Download e-book for kindle: A History of Pi by Petr Beckmann

By Petr Beckmann

ISBN-10: 1466887168

ISBN-13: 9781466887169

The background of pi, says the writer, even though a small a part of the historical past of arithmetic, is however a reflect of the background of guy. Petr Beckmann holds up this replicate, giving the heritage of the days while pi made growth -- and likewise while it didn't, simply because technology used to be being stifled by means of militarism or spiritual fanaticism.

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Extra resources for A History of Pi

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2. The generic-multiverse of sets. Let the multiverse (of sets) refer to the collection of possible universes of sets. The truths of the Set Theory are the sentences which hold in each universe of the multiverse. The multiverse is the generic-multiverse if it is generated from each universe of the collection by closing under generic extensions (enlargements) and under generic refinements (inner models of a universe which the given universe is a generic extension of). To illustrate the concept of the generic-multiverse, suppose that M is a countable transitive set with the property that M ZFC.

By (2) and Lemma 30(1), for each ∈ I , for each N ∈ VM ∩V , N Therefore by Lemma 31, for each ∈ IX , for each N ∈ VMX ∩V , N φ. Suppose N ∈ VMX ∩ M . Then by Lemma 30(2), there exists 0 ∈ IX such that N ∩ V 0 ∈ VMX ∩V 0 and such that N ∩ V 0 ≺Σn N. THE CONTINUUM HYPOTHESIS, THE GENERIC-MULTIVERSE OF SETS . . 41 Since N ∩ V 0 ∈ VMX ∩V 0 , N ∩ V 0 φ and therefore since N ∩ V 0 ≺Σn N , N φ. Therefore , for all N ∈ VMX ∩ M , N φ. This proves (1). I finish by discussing the generalization of Lemma 32 to any restricted generic-multiverse of the following form.

Suppose that N1 is a countable transitive set, N1 ZC(VN) + Σ1 -Replacement, and that k ≥ 1. Let I = { ∈ N1 ∩ Ord | N1 ∩ V ≺Σk N1 } and suppose that I is cofinal in N1 ∩ Ord. (1) Suppose ∈ I . Then for each N ∈ VN1 ∩V there exists N ∗ ∈ VN1 such that N = N ∗ ∩ V and N ≺Σk N ∗ . (2) Suppose that N ∈ VN1 . Then there exists ∈ I\ , N ∩ V ∈ VN ∩V ∈ N1 ∩ Ord such that for all and N ∩ V ≺Σk N . Proof. We first prove (1). Fix ∈ I , let N2 = N1 ∩ V and fix N ∈ VN2 . 3) for all i + 1 ≤ m either a) Pi ∈ Mi , Gi ⊂ Pi is Mi -generic and Mi+1 = Mi [Gi ], or b) Pi ∈ Mi+1 , Gi ⊂ Pi is Mi+1 -generic and Mi = Mi+1 [Gi ].

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A History of Pi by Petr Beckmann

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