Chapter 3. Formal Number Theory
|
|
- Rudolf Preston
- 6 years ago
- Views:
Transcription
1 Chapter 3. Formal Number Theory 1. An Axiom System for Peano Arithmetic (S) The language L A of Peano arithmetic has a constant 0, a unary function symbol, a binary function symbol +, binary function symbol, and a binary relation symbol =. There are two versions, both containing these axioms: S1 x = y (x = z y = z) S2 x = y x = y S3 (0 = x ) S4 S5 x = y x = y x + 0 = x S6 (x + y) = x + y S7 x 0 = 0 S8 x y = x y + x Second-order S has induction for all subsets Q, stated with a second-order quantifier Q, where Q(x) means x is an element of Q. (1) Q(Q(0)( x(q(x) Q(x )) xq(x))). First-order S has induction only for definable subsets: (2) B(0)( x(b(x) B(x )) xb(x)). for every formula B(x) in L A. S is a theory in L A, but the second-order version is not. In any model M for the language of Peano arithmetic, the binary relation symbol = is interpreted as the binary relation = M on the nonempty domain D M, the binary function symbol + is interpreted as the binary operation + M on D M, the binary function symbol is interpreted as the binary operation M on D M, and the unary function symbol is interpreted as the function M from D M to D M. The standard interpretation/model N has domain ω = {0, 1, 2, 3, }, interprets as +1 (the function that adds 1 to its input), inteprets the symbol as the binary operation of multiplication of natural numbers, and interprets the symbol + as the binary operation + of addition of natural numbers, and interprets the relation symbol = as the equality relation between natural numbers. A nonstandard model is any model that interprets the relation symbol = as the equality relation on its universe and yet is not isomorphic to the standard model. Second-order S is categorical: any two models are isomorphic. (Prove this.) S has only infinite models (this follows from just S1 S4), but also has models of all infinite cardinalities (by the upward Löwenheim-Skolem Theorem), and has 2 ℵ 0 pairwise nonisomorphic models of cardinality ℵ 0 (Ehrenfeucht). 1
2 2 Lemma 3.1 Some theorems of S obtained by replacing variables in the axioms with terms. They follow from S1 S8 by Gen and the instantiation schema (A4). S1 t = r (t = s r = s) S2 t = r t = r S3 (0 = t ) S4 S5 t = r t = r t + 0 = t S6 (t + r) = t + r S7 t 0 = 0 S8 t r = t r + t Prop. 3.2 All the following formulas are theorems of S. They express basic properties of the operations + M and M and the relation = M in any model M of S. (a) S x = x, so = M is reflexive on D M. (b) S x = y y = x, so = M is symmetric. (c) S x = y (y = z x = z), so = M is transitive. (d) S y = x (z = x y = z). (e) S x = y (x + z = y + z). (f) S x = 0 + x. (g) S x + y = (x + y). (h) S x + y = y + x, so + M is commutative. (i) S x = y (z + x = z + y). (j) S x + (y + z) = (x + y) + z, so + M is associative. (k) S x = y x z = y z. (l) S 0 x = 0. (m) S x y = x y + y. (n) S x y = y x, so M is commutative. (o) S x = y z x = z y. The following proof is not completely formalized within first-order logic. (The proof in the text is somewhat more formal.) That the proof below can be so formalized follows from the Deduction Theorem and other results from the text. (a): x + 0 = x (x + 0 = x x = x) x + 0 = x x = x S1 S5 MP twice (b): x = y (x = x y = x) x = y y = x S1 (a), PC
