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The Development of Mathematical Logic from Russell to Tarski: 19001935
"... this paper that from the logical point of view set theory is the proper foundation of the mathematical sciences. Thus, he adds, if one wants to give general definitional principles that hold for all of mathematics it is necessary to account for the definitional principles of set theory. First, he b ..."
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this paper that from the logical point of view set theory is the proper foundation of the mathematical sciences. Thus, he adds, if one wants to give general definitional principles that hold for all of mathematics it is necessary to account for the definitional principles of set theory. First, he begins his definitional analysis with geometry. Relying on Pieri's work on the foundations of geometry he starts with two relations, x y and E(x, y , z). E(x, y , z) means that y and z are equidistant from x. Then he adds that all definitions in Pieri's geometry can be obtained by closing the basic relationships under five principles: 1. Permutation of variables: if A(x,y , z) is a ternary relation so is A(x,z,y)
MATHEMATICAL LOGIC QUARTERLY
, 2007
"... The axiomofchoice and the law of excluded middle in weak set theories ..."
Zermelo's WellOrdering Theorem in Type Theory
"... Abstract. Taking a `set ' to be a type together with an equivalence relation and adding an extensional choice axiom to the logical framework (a restricted version of constructive type theory) it is shown that any `set' can be wellordered. Zermelo's rst proof from 1904 is followed, with a simpli cat ..."
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Abstract. Taking a `set ' to be a type together with an equivalence relation and adding an extensional choice axiom to the logical framework (a restricted version of constructive type theory) it is shown that any `set' can be wellordered. Zermelo's rst proof from 1904 is followed, with a simpli cation to avoid using comparability of wellorderings. The proof has been formalised in the system AgdaLight. 1