[FOM] 242:4th Pi01 Update
Harvey Friedman
friedman at math.ohio-state.edu
Sat Dec 18 21:47:33 EST 2004
1. We prefer to give forms without bounds that are easily equivalent to
forms with bounds. We prefer to give the forms with bounds separately, and
use the bound [2pk^2] on A. This is a presentation issue only.
2. There is nothing wrong with the symmetric difference versions. We put
them back.
3. We have modified the very strong statement, as we grasp what is really
needed to get lots of ranks into themselves.
I *think* that this round has settled down.
These developments in no way, shape, or form obsolete BRT.
PS: At this point, I move to finish the largely completed BRT book for
submission. The BRT book is not expected to include this Pi01 material, but
the final version of the book will include a summary of the state of the art
in Pi01.
##########################################
Let N be the set of all nonnegative integers. For k >= 1, N^k is the set of
all k-tuples from N.
Let T:N^k into N, and V containedin N^k. We define the upper image of T on V
by
T<[V] = {T(x): x in V and T(x) > max(x)}.
We use U. for disjoint union. Thus A U. B is A U B if A,B are disjoint;
undefined otherwise.
For n in N, we write [n] = {0,...,n}.
THEOREM 1. For all T:N^k into N, some A U. T<[A^k] = N. In fact, A is
unique.
Theorem 1 has the following finite form.
THEOREM 2. For all T:[p]^k into N, some A U. T<[A^k] = [max(rng(T))]. In
fact, A is unique and nonmepty.. Furthermore, we can use any integer >=
max(rng(T)).
We will use two classes of functions.
We write PL([p]^k,E) for the set of all piecewise linear transformations
T:[p]^k into N over E. These are the T:[p]^k into N defined by finitely many
cases, where each case is given by a finite set of linear inequalities, and
T is given by an affine expression with coefficients in each case, and where
all coefficients used in the inequalities and affine expressions lie in E.
We write RL([p]^k,E) for the set of all restricted linear transformations
T:[p]^k into N over E. These are the partial T:[p]^k into N defined by a
linear transformation restricted to a semilinear set, where all coefficients
used in the semilinear set and the linear expression lie in E. Here a
semilinear set is given by a Boolean combination of linear inequalities).
For vectors y in N^k, we define y! = (y_1!,...,y_k!). We take min(emptyset)
= 0.
PROPOSITION 3. For all T in PL([p]^2k,[k]), some A U. T<[A^2k] contains all
min(T[A^k x {y!}]) and omits (8k)!!-1.
PROPOSITION 4. For all T in RL([p]^2k,[k]), some A U. T<[A^2k] contains all
min(T[A^k x {y!}]) and omits (8k)!!-1.
PROPOSITION 5. For all T in PL([p]^2k,[k]), some A delta T<[A^2k] contains
all min(T[A^k x {y!}]) and omits (8k)!!-1.
PROPOSITION 6. For all T in RL([p]^2k,[k]), some A delta T<[A^2k] contains
all min(T[A^k x {y!}]) and omits (8k)!!-1.
It is clear that we need only consider A containedin [2pk^2] and y in
[p+1]^k, in Propositions 3-6. So at the cost of cluttering up the
statements, we can write them in explicitly Pi01 form as follows.
PROPOSITION 3'. For all k,p >= 1 and T in PL([p]^2k,[k]), some A U.
T<[A^2k], A containedin [2pk^2], contains all min(T[A^k x {y!}]), y in
[p+1]^k, and omits (8k)!!-1.
PROPOSITION 4'. For all k,p >= 1 and T in RL([p]^2k,[k]), some A U.
T<[A^2k], A containedin [2pk^2], contains all min(T[A^k x {y!}]), y in
[p+1]^k, and omits (8k)!!-1.
PROPOSITION 5'. For all k,p >= 1 and T in PL([p]^2k,[k]), some A delta
T<[A^2k], A containedin [2pk^2], contains all min(T[A^k x {y!}]), y in
[p+1]^k, and omits (8k)!!-1.
