered postings, most recently in my
reply to Baldwin at
https://cs.nyu.edu/pipermail/fom/2018-January/020776.html
I want to start with reviewing and commenting on what I said there,
correcting some misstatements along the way.
Although 1,2 gives us a huge amount of future possibilities and upheavals
(particularly of conventional wisdoms), I would like to later venture far
beyond these old iconic 1,2.
1. No less than Carl Ludwig Siegel (CLS) is famously created for
> establishing a decision procedure for the solvability of quadratic
> equations in the integers. (By the way, somebody questioned whether
> this has been done for the rationals. References?)
>
I found this:
Over integers:
https://www.researchgate.net/publication/231866005_How_to_solve_a_quadratic_equation_in_integers
Over rationals:
https://academic.oup.com/blms/article-abstract/30/1/24/271972?redirectedFrom=fulltext
The former gives a decision procedure over Z, but I don't think the latter
does over Q in full generality.
> 2. We know that the solvability of systems of quadratic equations in
> the integers is Goedelian. So CLS is banging his head potentially
> against the Goedelian. POSSIBLE FUTURE: Very few simultaneous
> quadratic equations in very few variables are already algorithmically
> unsolvable.
>
EXTREME POSSIBILITY. Solving two simultaneous quadratic equations over the
integers or over the rationals is algorithmically unsolvable.
MORE EXTREME. This is the case even with, say, five unknowns. Also maybe
over Q.
> 3. At the most EXTREME that I know of, there is the possibility that
> {n^2 + m^3: n,m in Z} is algorithmically unsolvable. If you find that
> unimaginable, fool with that expression a tiny bit, and then see what
> you think.
>
This is NOT possible, as bounds are well known on the n,m over Z. Also if
coefficients are used. I.e., it is algorithmically solvable in the
coefficients, whether an^2 + bm^3 = c is solvable in integers a,b. However
EXTREME POSSIBILITY. {p^2 + q^3: p,q in Q} is algorithmically unsolvable.
EXTREME INCOMPLETENESS. There exists integers a,b,c of magnitudes at most
2^1000 such that the statement "ap^2 + bq^3 = c is solvable in rationals
p,q" is provably equivalent over EFA to the consistency of ZFC. Same with
ZFC replaced by "ZFC + various large cardinal axioms such as measurable
cardinals".
MORE EXTREME INCOMPLETENESS. a,b,c can be taken to be less than 1000.
Now let's be less extreme. Revisit all of the above, both nonrecursiveness
and incompleteness, for equations of these forms over Z or Q:
x^i + y^j = t
x^i + y^j + z^k = t
4. There is an integer 0 < r < 2^30 such that the solvability in
> integers of n^2 + m^3 = r (or use your favorite alternative very very
> simple expression, if you like, maybe with coeffieicnets also < 2^30 -
> is independent of the ZFC axioms. Provable from large cardinals, but
> not in ZFC.
>
See above.
> 5. It is possible to take, rather generally, simple combinatorial
> properties of long well orderings (too long for ZFC), which are
> sharper than well known such provable in the nonnegative integers,
> and, in a simple strategic uniform way, restate them in the rational
> numbers, so that the resulting statements can and can only be proved
> using large cardinals. As of December 2016, this is more or less
> essentially here.
>
CORRECTION: Replace December 2016 by the more recent December 2017. See
https://cs.nyu.edu/pipermail/fom/2017-December/020746.html See Emulation
Shift Theorem there.
> 6. 5 will be further pushed down into elementary number theoretic
> relations between finite sets of integers that will gradually be
> extremely simple and totally strategic. A probably significant but
> small step is in my recent series on 1-dimensional incompleteness.
>
See https://cs.nyu.edu/pipermail/fom/2018-January/020756.html
> 7. These Goedelian combinatorial matters will be slowly moved into the
> realm of finitely presented groups. Then the longstanding connection
> with geometry/topology will be firmed up for this, to put Goedel
> inexorably into geometry/topology.
>
E.g., there are two not too big combinatorially presented geometric
objects whose "equivalence" is provably equivalent to the consistency of
ZFC or ZFC + large cardinals.
