[FOM] 648: Necessary Irrelevance 1

[Harvey Friedman]

Let PA(prim) be the usual formulation of Peano Arithmetic with all
primitive recursive function symbols.

For primitive recursive function symbols F, let PA(F) be the fragment
corresponding to F. This consists of the usual axioms for successor,
the equational axioms for F and all primitive recursive function
symbols used for the introduction of F and all instances of induction
that use only successor, F, and the primitive recursive function
symbols used for the introduction of F.

THEOREM 1. There is a unary primitive recursive function symbol F such
that the statement "F is positive everywhere" is
i. Provable in PA(prim).
ii. Not provable in PA(F).

We can sharpen i) to HA(prim), since HA(prim) and PA(prim) prove the
same sentences of the form "F is positive everywhere".

The lesson is that in order to prove the PA(prim) theorem "F is
positive everywhere", we are forced to use "irrelevant" parts of
PA(prim).

There is a yet sharper form of Theorem 1. Let PA(F,G)* be PF(F)
together with the set of all true sentences that involve only the
primitive recursive function symbols used for the introduction of F
(including successor but excluding F itself).

THEOREM 2. There exists a (unary) primitive recursive function symbol
F such that the statement "F is positive everywhere" is
i. Provable in PA(prim).
ii. Not provable in PA(F)*.

The idea of the proof is to first introduce an infrastructure of
finitely many primitive recursive function symbols that treat a
convenient fragment of (N,<,+,base 2 exponentiation), whose first
order theory is known to be tame, and completely handled well within
PRA. Note that this structure has a convenient pairing function.
We then introduce binary F, simulating the search for an inconsistency
in PRA. Finding an inconsistency in PRA corresponds to F
having a zero, and things are set up so that if F(n) = 0 then
automatically F(m) = 0 for m > n. Then in any model of PRA + not
Con(PRA), we see that PA(F) plus the true sentences of (N,<,+,base 2
exponentiation) + not "F is positive everywhere" holds.

This argument establishes the following sharper formulation.

THEOREM 3. Let k >= 1. There exists a (unary) primitive recursive
function symbol
F such that the statement "F is positive everywhere" is
i. Provable in PA(prim).
ii. Not provable in PA(F)* + Sigma-0-k Induction.

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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 648th 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 #1-599 can be found at the FOM posting
http://www.cs.nyu.edu/pipermail/fom/2015-August/018887.html

600: Removing Deep Pathology 1  8/15/15  10:37PM
601: Finite Emulation Theory 1/perfect?  8/22/15  1:17AM
602: Removing Deep Pathology 2  8/23/15  6:35PM
603: Removing Deep Pathology 3  8/25/15  10:24AM
604: Finite Emulation Theory 2  8/26/15  2:54PM
605: Integer and Real Functions  8/27/15  1:50PM
606: Simple Theory of Types  8/29/15  6:30PM
607: Hindman's Theorem  8/30/15  3:58PM
608: Integer and Real Functions 2  9/1/15  6:40AM
609. Finite Continuation Theory 17  9/315  1:17PM
610: Function Continuation Theory 1  9/4/15  3:40PM
611: Function Emulation/Continuation Theory 2  9/8/15  12:58AM
612: Binary Operation Emulation and Continuation 1  9/7/15  4:35PM
613: Optimal Function Theory 1  9/13/15  11:30AM
614: Adventures in Formalization 1  9/14/15  1:43PM
615: Adventures in Formalization 2  9/14/15  1:44PM
616: Adventures in Formalization 3  9/14/15  1:45PM
617: Removing Connectives 1  9/115/15  7:47AM
618: Adventures in Formalization 4  9/15/15  3:07PM
619: Nonstandardism 1  9/17/15  9:57AM
620: Nonstandardism 2  9/18/15  2:12AM
621: Adventures in Formalization  5  9/18/15  12:54PM
622: Adventures in Formalization 6  9/29/15  3:33AM
623: Optimal Function Theory 2  9/22/15  12:02AM
624: Optimal Function Theory 3  9/22/15  11:18AM
625: Optimal Function Theory 4  9/23/15  10:16PM
626: Optimal Function Theory 5  9/2515  10:26PM
627: Optimal Function Theory 6  9/29/15  2:21AM
628: Optimal Function Theory 7  10/2/15  6:23PM
629: Boolean Algebra/Simplicity  10/3/15  9:41AM
630: Optimal Function Theory 8  10/3/15  6PM
631: Order Theoretic Optimization 1  10/1215  12:16AM
632: Rigorous Formalization of Mathematics 1  10/13/15  8:12PM
633: Constrained Function Theory 1  10/18/15 1AM
634: Fixed Point Minimization 1  10/20/15  11:47PM
635: Fixed Point Minimization 2  10/21/15  11:52PM
636: Fixed Point Minimization 3  10/22/15  5:49PM
637: Progress in Pi01 Incompleteness 1  10/25/15  8:45PM
638: Rigorous Formalization of Mathematics 2  10/25/15 10:47PM
639: Progress in Pi01 Incompleteness 2  10/27/15  10:38PM
640: Progress in Pi01 Incompleteness 3  10/30/15  2:30PM
641: Progress in Pi01 Incompleteness 4  10/31/15  8:12PM
642: Rigorous Formalization of Mathematics 3
643: Constrained Subsets of N, #1  11/3/15  11:57PM
644: Fixed Point Selectors 1  11/16/15  8:38AM
645: Fixed Point Minimizers #1  11/22/15  7:46PM
646: Philosophy of Incompleteness 1  Nov 24 17:19:46 EST 2015
647: General Incompleteness almost everywhere 1  11/30/15  6:52PM

Harvey Friedman