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Lecture Notes from My Topics Course, Spring 2004 Topics in Logic and Foundations: Spring 2004 Stephen G. Simpson Copyright c 2004 First Draft: April 30, 2004 This Draft: November 1, 2005 The latest version is available at http://www.math.psu.edu/simpson/notes/. Please send corrections to <[email protected]>. This is a set of lecture notes from a 15-week graduate course at the Penn- sylvania State University taught as Math 574 by Stephen G. Simpson in Spring 2004. The course was intended for students already familiar with the basics of mathematical logic. The course covered some topics which are important in contemporary mathematical logic and foundations but usually omitted from introductory courses at Penn State. These notes were typeset by the students in the course: Robert Dohner, Esteban Gomez-Riviere, Christopher Griffin, David King, Carl Mummert, Heiko Todt. In addition, the notes were revised and polished by Stephen Simpson. Contents Contents 1 1 Computability in core mathematics 4 1.1 Reviewofcomputablefunctions. 4 1.1.1 Registermachines ...................... 4 1.1.2 Recursive and partial recursive functions . 5 1.1.3 The µ-operator........................ 7 1.2 Introductiontocomputablealgebra. 7 1.2.1 Computablegroups ..................... 7 1.2.2 Computablefields ...................... 9 1.3 Finitely presented groups and semigroups . 10 1.3.1 Freegroups .......................... 10 1.3.2 Grouppresentationsand wordproblems . 11 1.3.3 Finitelypresentedsemigroups . 13 1.3.4 Unsolvability of the word problem for semigroups . 14 1.4 Moreoncomputablealgebra . 15 1.4.1 Splittingalgorithms . 15 1.4.2 Computablevectorspaces . 16 1.5 Computableanalysisandgeometry . 16 1.5.1 Computablerealnumbers . 16 1.5.2 Computablesequencesofrealnumbers . 17 1.5.3 EffectivePolishspaces . 18 1.5.4 ExamplesofeffectivePolishspaces . 19 1.5.5 Effective topology and effective continuity . 20 2 Degrees of unsolvability 22 2.1 Oraclecomputations . 22 2.2 Relativization............................. 23 2.3 Thearithmeticalhierarchy. 24 2.4 Turingdegrees ............................ 26 2.5 Thejumpoperator .......................... 27 2.6 Finiteapproximations . 29 2.7 Post’sTheoremanditscorollaries . 32 2.8 AminimalTuringdegree . 34 1 2.9 Sacksforcing ............................. 37 2.10 HomogeneityofSacksforcing . 42 2.11 Cohengenericity ........................... 43 3 Models of set theory 44 3.1 Countabletransitivemodels . 44 3.2 Modelsconstructedbyforcing. 44 3.3 Anexample:Cohenforcing . 46 3.4 Propertiesofgenericextensions . 47 3.5 Blowingupthecontinuum. 48 3.6 Preservationofcardinalsandthec.c.c. .. 49 3.7 ForcingtheContinuumHypothesis . 51 3.8 Additional models obtained by forcing . 52 4 Absoluteness and constructibility 54 4.1 Absoluteness ............................. 54 4.2 Treesandwellfoundedness . 55 4.3 TheShoenfieldAbsolutenessTheorem . 58 4.4 Someexamples ............................ 60 4.5 Constructiblesets. .. .. .. .. .. .. .. .. .. .. .. 61 4.6 Constructiblereals .. .. .. .. .. .. .. .. .. .. .. 62 4.7 Relativeconstructibility . 63 5 Measurable cardinals 65 5.1 Filters ................................. 65 5.2 Theclosedunboundedfilter . 66 5.3 Ultrafilters .............................. 67 5.4 Ultraproductsandultrapowers . 68 5.5 An elementary embedding of V ................... 70 5.6 Largenessandnormality. 72 5.7 Ramsey’sTheorem .......................... 74 5.8 IndiscerniblesandEM-sets . 75 5.9 Measurable cardinals and L ..................... 78 5.10The#operator............................ 81 6 Determinacy 84 6.1 Gamesanddeterminacy . 84 6.2 OpenandBoreldeterminacy . 86 6.3 Projectivesets ............................ 88 6.4 Consequencesofprojectivedeterminacy . 89 6.5 Turingdegreedeterminacy. 93 1 6.6 Σ1 determinacy............................ 94 6.7 Hyperarithmetictheory . 95 6.8 Admissiblesets ............................ 98 6.9 Admissibility and cardinal collapsing . 100 2 Bibliography 103 3 Chapter 1 Computability in core mathematics 1.1 Review of computable functions For the sake of completeness, we will review the basic definitions from elemen- tary recursion theory that will be necessary for our study of computability in core mathematics. We write ω = 0, 1, 2,... for the set of nonnegative integers. For k 1, ωk denotes the set of sequences{ of} elements of ω of length k. ≥ 1.1.1 Register machines Recall that a register machine is composed of an infinite number of registers, usually denoted by bins as shown in Figure 1.1. Each register holds a natural number. Register machines execute register machine programs. Register ma- chine programs are flow-charts of four types of instructions shown in Figure 1.2. The increment instruction adds 1 to the value of the register specified in the instruction. Program flow then continues along the out arrow. The decrement