3 (c): By S1, (b), PC, so (c) holds. y = x (x = z y = z) (d): holds by (a),(b),(c). From now on we use the fact that = M is transitive without any explicit mention. (e): S x = y (x + z = y + z). Let B(z) be x = y (x + z = y + z)). B(0) holds because if x = y then Show: B(z) B(z ). x + 0 = S5 x = y = S5 y + 0. Hyp: x = y (x + z = y + z)). Show: x = y (x + z = y + z )). Assume x = y. Then x + z = y + z by Hyp, so x + z = S6 (x + z) = S2 (y + z) = S6 y + z Since x and y were arbitrary, we have shown S B(0) and S B(z) B(z ). By (2) we have S B(0)( z(b(z) B(z )) zb(z)), hence, by MP, S zb(z)), hence, by (A4), S B(z). Below, in proofs of the other parts, these last steps will be omitted. (f): S x = 0 + x. Let B(x) be x = 0 + x. B(0) holds because 0 = by S5 and (b). Show: B(x) B(x ). Hyp: x = 0 + x. Show: x = 0 + x. x = Hyp,S2 (0 + x) = S6 0 + x (g): S x + y = (x + y). Let B(y) be x + y = (x + y). B(0) holds because x + 0 = S5 x = S2,S5 (x + 0). Show: B(y) B(y ). Hyp: x + y = (x + y). Show: x + y = (x + y ). x + y = S6 (x + y) = Hyp,S2 (x + y) = S6,S2 (x + y ) (h): S x + y = y + x. Let B(y) be x + y = y + x. B(0) holds because Show: B(y) B(y ). x + 0 = S5 x = (f) 0 + x 3
4 4 Hyp: x + y = y + x. Show: x + y = y + x. x + y = S6 (x + y) = Hyp,S2 (y + x) = S6 x + y (i): S x = y (z + x = z + y). This can be proved by invoking parts (h) and (e), but for an inductive proof via S9, let B(z) be x = y (z + x = z + y). Then B(0) holds because if x = y then 0 + x = (f) x = Hyp y = (f) 0 + y. Show: B(z) B(z ). Hyp: x = y (z + x = z + y). Show: x = y (z + x = z + y). Assume x = y. Then z + x = z + y by Hyp, so z + x = (g) (z + x) = S2 (z + y) = (g) z + y (j): S x + (y + z) = (x + y) + z. Let B(z) be x + (y + z) = (x + y) + z. B(0) holds because Show: B(z) B(z ). x + (y + 0) = S5,(i) x + y = S5 (x + y) + 0. Hyp: x + (y + z) = (x + y) + z. Show: x + (y + z ) = (x + y) + z. x + (y + z ) = S6,(i) x + (y + z) = S6 (x + (y + z)) = Hyp,S2 ((x + y) + z) = (x + y) + z (k): S x = y x z = y z. Let B(z) be x = y x z = y z. B(0) holds because if x = y then x 0 = S8 x 0 + x = S7,(e) 0 + x = (f) x = Hyp y = (f) 0 + y = S7,(e) 0 y + y = S8 0 y Show: B(z) B(z ). Hyp: x = y x z = y z. Show: x = y x z = y z. Assume x = y. Then x z = y z, so x z = S8 x z + x = (e) y z + x = (i) y z + y = S8 y z (l): S 0 x = 0. Let B(x) be 0 x = 0. B(0) holds because 0 0 = 0 by S7. Show: B(x) B(x ). Hyp: 0 x = 0. Show: 0 x = 0. 0 x = S8 0 x + 0 = S5 0 x = S7 0 (m): S x y = x y + y. Let B(y) be x y = x y + y. B(0) holds because Show: B(y) B(y ). x 0 = S7 0 = S = S7,(e) x Hyp: x y = x y + y. Show: x y = x y + y. x y = S8 x y +x = Hyp,(e) (x y +y)+x = (j) x y +(y +x ) = S6,(i) x y +(y +x) = (h),s2,(i) x y + (x + y) = S6,(i) x y + (x + y ) = (j) (x y + x) + y = S8,(e) x y + y
5 5 (n): S x y = y x. Let B(y) be x y = y x. B(0) holds because Show: B(y) B(y ). Hyp: x y = y x. Show: x y = y x. x 0 = S7 x = (l) 0 x x y = S8 x y + x = Hyp,(e) y x + x = (m) y x (o): S x = y z x = z y. Let B(z) be x = y z x = z y. B(0) holds because if x = y then Show: B(z) B(z ). 0 x = (l) 0 = (l) 0 y Hyp: x = y z x = z y. Show: x = y z x = z y. Assume x = y. Then z x = z y, so z x = (m) z x + x = (e) z y + x = (i) z y + y = (m) z y Cor. 3.3 S is a theory with equality, that is, the equality axioms A6 and A7 are provable in S. S proves A6 by 3.2(a), and S proves A7 by various other considerations, starting with 3.2(a)(b)(c), which show that = M is an equivalence relation on the universe D M of any model, and S2, 3.2(e), 3.2(i), 3.2(k), 3.2(o), which show that = M is a congruence relation with respect to the functions M, + M, and M. Prop. 3.4 These arithmetical laws are provable in S: left and right distributivity of over +, associativity of, and the cancellation law for +. (a) S x (y + z) = x y + x z, (b) S (x + y) z) = x z + y z, (c) S (x y) z = x (y z), (d) S x y = x z y = z. (a): Use (2) where B is x (y + z) = x y + x z. Assume x (y + z) = x y + x z. x (y + 0) = x y = x y + 0 = x y + x 0 x (y + z ) = x ((y + z) ) = x (y + z) + x = (x y + x z) + x = x y + (x z + x) = x y + x z Define the terms called numerals, so that there is a term for each natural number, as follows. Let 0 = 0, and for every positive n ω let n {}}{ n = 0. For example, 1 = 0, 2 = 0, 3 = 0, 4 = 0, etc.
