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We give a complete proof that all 3-quantifier sentences in the primitive notation of set theory (∈, =), are decided in ZFC, and in fact in a weak fragment of ZF without the power set axiom. We obtain information concerning witnesses of 2-quantifier formulas with one free variable. There is a 5-quantifier sentence that is not decided in ZFC (see [2]).
@article{HarveyM2003, abstract = {We give a complete proof that all 3-quantifier sentences in the primitive notation of set theory (∈, =), are decided in ZFC, and in fact in a weak fragment of ZF without the power set axiom. We obtain information concerning witnesses of 2-quantifier formulas with one free variable. There is a 5-quantifier sentence that is not decided in ZFC (see [2]).}, author = {Harvey M. Friedman}, journal = {Fundamenta Mathematicae}, keywords = {3-quantifier sentences; primitive notation of set theory; ZFC; weak fragment of ZF; witnesses of 2-quantifier formulas; 5-quantifier sentence}, language = {eng}, number = {3}, pages = {213-240}, title = {Three-quantifier sentences}, url = {http://eudml.org/doc/283350}, volume = {177}, year = {2003}, }
TY - JOUR AU - Harvey M. Friedman TI - Three-quantifier sentences JO - Fundamenta Mathematicae PY - 2003 VL - 177 IS - 3 SP - 213 EP - 240 AB - We give a complete proof that all 3-quantifier sentences in the primitive notation of set theory (∈, =), are decided in ZFC, and in fact in a weak fragment of ZF without the power set axiom. We obtain information concerning witnesses of 2-quantifier formulas with one free variable. There is a 5-quantifier sentence that is not decided in ZFC (see [2]). LA - eng KW - 3-quantifier sentences; primitive notation of set theory; ZFC; weak fragment of ZF; witnesses of 2-quantifier formulas; 5-quantifier sentence UR - http://eudml.org/doc/283350 ER -