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Choice principles in constructive and classical set theories
 POHLERS (EDS.): PROCEEDINGS OF THE LOGIC COLLOQUIUM 2002
, 2002
"... The objective of this paper is to assay several forms of the axiom of choice that have been deemed constructive. In addition to their deductive relationships, the paper will be concerned with metamathematical properties effected by these choice principles and also with some of their classical models ..."
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The objective of this paper is to assay several forms of the axiom of choice that have been deemed constructive. In addition to their deductive relationships, the paper will be concerned with metamathematical properties effected by these choice principles and also with some of their classical models.
Contemporary Mathematics Permutations and Wellfoundedness: the True Meaning of the Bizarre Arithmetic of Quine’s NF
"... It is shown that, according to NF, many of the assertions of ordinal arithmetic involving the Tfunction which is peculiar to NF turn out to be equivalent to the truthincertainpermutationmodels of assertions which have perfectly sensible ZFstyle meanings, such as: the existence of wellfounded s ..."
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It is shown that, according to NF, many of the assertions of ordinal arithmetic involving the Tfunction which is peculiar to NF turn out to be equivalent to the truthincertainpermutationmodels of assertions which have perfectly sensible ZFstyle meanings, such as: the existence of wellfounded sets of great size or rank, or the nonexistence of small counterexamples to the wellfoundedness of ∈. Everything here holds also for NFU if the permutations are taken to fix all urelemente. NF is Quine’s system of set theory, axiomatised by extensionality and those instances of the naïve set existence scheme that are stratified. A formula of the language of set theory is stratified if the variables within it can be labelled with integers in such a way that all occurrences of each variable receive the same label, and if x ∈ y appears in it then the label of x must be one less than the label of y, and if x = y occurs then the label of x must be the same as the label of y.
A.Miller Long Borel Hierarchies 1 Long Borel
, 704
"... We show that it is relatively consistent with ZF that the Borel hierarchy on the reals has length ω2. This implies that ω1 has countable cofinality, so the axiom of choice fails very badly in our model. A similar argument produces models of ZF in which the Borel hierarchy has length any given limit ..."
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We show that it is relatively consistent with ZF that the Borel hierarchy on the reals has length ω2. This implies that ω1 has countable cofinality, so the axiom of choice fails very badly in our model. A similar argument produces models of ZF in which the Borel hierarchy has length any given limit ordinal less than ω2, e.g., ω or ω1 + ω1.