A start would be to write down a small semigroup presentation whose
nontrivially is equivalent to the consistency of ZFC or ZFC + large
cardinals.
Also see my PIXELIZATION stuff at
http://u.osu.edu/friedman.8/foundational-adventures/downloadable-manuscripts/
#93
Much further progress in this is on the horizon.
> 8. At a more abstract level, we already know that for consistent
> single sentences in predicate calculus, there is no maximal such under
> interpretability. But the natural proof, and in fact the only proof I
> am aware of, passes through the heart of Goedel. OK, sure, you want to
> call predicate calculus Goedelian and remove it as well (at least the
> extreme form of Baldwin). But what about finite systems of equations,
> with the tacit "there exists at least two objects" - same question
> about maximal interpretation power. This also passes through Goedel.
> And this can be made even more algebraic.
>
I am referring here in 8 to simply a very convenient use of Goedelian
machinery to prove some fundamental algebraic/model theoretic facts - no
maximality theorems.
> 9. There will be an intelligible sentence in the usual (decidable)
> ordered field of real numbers with the following properties.
> i. It is true.
> ii. Any proof of it in ZFC is too large to fit in the existing worldwide
> web.
I.e., Quantitative Incompleteness. This has already started in several
contexts, but not nearly this strong. The idea is that there is no way to
take refuge even in very tame classical structures. This is already well
known from the asymptotic computational complexity point of view, much
less mined out for the finite computational complexity point of view, and
even less understood from the Incompleteness point of view.
************************************************************************
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 78
2nd
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 #1-699 can be found at
http://u.osu.edu/friedman.8/foundational-adventures/fom-email-list/
700: Large Cardinals and Continuations/14 8/1/16 11:01AM
701: Extending Functions/1 8/10/16 10:02AM
702: Large Cardinals and Continuations/15 8/22/16 9:22PM
703: Large Cardinals and Continuations/16 8/26/16 12:03AM
704: Large Cardinals and Continuations/17 8/31/16 12:55AM
705: Large Cardinals and Continuations/18 8/31/16 11:47PM
706: Second Incompleteness/1 7/5/16 2:03AM
707: Second Incompleteness/2 9/8/16 3:37PM
708: Second Incompleteness/3 9/11/16 10:33PM
709: Large Cardinals and Continuations/19 9/13/16 4:17AM
710: Large Cardinals and Continuations/20 9/14/16 1:27AM
700: Large Cardinals and Continuations/14 8/1/16 11:01AM
701: Extending Functions/1 8/10/16 10:02AM
702: Large Cardinals and Continuations/15 8/22/16 9:22PM
703: Large Cardinals and Continuations/16 8/26/16 12:03AM
704: Large Cardinals and Continuations/17 8/31/16 12:55AM
705: Large Cardinals and Continuations/18 8/31/16 11:47PM
706: Second Incompleteness/1 7/5/16 2:03AM
707: Second Incompleteness/2 9/8/16 3:37PM
708: Second Incompleteness/3 9/11/16 10:33PM
709: Large Cardinals and Continuations/19 9/13/16 4:17AM
710: Large Cardinals and Continuations/20 9/14/16 1:27AM
711: Large Cardinals and Continuations/21 9/18/16 10:42AM
712: PA Incompleteness/1 9/23/16 1:20AM
713: Foundations of Geometry/1 9/24/16 2:09PM
714: Foundations of Geometry/2 9/25/16 10:26PM
715: Foundations of Geometry/3 9/27/16 1:08AM