instruction checks to see if the value in the specified register is greater than zero, if not the program continues execution along the arrow labeled e. Otherwise, the value in the specified register is decremented and the program continues execution along the unlabeled arrow. For every program there is one and only one start instruction. The stop instruction halts program execution. There may be many stop instructions in a program flow chart. A simple program is shown in Figure 1.2. Let denote a register machine program. When we run with inputs P P x1,...,xk, we assume that these inputs are initially placed into the registers R ,...,R and that all other registers are empty. If and when halts (i.e., 1 k P completes its execution), the output is stored in register Rk+1. We denote a run of a register machine with program on inputs x ,...,x by (x ,...,x ). P 1 k P 1 k The k-place number-theoretic function computed by is denoted f (k). Thus P P 4 . x1 x2 xk 0 R1 R2 Rk Rk+1 P ? ? . ? y . R1 R2 Rk Rk+1 (k) Figure 1.1: A register machine computing y = fP (x1,...,xk). Here y is the result stored in register R after running (x ,...,x ). k+1 P 1 k f (k)(x ,...,x ) = y means that y is the output after running with inputs P 1 k P x1,...,xk. For a more detailed review of register machines and register machine programs, see [18]. Each register machine program can be assigned a natural number #( ), in a recursive one-to-one way. ThisP number is called the G¨odel number of ,P or (k) (k) P an index of fP . We write y = ϕe (x1,...,xk) to indicate that the program with G¨odel number e with inputs x1,...,xk eventually halts with output y. 1.1.2 Recursive and partial recursive functions Definition 1.1.1 (recursive function). A k-place function f : ωk ω is computable (equivalently, recursive) if there exists a register machine program→ that computes it, i.e., for all x1,...,xk ω, (x1,...,xk) eventually halts P (k) ∈ P with y = f(x1,...,xk)= fP (x1,...,xk) as the output. A k-place partial function is a function from a subset of ωk to ω. We write ψ : ωk P ω to indicate that ψ is a k-place partial function. The fact that a → partial function ψ may not be defined for some arguments x1,...,xk ω leads us to consider expressions that may or may not have a numerical value.∈ Let E be such an expression. We say that E is defined if E has a numerical value, and in this case, we write E . Otherwise, we say that E is undefined and we write ↓ E . Furthermore, we write E1 E2 to mean that E1 and E2 are either both defined↑ and equal or both undefined.≃ The symbol is called strong equality. ≃ Definition 1.1.2 (partial recursive function). A partial function ψ : ωk P → 5 + START Ri Starting Instruction Increment Instruction e − STOP Ri Halting Instruction Decrement Instruction − e + START R1 R2 STOP + R2 Add one to the input in register one. Figure 1.2: The four register machine instructions and a sample program. 6 ω is partial recursive if and only if there is a register machine program such P that x ,...,x ψ(x ,...,x ) f (k)(x ,...,x ). ∀ 1 k 1 k ≃ P 1 k Definition 1.1.3. A set (or relation) R ωk is said to be computable (or ⊆ k equivalently recursive) if the characteristic function χR : ω 0, 1 of R is a computable function. →{ } Definition 1.1.4. A set A ω is said to be recursively enumerable (abbreviated r.e.) if it is the range of a partial⊆ recursive function. By a problem we mean a set A ω. We often can use G¨odel numberings to express problems in other areas of⊆ mathematics in this more formal sense. We will see an example of this when we discuss the word problem for groups and semigroups. A problem is said to be solvable if and only if it is a recursive set, otherwise unsolvable. The classification of unsolvable problems is an important theme in recursion theory. 1.1.3 The µ-operator Let R ωk+1 be a computable relation. Suppose we wish to search for the least y ω such⊆ that R(x ,...,x ,y) holds. We write this as ∈ 1 k ψ(x ,...,x ) µy R(x ,...,x ,y). 1 k ≃ 1 k The operator µ is called the minimization operator or the least number operator and can be used to create partial recursive functions from recursive sets. Since our search for a y satisfying R(x1,...,xk,y) may not halt, ψ(x1,...,xk) may be undefined. For example, if R is a recursive subset of ω ω and for each m there is an n such that m,n R, then the function m ×µn m,n R is recursive. h i ∈ 7→ h i ∈ 1.2 Introduction to computable algebra 1.2.1 Computable groups Definition 1.2.1. A computable group is a group G, whose elements form a recursive subset of ω and whose multiplication is ah recursive·i function. Proposition 1.2.2. Let G, be a computable group. Then the function a a−1 is computable. h ·i 7→ −1 Proof. To compute a for an element a G, we first find 1G, then search for the unique element b such that a b =1 .∈ That is, a−1 = µb (a b =1 ).
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