893: Remarks on Reverse Mathematics/3
Harvey Friedman
hmflogic at gmail.com
Thu Sep 23 22:04:10 EDT 2021
MODIFICATION OF 891, Remarks on Reverse Mathematics/1
We wrote:
"1. Identify an appropriate language for expressing statements in
finite mathematics. I do not like an open ended series of notions one
after the other. I have at least temporarily fixed on an appropriate
universal language for the expression of finite mathematics. See below
for my current thoughts."
Replace this with:
1. Identify an appropriate language for expressing statements in
finite mathematics. I do not like an open ended series of notions one
after the other. I am working on an appropriate universal language for
the expression of finite mathematics. There are some unresolved
issues.
MORE ON DEALING WITH THIRD ORDER OBJECTS IN ORDINARY RM
FIRST OBSERVATION: The way third order objects are presented in a
statement about third order objects will generally affect the logical
status of the statement.
SECOND OBSERVATION. It is an easy matter for ordinary RM to fully
accommodate the first observation.
For example. Let us consider a statement about all continuous
functions from R into R. In RM one usually just assumes a particular
way of presenting such functions which I call sequential
presentations. There are many ways to formalize sequential
presentations of continuous functions. There should be a big
robustness theorem that gives us confidence that any way of talking
about sequential presentations of these functions are equivalent. So
the entiere RM treatment of such functions is entirely preserved even
under the First Observation. How? We simply use the phrase
"sequentially presented continuous function" in the RM results.
The only alternative presentations of continuous functions from R into
R of a similar fundamental character that I am aware of are through
the Baire classes. I.e., Baire class alpha, where 1 <= alpha <=
omega_1.
So for example, there is the reverse mathematics of Baire class
omega_1 presented continuous functions. There should be a big
robustness theorem that any way of presenting Baire class alpha
functions, 1 <= alpha <= omega_1, are equivaelnt in an appropriate
sense. Then there is a perfectly robust important subject called
the theory of Baire class alpha presented continuous functions
where one can use various definitions of continuous here, perhaps most
naturally just the usual epsilon delta definition. The most
interesting cases are probably alpha = 1, 2, omega_1.
Now of course one should in general use Polish spaces rather than R.
So we have this kind of terminology:
sequentially continuous Polish functions
Baire class alpha presented continuous Polish functions
So this fully accommodates the vast bulk of mathematics commonly
presented using third order notions like functions from one Polish
space into another, within the usual RM framework.
So let's review the A-I from
https://cs.nyu.edu/pipermail/fom/2021-September/022876.html
A. Every sequence from X misses an element of X. Here X is a Polish
space (with an appropriate nontriviality condition).
B. No function from X into Y is injective. Here X is a nontrivial
Polish space and Y is an appropriately countable space.
C. No function from X into Y is bijective. Here X is a nontrivial
Polish space and Y is an appropriately countable space, but even wider
contexts may be very interesting.
D. Every monotone function has at most countably many discontinuities.
Here use R but maybe some other ordered spaces.
E. Every function of bounded variation is the difference between two
monotone functions.
F. A function is Riemann integrable if and only if it is continuous
almost everywhere.
G. A monotone function is differentiable almost everywhere.
H. The union of a sequence of sets of measure zero is of measure 0.
I. Countable additivity of Lebesgue measure.
We can do the reverse mathematics of A-I for the Baire class alpha
presented functions.
I DO NOT BELIEVE THAT ANY UNUSUALLY HIGH LOGICAL STRENGTHS ARISE FROM
SUCH A STUDY.
##########################################
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 890th 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 #1-799 can be found at
http://u.osu.edu/friedman.8/foundational-adventures/fom-email-list/
800: Beyond Perfectly Natural/6 4/3/18 8:37PM
801: Big Foundational Issues/1 4/4/18 12:15AM
802: Systematic f.o.m./1 4/4/18 1:06AM
803: Perfectly Natural/7 4/11/18 1:02AM
