[FOM] 180:Provable Functions of PA
Harvey Friedman
friedman at math.ohio-state.edu
Mon Jun 16 00:42:13 EDT 2003
The provably recursive functions of PA have been well studied and are
well understood. They are the <epsilon_0 recursive functions.
For the purposes of this posting, we will assume that PA has been
formulated with primitive recursive function symbols. Also, here the
Sigma_n and Pi_n formulas are those with n alternating quantifiers
starting with an existential and universal quantifier, followed by a
quantifier free matrix. The Delta_0, Sigma_0, Pi-0 formulas, are the
quantifier free formulas.
In the general framework of this posting, the provably recursive
functions of PA are the functions f:N into N which are given by a
provable sentence in PA of the form
(forall n)(therexists unique m)(therexists r)(A(n,m,r))
where A is quantifier free.
I.e., the f:N into N such that there is a sentence of that form,
provable in PA, for which f is the unique outer Skolem function.
More generally, for a natural class K of formulas, we wish to
consider those f:N into N which are given by a provable sentence in
PA of the form
(forall n)(therexists unique m)(A(n,m))
where A is in K.
We call such a function a PA provable K function. Thus the provably
recursive functions of PA are the PA provable Sigma_1 functions.
If we omit the requirement that the sentence above is provable in PA,
then we say that such a function is a K function. Thus the recursive
functions are the Sigma_1 functions.
THEOREM 1. (Obvious). The Delta_0 characteristic functions are
exactly the primitive recursive characteristic functions.
THEOREM 2. The PA provable Delta_0 functions are exactly the Delta_0
functions that are bounded by a <epsilon_0 recursive function.
We now consider the PA provable Sigma_2 functions. The Sigma_2
functions are exactly the functions that are recursive in 0'. These
are also the same as the Delta_2 functions.
The first thing to realize is this.
THEOREM 3. Any recursive function is a PA provable Sigma_2 function.
This can be sharpened as follows.
THEOREM 4. Any function whose graph is a Boolean combination of r.e.
sets, is a PA provable Sigma_2 function.
Now for the characterization.
THEOREM 5. The PA provable Sigma_2 functions are exactly the
functions that are <epsilon_0 recursive in some Pi_1 function. In
particular, we can use the function H(n) = the number of steps it
takes for the n-th Turing machine to converge at the empty input tape
if it exists; 0 otherwise.
More generally, we have the following.
THEOREM 6. Let k >= 1. The PA provable Sigma_k functions are exactly
the functions that are <epsilon_0 recursive in some Pi_k-1 function.
*********************************************
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This is the 180th 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 #1-149 can be found at
http://www.cs.nyu.edu/pipermail/fom/2003-May/006563.html in the FOM
archives, 5/8/03 8:46AM. Previous ones counting from #150 are:
150:Finite obstruction/statistics 8:55AM 6/1/02
151:Finite forms by bounding 4:35AM 6/5/02
152:sin 10:35PM 6/8/02
153:Large cardinals as general algebra 1:21PM 6/17/02
154:Orderings on theories 5:28AM 6/25/02
155:A way out 8/13/02 6:56PM
156:Societies 8/13/02 6:56PM
157:Finite Societies 8/13/02 6:56PM
158:Sentential Reflection 3/31/03 12:17AM
159.Elemental Sentential Reflection 3/31/03 12:17AM
160.Similar Subclasses 3/31/03 12:17AM
161:Restrictions and Extensions 3/31/03 12:18AM
162:Two Quantifier Blocks 3/31/03 12:28PM
163:Ouch! 4/20/03 3:08AM
164:Foundations with (almost) no axioms, 4/22/0 5:31PM
165:Incompleteness Reformulated 4/29/03 1:42PM
166:Clean Godel Incompleteness 5/6/03 11:06AM
167:Incompleteness Reformulated/More 5/6/03 11:57AM
168:Incompleteness Reformulated/Again 5/8/03 12:30PM
169:New PA Independence 5:11PM 8:35PM
170:New Borel Independence 5/18/03 11:53PM
171:Coordinate Free Borel Statements 5/22/03 2:27PM
172:Ordered Fields/Countable DST/PD/Large Cardinals 5/34/03 1:55AM
173:Borel/DST/PD 5/25/03 2:11AM
174:Directly Honest Second Incompleteness 6/3/03 1:39PM
175:Maximal Principle/Hilbert's Program 6/8/03 11:59PM
176:Count Arithmetic 6/10/03 8:54AM
177:Strict Reverse Mathematics 1 6/10/03 8:27PM
178:Diophantine Shift Sequences 6/14/03 6:34PM
179:Polynomial Shift Sequences/Correction 6/15/03 2:24PM
Harvey Friedman
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