Computational Prospects of Infinity, Part I: Tutorials: by Chitat Chong, Chitat Chong, Qi Feng, Theodore A. Slaman, W.

By Chitat Chong, Chitat Chong, Qi Feng, Theodore A. Slaman, W. Hugh Woodin, Yue Yang

This quantity offers the written models of the academic lectures given on the Workshop on Computational customers of Infinity, held from 18 June to fifteen August 2005 on the Institute for Mathematical Sciences, nationwide collage of Singapore. It involves articles via 4 of the top specialists in recursion idea (computability thought) and set concept. The survey paper of Rod Downey offers a finished creation to algorithmic randomness, probably the most energetic components of present learn in recursion thought. Theodore A Slaman's article is the 1st revealed account of the ground-breaking paintings of Slaman Woodin and Slaman Shore at the definability of the Turing leap. John metal offers a few effects at the homes of derived versions of mice, and at the lifestyles of mice with huge derived types. The learn used to be prompted via the various recognized Holy Grails in internal version concept, together with the Mouse Set Conjecture. In his presentation, W Hugh Woodin supplies an summary of an multiplied model (unpublished) on appropriate extender sequences, a topic that used to be constructed within the try to comprehend internal version thought for giant cardinals past the extent of superstrong cardinals.

the amount serves as an invaluable advisor for graduate scholars and researchers in recursion conception and set idea to a couple of crucial and demanding advancements in those matters in recent times.

Contents: 5 Lectures on Algorithmic Randomness (R Downey); worldwide homes of the Turing levels and the Turing leap (T A Slaman); Derived types linked to Mice (J R Steel); instructional define: compatible Extender Sequences (W H Woodin).

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Tests, {We,j,s : e, j, s ∈ N} and stops the enumeration of one if the measure µ(We,j,s ) threatens to exceed 2−(j+1) at any stage s of the simultaneous enumeration. Then we can let Un = ∪e∈N We,n+e+1 . of test, and moreover, a real A passes Then we note that Un is a Martin-L¨ all Martin-L¨ of tests iff A ∈ ∩n∈N Un . We have established the following. 1: (Martin-L¨ of [53]) There exist universal Martin-L¨ of tests: That is there is a Martin-L¨ of test {Un : n ∈ N} such that, for any MartinL¨of test {Vn : n ∈ N}, x ∈ ∩n∈N Vn implies x ∈ ∩n∈N Un .

Hence, x is not Levin-G´acs-Chaitin random. We can use Schnorr’s Theorem to prove that there are many strings that are (weakly) Chaitin random yet are not Kolmogorov random. 2: There are infinitely many n and strings x of length n such that (i) K(x) ≥ n and (ii) C(x) ≤ n − log n. Proof: Let α be Martin-L¨ of random. 4, for all n, K(α n) ≥ n − O(1). 1, for infinitely many n, C(α n) ≤ n − log n. Schnorr’s Theorem also allows us to define some specific kinds of random reals. For instance, the class R = ∪c Rc where Rc = {A : ∀n(K(A n) ≥ n − c}, is the Σ02 class of all Martin-L¨of random reals.

We refer the reader to [39] or Downey and Hirschfeldt [18]. Strictly speaking the proof only uses weak n + 1-randomness. 7. van Lambalgen’s Theorem In this section, we will study independence results for random reals. We begin with a central result concerning randomness, which has turned out to play a very important role in the theory. We begin with a simple Lemma. 14: (van Lambalgen, Kurtz, Kautz) (i) If A ⊕ B is n-random so is A. (ii) If A is n-random so is A[n] , the n-th column of A. (iii) If A ⊕ B is n-random, then A is n − B-random.

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