804: Beyond Perfectly Natural/8 4/12/18 11:23PM
805: Beyond Perfectly Natural/9 4/20/18 10:47PM
806: Beyond Perfectly Natural/10 4/22/18 9:06PM
807: Beyond Perfectly Natural/11 4/29/18 9:19PM
808: Big Foundational Issues/2 5/1/18 12:24AM
809: Goedel's Second Reworked/1 5/20/18 3:47PM
810: Goedel's Second Reworked/2 5/23/18 10:59AM
811: Big Foundational Issues/3 5/23/18 10:06PM
812: Goedel's Second Reworked/3 5/24/18 9:57AM
813: Beyond Perfectly Natural/12 05/29/18 6:22AM
814: Beyond Perfectly Natural/13 6/3/18 2:05PM
815: Beyond Perfectly Natural/14 6/5/18 9:41PM
816: Beyond Perfectly Natural/15 6/8/18 1:20AM
817: Beyond Perfectly Natural/16 Jun 13 01:08:40
818: Beyond Perfectly Natural/17 6/13/18 4:16PM
819: Sugared ZFC Formalization/1 6/13/18 6:42PM
820: Sugared ZFC Formalization/2 6/14/18 6:45PM
821: Beyond Perfectly Natural/18 6/17/18 1:11AM
822: Tangible Incompleteness/1 7/14/18 10:56PM
823: Tangible Incompleteness/2 7/17/18 10:54PM
824: Tangible Incompleteness/3 7/18/18 11:13PM
825: Tangible Incompleteness/4 7/20/18 12:37AM
826: Tangible Incompleteness/5 7/26/18 11:37PM
827: Tangible Incompleteness Restarted/1 9/23/19 11:19PM
828: Tangible Incompleteness Restarted/2 9/23/19 11:19PM
829: Tangible Incompleteness Restarted/3 9/23/19 11:20PM
830: Tangible Incompleteness Restarted/4 9/26/19 1:17 PM
831: Tangible Incompleteness Restarted/5 9/29/19 2:54AM
832: Tangible Incompleteness Restarted/6 10/2/19 1:15PM
833: Tangible Incompleteness Restarted/7 10/5/19 2:34PM
834: Tangible Incompleteness Restarted/8 10/10/19 5:02PM
835: Tangible Incompleteness Restarted/9 10/13/19 4:50AM
836: Tangible Incompleteness Restarted/10 10/14/19 12:34PM
837: Tangible Incompleteness Restarted/11 10/18/20 02:58AM
838: New Tangible Incompleteness/1 1/11/20 1:04PM
839: New Tangible Incompleteness/2 1/13/20 1:10 PM
840: New Tangible Incompleteness/3 1/14/20 4:50PM
841: New Tangible Incompleteness/4 1/15/20 1:58PM
842: Gromov's "most powerful language" and set theory 2/8/20 2:53AM
843: Brand New Tangible Incompleteness/1 3/22/20 10:50PM
844: Brand New Tangible Incompleteness/2 3/24/20 12:37AM
845: Brand New Tangible Incompleteness/3 3/28/20 7:25AM
846: Brand New Tangible Incompleteness/4 4/1/20 12:32 AM
847: Brand New Tangible Incompleteness/5 4/9/20 1 34AM
848. Set Equation Theory/1 4/15 11:45PM
849. Set Equation Theory/2 4/16/20 4:50PM
850: Set Equation Theory/3 4/26/20 12:06AM
851: Product Inequality Theory/1 4/29/20 12:08AM
852: Order Theoretic Maximality/1 4/30/20 7:17PM
853: Embedded Maximality (revisited)/1 5/3/20 10:19PM
854: Lower R Invariant Maximal Sets/1: 5/14/20 11:32PM
855: Lower Equivalent and Stable Maximal Sets/1 5/17/20 4:25PM
856: Finite Increasing reducers/1 6/18/20 4 17PM :
857: Finite Increasing reducers/2 6/16/20 6:30PM
858: Mathematical Representations of Ordinals/1 6/18/20 3:30AM
859. Incompleteness by Effectivization/1 6/19/20 1132PM :
860: Unary Regressive Growth/1 8/120 9:50PM
861: Simplified Axioms for Class Theory 9/16/20 9:17PM
862: Symmetric Semigroups 2/2/21 9:11 PM
863: Structural Mapping Theory/1 2/4/21 11:36PM
864: Structural Mapping Theory/2 2/7/21 1:07AM
865: Structural Mapping Theory/3 2/10/21 11:57PM
866: Structural Mapping Theory/4 2/13/21 12:47AM
867: Structural Mapping Theory/5 2/14/21 11:27PM
868: Structural Mapping Theory/6 2/15/21 9:45PM
869: Structural Proof Theory/1 2/24/21 12:10AM
870: Structural Proof Theory/2 2/28/21 1:18AM
871: Structural Proof Theory/3 2/28/21 9:27PM
872: Structural Proof Theory/4 2/28/21 10:38PM
873: Structural Proof Theory/5 3/1/21 12:58PM
874: Structural Proof Theory/6 3/1/21 6:52PM
875: Structural Proof Theory/7 3/2/21 4:07AM
876: Structural Proof Theory/8 3/2/21 7:27AM
877: Structural Proof Theory/9 3/3/21 7:46PM
878: Structural Proof Theory/10 3/3/21 8:53PM
879: Structural Proof Theory/11 3/4/21 4:22AM
880: Tangible Updates/1 4/15/21 1:46AM
881: Some Logical Thresholds 4/29/21 11:49PM
882: Logical Strength Comparability 5/8/21 5:49PM
883: Tangible Incompleteness Lecture Plans 5/16/21 1:29:44
884: Low Strength Zoo/1 5/16/21 1:34:
885: Effective Forms 5/16/21 1:47AM
886: Concerning Natural/1 5/16/21 2:00AM
887: Updated Adventures 9/9/21 9:47AM 2021
888: New(?) kinds of questions 9/9/21 12:32PM
889: Generating r.e. sets 9/12/21 3:38PM
890: Update on Tangible Incompleteness 9/18/21 9:50AM
891: Comments on Reverse Mathematics/1 9/23/21 3:45AM
892: Comments on Reverse Mathematics/2 9/21/21 8:37AM
Harvey Friedman
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