937: Logic of Real Numbers/1

[Harvey Friedman]

I am improving and strengthening the motivation of Tangible Incompleteness
from Core Mathematics beyond 936: Stable Maximality/Tangible
Incompleteness/2
https://cs.nyu.edu/pipermail/fom/2022-June/023390.html

and will be reporting on this soon. I'm now calling this Embedded
Maximality.

But now I just posted

https://u.osu.edu/friedman.8/foundational-adventures/downloadable-lecture-notes-2/

79. Strict Reverse Mathematics, 6/21/22, 15 pages.

This is the revised lecture notes for the talk delivered 6/14/22 at the
Workshop on Reverse Mathematics and its Philosophy, June 13-17, 2022,
Paris, France.SRM062122Paris’
<https://u.osu.edu/friedman.8/files/2022/06/SRM062122Paris.pdf>

I discussed the base theory for Countable SRM and the base theory for
Finite SRM. Also how SRM differs from RM and how I see the development of
SRM springing off of the base theories.

I stopped short getting into the SRM of real numbers. It appears that the
RM and SRM of even very basic real analysis is something that has not
really been gone into systematically, with much left to clarify both in RM
and in SRM.

Already there are several ways to even set up the reals that I'm not sure
have been explored deeply enough.

1. Reals as certain sets/functions in and between basic countable sets
(such as omega,Q). Equality is identity. Primary examples are left cuts,
and base n expansions not ending in all n-1. Infinite sequences of reals
are taken literally, using the standard flattening into a sets/functions
between basic countable sets. (such as omega,Q).

2. Reals as in 1 except equality is an appropriate equivalence relation.
Infinite sequences handled the same way as in 1. The conventional RM
approach is in this category, with explicit convergence estimates. An
alternative is merely Cauchy sequences.

3. Reals are taken as primitive - an SRM approach, over base theory ETF in
lecture note 78. Various systems of strictly mathematical theorems are
investigated. Equality is the identity.

1,2 can be pursued as a kind of countable combinatorics over omega,Q, of
course different than the usual countable combinatorics heavily
investigated in RM in that the field operations on Q play such a crucial
role. I expect 3 to break some entirely new ground in various respects.

##########################################

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 937th 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 #1-899 can be found at
http://u.osu.edu/friedman.8/foundational-adventures/fom-email-list/

900: Ultra Convergence/2  10/3/21 12:35AM
901: Remarks on Reverse Mathematics/6  10/4/21 5:55AM
902: Mathematical L and OD/RM  10/7/21  5:13AM
903: Foundations of Large Cardinals/1  10/12/21 12:58AM
904: Foundations of Large Cardinals/2  10/13/21 3:17PM
905: Foundations of Large Cardinals/3  10/13/21 3:17PM
906: Foundations of Large Cardinals/4  10/13/21  3:17PM
907: Base Theory Proposals for Third Order RM/1  10/13/21 10:22PM
908: Base Theory Proposals for Third Order RM/2  10/17/21 3:15PM
909: Base Theory Proposals for Third Order RM/3  10/17/21 3:25PM
910: Base Theory Proposals for Third Order RM/4  10/17/21 3:36PM
911: Ultra Convergence/3  1017/21  4:33PM
912: Base Theory Proposals for Third Order RM/5  10/18/21 7:22PM
913: Base Theory Proposals for Third Order RM/6  10/18/21 7:23PM
914: Base Theory Proposals for Third Order RM/7  10/20/21 12:39PM
915: Base Theory Proposals for Third Order RM/8  10/20/21 7:48PM
916: Tangible Incompleteness and Clique Construction/1  12/8/21   7:25PM
917: Proof Theory of Arithmetic/1  12/8/21  7:43PM
918: Tangible Incompleteness and Clique Construction/1  12/11/21  10:15PM
919: Proof Theory of Arithmetic/2  12/11/21  10:17PM
920: Polynomials and PA  1/7/22  4:35PM
921: Polynomials and PA/2  1/9/22  6:57 PM
922: WQO Games  1/10/22 5:32AM
923: Polynomials and PA/3  1/11/22  10:30 PM
924: Polynomials and PA/4  1/13/22  2:02 AM
925: Polynomials and PA/5  2/1/22  9::04PM
926: Polynomials and PA/6  2/1/22 11:20AM
927: Order Invariant Games/1  03/04/22  9:11AM
928: Order Invariant Games/2  03/7/22  4:22AM
929: Physical Infinity/randomness  3/21/22 02:14AM
930: Tangible Indiscernibles/1 05/07/22  7:46PM
931: Tangible Indiscernibles/2 5/14/22  1:34PM
932: Tangible Indiscernibles/3  5/14/22  1:34PM
933: Provable Functions of Set Theories/1 5/16/22  7/11AM
934: Provable Ordinals of Set Theories/1  5/17/22  8:35AM
935: Stable Maximality/Tangible Incompleteness/1  6/3/22  7:05PM
936: Stable Maximality/Tangible Incompleteness/2  *6/4/22  11:31PM*

Harvey Friedman
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