[FOM] 189:Some Model theoretic Pi-0-1 statements
Harvey Friedman
friedman at math.ohio-state.edu
Thu Sep 25 11:04:22 EDT 2003
SOME MODEL THEORETIC Pi-0-1 STATEMENTS
by
Harvey M. Friedman
September 21, 2003
We give some statements of a simple model theoretic nature, which assert the
existence of a relational structure satisfying certain conditions. The
conditions can easily be seen to be in the form of a first order theory.
Hence these sentences are provably equivalent to Pi-0-1 sentences. In fact,
they are provably equivalent to the consistency of very strong set theories.
Let R be a binary relation on a set A. A cross section of R is a set of the
form {y: R(x,y)}, for some x in A. A unique cross section of R is a set of
is of the form {y: R(x,y)}, for some unique x in A.
PROPOSITION 1. There is a relational structure (A,R,j), where j is an
elementary embedding from (A,R) into (A,R), such that every (A,R,j)
definable set of fixed points of j is a unique cross section of R.
PROPOSITION 2. There is a relational structure (A,R,j,k), where j is an
elementary embedding from (A,R) into (A,R), k is an elementary embedding
from (A,R,j) into (A,R,j), and every (A,R,j,k) definable set of fixed points
of k is a unique cross section of R.
PROPOSITION 3. There is a relational structure (A,R,j,k), where j is an
elementary embedding from (A,R) into (A,R) whose fixed points form an
elementary substructure of (A,R), k is an elementary embedding from (A,R,j)
into (A,R,j), and every (A,R,j,k) definable set of fixed points of k is a
unique cross section of R.
THEOREM 4. Proposition 1 is provably equivalent, over ACA, to the
consistency of ZF + {there exists an elementary embedding j:V(alpha) into
V(beta) with critical point kappa such that j^n(kappa) < alpha}_n. By work
of Woodin, we can replace ZF with ZFC.
THEOREM 5. Proposition 2 implies, over ACA, the consistency of ZF + there
exists a nontrivial elementary embedding from some V(alpha) into V(alpha),
where alpha is an inaccessible cardinal (and more). By work of Woodin, the
latter implies the consistency of ZFC + there exists a nontrivial elementary
embedding from some V(alpha) into V(alpha).
THEOREM 6. Proposition 3 implies, over ACA, ZF + there exists a nontrivial
elementary embedding from some V(alpha) into V(alpha), where V(critical
point) is an elementary substructure of V(alpha). By work of Woodin, the
latter implies the consistency of MKC + there exists a nontrivial elementary
embedding from some L(V(alpha+1)) into L(V(alpha+1)).
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This is the 188th 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
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
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