716: Foundations of Geometry/4 9/27/16 10:25PM
717: Foundations of Geometry/5 9/30/16 12:16AM
718: Foundations of Geometry/6 101/16 12:19PM
719: Large Cardinals and Emulations/22
720: Foundations of Geometry/7 10/2/16 1:59PM
721: Large Cardinals and Emulations//23 10/4/16 2:35AM
722: Large Cardinals and Emulations/24 10/616 1:59AM
723: Philosophical Geometry/8 10/816 1:47AM
724: Philosophical Geometry/9 10/10/16 9:36AM
725: Philosophical Geometry/10 10/14/16 10:16PM
726: Philosophical Geometry/11 Oct 17 16:04:26 EDT 2016
727: Large Cardinals and Emulations/25 10/20/16 1:37PM
728: Philosophical Geometry/12 10/24/16 3:35PM
729: Consistency of Mathematics/1 10/25/16 1:25PM
730: Consistency of Mathematics/2 11/17/16 9:50PM
731: Large Cardinals and Emulations/26 11/21/16 5:40PM
732: Large Cardinals and Emulations/27 11/28/16 1:31AM
733: Large Cardinals and Emulations/28 12/6/16 1AM
734: Large Cardinals and Emulations/29 12/8/16 2:53PM
735: Philosophical Geometry/13 12/19/16 4:24PM
736: Philosophical Geometry/14 12/20/16 12:43PM
737: Philosophical Geometry/15 12/22/16 3:24PM
738: Philosophical Geometry/16 12/27/16 6:54PM
739: Philosophical Geometry/17 1/2/17 11:50PM
740: Philosophy of Incompleteness/2 1/7/16 8:33AM
741: Philosophy of Incompleteness/3 1/7/16 1:18PM
742: Philosophy of Incompleteness/4 1/8/16 3:45AM
743: Philosophy of Incompleteness/5 1/9/16 2:32PM
744: Philosophy of Incompleteness/6 1/10/16 1/10/16 12:15AM
745: Philosophy of Incompleteness/7 1/11/16 12:40AM
746: Philosophy of Incompleteness/8 1/12/17 3:54PM
747: PA Incompleteness/2 2/3/17 12:07PM
748: Large Cardinals and Emulations/30 2/15/17 2:19AM
749: Large Cardinals and Emulations/31 2/15/17 2:19AM
750: Large Cardinals and Emulations/32 2/15/17 2:20AM
751: Large Cardinals and Emulations/33 2/17/17 12:52AM
752: Emulation Theory for Pure Math/1 3/14/17 12:57AM
753: Emulation Theory for Math Logic 3/10/17 2:17AM
754: Large Cardinals and Emulations/34 3/12/17 12:34AM
755: Large Cardinals and Emulations/35 3/12/17 12:33AM
756: Large Cardinals and Emulations/36 3/24/17 8:03AM
757: Large Cardinals and Emulations/37 3/27/17 2:39AM
758: Large Cardinals and Emulations/38 4/10/17 1:11AM
759: Large Cardinals and Emulations/39 4/10/17 1:11AM
760: Large Cardinals and Emulations/40 4/13/17 11:53PM
761: Large Cardinals and Emulations/41 4/15/17 4:54PM
762: Baby Emulation Theory/Expositional 4/17/17 1:23AM
763: Large Cardinals and Emulations/42 5/817 2:18AM
764: Large Cardinals and Emulations/43 5/11/17 12:26AM
765: Large Cardinals and Emulations/44 5/14/17 6:03PM
766: Large Cardinals and Emulations/45 7/2/17 1:22PM
767: Impossible Counting 1 9/2/17 8:28AM
768: Theory Completions 9/4/17 9:13PM
769: Complexity of Integers 1 9/7/17 12:30AM
770: Algorithmic Unsolvability 1 10/13/17 1:55PM
771: Algorithmic Unsolvability 2 10/18/17 10/15/17 10:14PM
772: Algorithmic Unsolvability 3 Oct 19 02:41:32 EDT 2017
773: Goedel's Second: Proofs/1 Dec 18 20:31:25 EST 2017
774: Goedel's Second: Proofs/2 Dec 18 20:36:04 EST 2017
775: Goedel's Second: Proofs/3 Dec 19 00:48:45 EST 2017
776: Logically Natural Examples 1 12/21 01:00:40 EST 2017
777: Goedel's Second: Proofs/4 12/28/17 8:02PM
778: Goedel's Second: Proofs/5 12/30/17 2:40AM
779: End of Year Claims 12/31/17 8:03PM
780: One Dimensional Incompleteness/1 1/4/18 1:14AM
781: One Dimensional Incompleteness/2 1/6/18 11:25PM
Harvey Friedman
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