[FOM] 768: Theory Completions
Harvey Friedman
hmflogic at gmail.com
Mon Sep 4 21:13:15 EDT 2017
SENTENCES IN PC(=) AS FINITE STRINGS IN A 16 LETTER ALPHABET
We work in PC(=), the first order predicate calculus with =. The
sentences in PC(=) are viewed as finite strings from a finite
alphabet. This finite alphabet consists of the following alphabet V of
16 letters:
c R F v not and or implies iff forall therexists 0 1 , ( )
The 0,1 are used for the base 2 representations of subscripts on
c,v,R,F, and for superscripts on R,F. Subscripts and superscripts are
from 1,2,3,... . Comma is used to separate the superscripts from the
subscripts. I.e., the m-th n-ary function symbol F^n_m is written Fn,m
where the latter n,m are in base 2 and n,m >= 1. Analogously for the
m-th n-ary relation symbol R^n_m.
Let gamma be a listing of V without repetition. We now have the strict
linear ordering <_gamma on the sentences in PC(=), which orders the
sentences first according to their length (as a string from A), and
then lexicographically according to gamma.
We use gamma because any choice of ordering of the 16 symbols in V
seems particularly ad hoc.
LISTING SENTENCES IN PC(=)
A theory T in PC(=) consists of a set of constant, relation, and
function symbols, including =, together with a set T of sentences in
those symbols.
Let gamma be a listing of V without repetition. Now list the sentences
in the language of T according to <gamma, as
A_1 <gamma A_2 <gamma A_3 <gamma ...
THE PREFERRED COMPLETION
Let T be a theory in PC(=) and gamma be a listing of V without
repetition. The gamma-completion of T is constructed as follows. Below
-A is not(A).
Suppose +-A_1,...,+-A_k have been defined, k >= 0. Use A_k+1 if T +
{+-A_1,...,+-A_k} is consistent. Otherwise use -A_k+1.
The negative gamma-completion of T is constructed as follows:
Suppose +-A_1,...,+-A_k have been defined, k >= 0. Use -A_k+1 if T +
{+-A_1,...,+-A_k} is consistent. Otherwise use A_k+1.
QUESTIONS
1. Let T be PA in the language c_1, F^1_1, F^2_1, F^2_2, =. What is
the nature of the gamma-completion of T and the negative
gamma-completion of T, where gamma is the default list above: c R F v
not and or implies iff forall therexists 0 1 , ( )., or for the
various 16! gammas? How strong are these gamma-completions of T? E.g,
do either or both of these prove or refute Con(PA)? Con(ZFC)?
Con(SRP)? Con(ZFC + I1)?
2. Let T be ZFC in the language R^2_1, =. What is the nature of the
gamma-completion of T and the negative gamma-completion of T, where
gamma is the default list above: c R F v not and or implies iff forall
therexists 0 1 , ( )., or for the various 16! gammas? How strong are
these gamma-completions of T? E.g, do either or both of these prove or
refute Con(ZFC)? Con(SRP)? Con(ZFC + I1)? CH? GCH? V = L?
************************************************************************
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 768th 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
Harvey Friedman
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