[FOM] 162:Two Quantifier Blocks

Harvey Friedman friedman at math.ohio-state.edu
Mon Mar 31 12:28:04 EST 2003

I proved that all 3 quantifier sentences in the primitive language of 
set theory (epsilon, =) can be decided in a weak fragment of ZF 
without the power set axiom. See

Three Quantifier Sentences, http://www.mathpreprints.com/math/Preprint/show/

for a complete proof of this result and various slightly stronger results.

I can now prove that all 2 quantifier block sentences in the 
primitive language of set theory (epsilon, =) can be decided in a 
weak fragment of ZF without the power set axiom. I.e., A...AE...E 
(or, if you like, AnEm).

One also gets information like this: if E...EA...A is true then there 
are hereditarily finite witnesses whose ranks are bounded by a 
reasonable function of the number of existential quantifiers.

I proved that there is a 5 quantifier sentence in the primitive 
language of set theory (epsilon, =) which is not decided in ZFC. In 
fact, the example is equivalent to the existence of a subtle 
cardinal, and is EAEEA. See

Primitive Independence Results, 

CONJECTURE: Various levels of set theory correspond, in some rigorous 
sense, to various syntactic classes in the primitive language of set 

Daniel Gogol had two papers in the 70's on this topic. I have 
politely commented about the second one:

D. Gogol, Sentences with three ¦uantifiers are decidable in set 
theory, Fundamenta Mathematicae, CII (1979), 1-8.

I have not commented on the first one:

D. Gogol, The AnE-Completeness of Zermelo-Fraenkel Set Theory, 
Zeitschrift f. math. Logik und Grundlagen d. Mathl, Bd. 24, 29-290 


I use http://www.mathpreprints.com/math/Preprint/show/ for manuscripts with
proofs. Type Harvey Friedman in the window.

This is the 162nd in a series of self contained postings to FOM covering
a wide range of topics in f.o.m. Previous ones counting from #100 are:

100:Boolean Relation Theory IV corrected  3/21/01  11:29AM
101:Turing Degrees/1  4/2/01  3:32AM
102: Turing Degrees/2  4/8/01  5:20PM
103:Hilbert's Program for Consistency Proofs/1 4/11/01  11:10AM
104:Turing Degrees/3   4/12/01  3:19PM
105:Turing Degrees/4   4/26/01  7:44PM
106.Degenerative Cloning  5/4/01  10:57AM
107:Automated Proof Checking  5/25/01  4:32AM
108:Finite Boolean Relation Theory   9/18/01  12:20PM
109:Natural Nonrecursive Sets  9/26/01  4:41PM
110:Communicating Minds I  12/19/01  1:27PM
111:Communicating Minds II  12/22/01  8:28AM
112:Communicating MInds III   12/23/01  8:11PM
113:Coloring Integers  12/31/01  12:42PM
114:Borel Functions on HC  1/1/02  1:38PM
115:Aspects of Coloring Integers  1/3/02  10:02PM
116:Communicating Minds IV  1/4/02  2:02AM
117:Discrepancy Theory   1/6/02  12:53AM
118:Discrepancy Theory/2   1/20/02  1:31PM
119:Discrepancy Theory/3  1/22.02  5:27PM
120:Discrepancy Theory/4  1/26/02  1:33PM
121:Discrepancy Theory/4-revised  1/31/02  11:34AM
122:Communicating Minds IV-revised  1/31/02  2:48PM
123:Divisibility  2/2/02  10:57PM
124:Disjoint Unions  2/18/02  7:51AM
125:Disjoint Unions/First Classifications  3/1/02  6:19AM
126:Correction  3/9/02  2:10AM
127:Combinatorial conditions/BRT  3/11/02  3:34AM
128:Finite BRT/Collapsing Triples  3/11/02  3:34AM
129:Finite BRT/Improvements  3/20/02  12:48AM
130:Finite BRT/More  3/21/02  4:32AM
131:Finite BRT/More/Correction  3/21/02  5:39PM
132: Finite BRT/cleaner  3/25/02  12:08AM
133:BRT/polynomials/affine maps  3/25/02  12:08AM
134:BRT/summation/polynomials  3/26/02  7:26PM
135:BRT/A Delta fA/A U. fA  3/27/02  5:45PM
136:BRT/A Delta fA/A U. fA/nicer  3/28/02  1:47AM
137:BRT/A Delta fA/A U. fA/beautification  3/28/02  4:30PM
138:BRT/A Delta fA/A U. fA/more beautification  3/28/02  5:35PM
139:BRT/A Delta fA/A U. fA/better  3/28/02  10:07PM
140:BRT/A Delta fA/A U. fA/yet better  3/29/02  10:12PM
141:BRT/A Delta fA/A U. fA/grammatical improvement  3/29/02  10:43PM
142:BRT/A Delta fA/A U. fA/progress  3/30/02  8:47PM
143:BRT/A Delta fA/A U. fA/major overhaul  5/2/02  2:22PM
144: BRT/A Delta fA/A U. fA/finesse  4/3/02  4:29AM
145:BRT/A U. B U. TB/simplification/new chapter  4/4/02  4:01AM
146:Large large cardinals  4/18/02  4:30AM
147:Another Way  7:21AM  4/22/02
148:Finite forms by relativization  2:55AM  5/15/02
149:Bad Typo  1:59PM  5/15/02
150:Finite obstruction/statisics  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:Two Quantifier Blocks  3/31/03  12:18AM

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