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Mathematical Notes
Extension of standard models of ZFC to models of Nelson’s nonstandard set theory IST1999 •
This paper continues the study of the use of different models of ZF set theory as carriers for the mathematics of quantum mechanics. The basic tool used here is the construction of Cohen extensions of ZFC models by use of Boolean valued ZFC models [C = axiom of choice]. Let M be a standard transitive ZFC model. Inside M, B(H M) is the algebra of all bounded linear operators over some Hilbert space HM • It is shown that with each state pin B(H M) and projection operator 0 in B(H M) one can associate a unique Boolean valued ZFC model M po' B:po is the algebra of all Borel subsets of ! 0,1)W, the set of all infinite 0-1 sequences. modulo sets of P po = ® Ppo measure zero with PpoC! 1)) = Trpo in M. Let IJIM and <PM be respective maps from the sets of state preparation and question measuring procedures into B(HM). Let M =~, the minimal standard transitive ZFC model. It is then shown that with each state preparation procedure s E Dom(IJI Mo) and each question measuring procedure q E Dom(<P Mo) and with each infinite repetition (tsq) of doing sand q at times teO), t(l), ... , if the definition of randomness is sufficiently strong, one can associate the Cohen extension Mo[I/Ir,q] ofMo by Iji,sq" Iji"q is the random outcome sequence associated with (tsq). A third condition, in addition to the two given in the previous paper, is then given which must be satisfied if a ZFC model M is to serve as a carrier for the mathematics of quantum mechanics. In essence it says that for each pair (tsq) and (wuk) of distinct infinite repetitions of doing sand q and of doing u and k with s, u E Dom('I'M) and q, k EDom(<P M). the two outcome sequences Iji"q and l)iwuk are mutually statistically independent. It is then shown that for a strong definition of independence, corresponding to the definition of randomness used previously, no Cohen extension ~[I)i,sq] of ~ can serve as the carrier for the mathematics of quantum mechanics.
Journal of Mathematical Physics
Models of Zermelo Frankel set theory as carriers for the mathematics of physics. I1976 •
Annals of the Alexandru Ioan Cuza University - Mathematics
An Extension of a Permutative Model of Set Theory2012 •
2011 •
Abstract In this elementary paper we establish a few novel results in set theory; their interest is wholly foundational-philosophical in motivation. We show that in Cantor-Von Neumann Set-Theory, which is a reformulation of Von Neumann's original theory of functions and things that does not introduce 'classes'(let alone 'proper classes'), developed in the 1920ies, both the Pairing Axiom and 'half'the Axiom of Limitation are redundant—the last result is novel.
2023 •
Another Universalism: Seyla Benhabib and the Future of Critical Theory
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Secondary Sex Ratio in a South-Western Nigerian Town2012 •
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