[FOM] 222:Pi01 Statements/getting it right again

[Harvey Friedman]

There was a minor bug in #221. Here is the corrected version.

NEW INDEPENDENT Pi01 STATEMENTS
Harvey M. Friedman
October 9, 2004

We present a new explicitly P01 statement provable from Mahlo cardinals of
finite order but not from ZFC. We first state two infinite statements
(Theorem 1 and Proposition 2).

Let N be the set of all nonnegative integers. We say that A containedin Nk
is nontrivial if and only if A is not the empty set and not Nk.

We treat Cartesian products of Cartesian products as a single Cartesian
product. In particular, we identify Nk x Nk with N2k. This identification is
important when we consider order invariance (see below).

Let R containedin Nk x Nk. We can think of R as a binary relation on
Nk, or as a subset of N2k.

We say that R is strictly dominating if and only if R(x,y) implies max(x) <
max(y).

For A containedin Nk, we write RA for the image of R on A as a binary
relation on Nk, which is given by RA = {y: (there exists x in A)(R(x,y))}.

THEOREM 1. Let k >=1 and R containedin Nk x Nk be strictly dominating. There
exists A containedin Nk such that RA = A'. Furthermore, A is unique.

Let x,y in Nr. We say that x,y are order equivalent if and only if for all 1
<= i,j <= r, 

x[i] < x[j] if and only if y[i] < y[j].

We say that x,y are factorial equivalent if and only if for all 1 <= i,j <=
r and p >= 0,

x[i] = p! if and only if y[i] = p!.

We say that x,y are order/factorial equivalent if and only x,y are order
equivalent and factorial equivalent.

We say that R containedin Nr is order invariant if and only if
for all order equivalent x,y in Nr, x in R iff y in R.

PROPOSITION 2. Let k >= 8 and R containedin Nk x Nk be strictly dominating
and order invariant. There exists nontrivial A containedin Nk such that
every element of A x A x RA x A' is order/factorial equivalent to an element
of A x A x A' x RA in which k!!-1 does not appear.
 
NOTE: Proposition 2 is a trivial consequence of Theorem 1 if we remove
"k!!! -1 does not appear". This is because if RA = A' then obviously A x A x
RA x A' = A x A x A' x RA.

NOTE: We have used factorials and k!!-1 only because the statement looks so
uncluttered. The k!!-1 and k >= 8 are not even remotely tight. We will
tighten them up later.

In a slight abuse of notation, we we say that R containedin [1,n]r is order
invariant if and only if for all order equivalent x,y in [1,n]r, x in R iff
y in R. 

Here is an explicitly Pi01 statement.

PROPOSITION 3. Let k,n >= 8 and R containedin [1,n]k x [1,n]k be strictly
dominating and order invariant. There exists nontrivial A containedin [1,n]k
such that every element of A x A x RA x A' is order/factorial equivalent to
an element of A x A x A' x RA in which k!!-1 does not appear.

THEOREM 4. Propositions 2,3 is provably equivalent, over ACA, to the
consistency of MAH = ZFC + {there exists a k-Mahlo cardinal}k. In
particular, Propositions 2,3 are provable in MAH+ = ZFC + "for all k there
exists a k-Mahlo cardinal", and cannot be proved in ZFC (provided ZFC is
consistent).

We now introduce a new parameter, r.

PROPOSITION 5. Let k,r >= 3 and R containedin Nk x Nk be strictly dominating
and order invariant. There exists nontrivial A containedin Nk such that
every element of A x A x RA x A' is order/factorial equivalent to an element
of A x A x A' x RA in which (kr)!!-1 does not appear.

PROPOSITION 6. Let k,r,n >= 3 and R containedin [1,n]k x [1,n]k be strictly
dominating and order invariant. There exists nontrivial A containedin [1,n]k
such that every element of A x A x RA x A' is order/factorial equivalent to
an element of A x A x A' x RA in which (kr)!!-1 does not appear.

THEOREM 7. Propositions 5.6 for any fixed k is provable in MAH. This is
false for ZFC together with any "there exists a k-Mahlo cardinal", k fixed.
Propositions 5.6 for k = 3 is not provable in ZFC (provided ZFC is
consistent).

These results are related to BRT = Boolean relation theory, but serve a
somewhat different purpose. BRT has a particularly strong thematic
character, with potential points of contact with perhaps all areas of
mathematics. These Propositions are simply the most mathematically natural
explicitly Pi01 statements independent of ZFC that we have been able to find
- yet.

*********************************************

I use www.math.ohio-state.edu/~friedman/ for downloadable manuscripts.
This is the 222nd 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 4/22/03  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
180:Provable Functions of PA  6/16/03  12:42AM
181:Strict Reverse Mathematics 2:06/19/03  2:06AM
182:Ideas in Proof Checking 1  6/21/03 10:50PM
183:Ideas in Proof Checking 2  6/22/03  5:48PM
184:Ideas in Proof Checking 3  6/23/03  5:58PM
185:Ideas in Proof Checking 4  6/25/03  3:25AM
186:Grand Unification 1  7/2/03  10:39AM
187:Grand Unification 2 - saving human lives 7/2/03 10:39AM
188:Applications of Hilbert's 10-th 7/6/03  4:43AM
189:Some Model theoretic Pi-0-1 statements  9/25/03  11:04AM
190:Diagrammatic BRT 10/6/03  8:36PM
191:Boolean Roots 10/7/03  11:03 AM
192:Order Invariant Statement 10/27/03 10:05AM
193:Piecewise Linear Statement  11/2/03  4:42PM
194:PL Statement/clarification  11/2/03  8:10PM
195:The axiom of choice  11/3/03  1:11PM
196:Quantifier complexity in set theory  11/6/03  3:18AM
197:PL and primes 11/12/03  7:46AM
198:Strong Thematic Propositions 12/18/03 10:54AM
199:Radical Polynomial Behavior Theorems
200:Advances in Sentential Reflection 12/22/03 11:17PM
201:Algebraic Treatment of First Order Notions 1/11/04 11:26PM
202:Proof(?) of Church's Thesis 1/12/04 2:41PM
203:Proof(?) of Church's Thesis - Restatement 1/13/04 12:23AM
204:Finite Extrapolation 1/18/04 8:18AM
205:First Order Extremal Clauses 1/18/04 2:25PM
206:On foundations of special relativistic kinematics 1 1/21/04 5:50PM
207:On foundations of special relativistic kinematics 2  1/26/04  12:18AM
208:On foundations of special relativistic kinematics 3  1/26/04  12:19AAM
209:Faithful Representation in Set Theory with Atoms 1/31/04 7:18AM
210:Coding in Reverse Mathematics 1  2/2/04  12:47AM
211:Coding in Reverse Mathematics 2  2/4/04  10:52AM
212:On foundations of special relativistic kinematics 4  2/7/04  6:28PM
213:On foundations of special relativistic kinematics 5  2/8/04  9:33PM
214:On foundations of special relativistic kinematics 6  2/14/04 9:43AM
215:Special Relativity Corrections  2/24/04 8:13PM
216:New Pi01 statements  6/6/04  6:33PM
217:New new Pi01 statements  6/13/04  9:59PM
218:Unexpected Pi01 statements  6/13/04  9:40PM
219:Typos in Unexpected Pi01 statements  6/15/04  1:38AM
220:Brand New Corrected Pi01 Statements  9/18/04  4:32AM
221:Pi01 Statements/getting it right  10/7/04  5:56PM

Harvey Friedman