[FOM] 768: Theory Completions

[Harvey Friedman]

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?

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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 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