PROPOSITION 6'. For all k,p >= 1 and T in RL([p]^2k,[k]), some A delta
T<[A^2k], A containedin [2pk^2], contains all min(T[A^k x {y!}]), y in
[p+1]^k, and omits (8k)!!-1.
As things stabilize, we will sharpen the (8k)!!.
THEOREM 7. Theorem 1 is provable in RCA0 and Theorem 2 is provable in EFA.
Propositions 3-6 are each provably equivalent, over ACA, to the consistency
of MAH = ZFC + {there exists an n-Mahlo cardinal}_n. If we remove
"omits (8k)!!-1", then they become immediate consequences of Theorem 2, and
hence provable in EFA.
If we set p to be certain simple functions of k, rather than arbitrary, then
we can control the strength of Propositions 3-6 somewhat. We should be able
to get PA and n-th order arithmetic, for various n, as well as significant
fragments of ZFC, ZFC itself, and levels of the Mahlo hierarchy.
We now present much stronger Pi01 statements.
Let R1,...,Rt,S1,...,St be multivariate relations on N, where each Ri,Si are
of the same arity. Let A,B,V containedin N. We say that
f embeds (A,R1,...,Rt) into (B,S1,...,St) over D if and only if
i) f is a one-one function, where f[A] containedin B;
ii) for all x1,...,xn in dom(M) intersect V, Ri(x1,...,xn) iff
Si(f(x1),...,f(xn)), where the arity of Ri,Si is n.
For B containedin N^k and n in N, we write B|<n for the set of all elements
of B all of whose coordinates are < n.
For y in N^k, we write y! = (y1!,...,yk!).
PROPOSITION 8. For all T in PL([p]^6k,[k]) there exists A containedin [p]^3
such that for all i! < p, A_i! embeds ([i!],T,T[A^2k],T<[A^2k]) into
([i!]\{(8k)!!},T,T[A^2k|<i!],A_00') over the min(T[A^k x {y!}]).
Proposition 8 is obviously explicitly Pi01.
THEOREM 9. Proposition 8 is provable in VBC + "there exists a nontrivial
elementary embedding from V into some transitive class M such that V(lambda)
in M, where lambda is the first fixed point above the critical point", but
not in ZFC + "there is a proper class of cardinals kappa such that there is
a nontrivial elementary embedding of V(kappa) into V(kappa)". Proposition 8
proves the consistency of this latter system.
*************************************
I use www.math.ohio-state.edu/~friedman/ for downloadable manuscripts.
This is the 242nd 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
222:Statements/getting it right again 10/9/04 1:32AM
223:Better Pi01 Independence 11/2/04 11:15AM
224:Prettier Pi01 Independence 11/7/04 8:11PM
225:Better Pi01 Independence 11/9/04 10:47AM
226:Nicer Pi01 Independence 11/10/04 10:43AM
227:Progress in Pi01 Independence 11/11/04 11:22PM
228:Further Progress in Pi01 Independence 11/12/04 2:49AM
229:More Progress in Pi01 Independence 11/13/04 10:41PM
230:Piecewise Linear Pi01 Independence 11/14/04 9:38PM
231:More Piecewise Linear Pi01 Independence 11/15/04 11:18PM
232:More Piecewise Linear Pi01 Independence/correction 11/16/04 8:57AM
233:Neatening Piecewise Linear Pi01 Independence 11/17/04 12:22AM
234:Affine Pi01 Independence 11/20/04 9:54PM
235:Neatening Affine Pi01 Independence 11/28/04 6:08PM
236:Pi01 Independence/Huge Cardinals 12/2/04 3:49PM
237:More Neatening Pi01 Affine Independence 12/6/04 12:56AM
238:Pi01 Independence/Large Large Cardinals/Correction 12/7/04 10:31PM
239:Pi01 Update 12/11/04 1:12PM
240:2nd Pi01 Update 12/13/04 2:49AM
241:3rd Pi01 Update 12/13/04 4:08AM
Harvey Friedman
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