6 6 Prop. 3.5 The following formulas about numerals are theorems of S. (a) S x + 1 = x (b) S x 1 = x (c) S x 2 = x + x (d) S x + y = 0 (x = 0 y = 0) (e) S x 0 (y x = 0 y = 0) (f) S x + y = 1 ((x = 0 y = 1) (x = 1 y = 0)) (g) S x y = 1 (x = 1 y = 1) (h) S x 0 y(x = y ) (i) S z 0 (x z = y z x = y) (j) S x 0 (x 1 z(x = z )). Prop. 3.6 (a) Let m, n ω. (i) If m n then S n m. (ii) S m + n = m + n and S mn m n. (b) All models of S are infinite. (c) S has models of all infinite cardinalities. Definitions of orderings >, <,,. For example, x < y is the formula z( (z = 0) x + z = y), where z is the first variable distinct from x and y. For terms t, s, t < s is the formula z( (z = 0) t + z = s), where z is the first variable that does not occur in t or s. (Such a variable does exist, but why?) Prop miscellaneous theorems of S about orderings. Prop. 3.8 More theorems of S on numerals. Prop. 3.9 (a) Complete induction (for formulas B) is provable in S: S ( y(y < x B(y)) B(x)) xb(x). (b) The least-number principle (for formulas B) is provable in S: S xb(x) m(b(m) x(x < m B(x))). Divisibility, a defined binary relation: for terms t and s, t s is the formula z(t z = s) where z is the first variable not occurring in t or s. Prop Some theorems of S about divisibility. Prop S proves a sentence that asserts the existence and uniqueness of quotient and remainder. Existence: Existence and uniqueness: q n d r(n = q d + r r < d) q n d r(n = q d + r r < d e s(n = q e + s s < e d = e r = s))
7 7 2. Number-Theoretic Functions and Relations Notation:! means there exists a unique..., that is,! x B(x) is an abbreviation of x(b(x) y(b(y) x = y)), where y is the first variable not occurring in B. Number-theoretic functions and relations are functions and relations on the actual natural numbers ω. Let K be a theory in the language L A. A number-theoretic relation R ω n is expressible in K iff there is a formula B(x 1,, x n ) of L A such that, for all k 1,..., k n ω, (1) if k 1,, k n R then K B(k 1,, k n ) (2) if k 1,, k n / R then K B(k 1,, k n ) For example, < is expressible in S by the formula z( (z = 0) x + z = y). Suppose that R ω n is expressible in K by B(x 1,, x n ). Then K is complete with respect to variable-free instances of B, in the sense that, for all k 1,..., k n ω, either K B(k 1,, k n ) or K B(k 1,, k n ), although both could hold in case K is inconsistent. Hence, if K B(k 1,, k n ) then K B(k 1,, k n ), if K B(k 1,, k n ) then K B(k 1,, k n ). A number-theoretic function f : ω n ω is representable in K iff there is a formula B(x 1,, x n, y) of L A such that, for all k 1,, k n, m ω, (1) if f(k 1,, k n ) = m then K B(k 1,, k n, m), (2) K! y B(k 1,, k n, y). A number-theoretic function f : ω n ω is strongly representable in a theory K iff there is a formula B(x 1,, x n, y) of L A such that, for all k 1,, k n, m ω, (1) if f(k 1,, k n ) = m then K B(k 1,, k n, m), 2. K! y B(x 1,, x n, y). Prop (V. Huber-Dyson) A number-theoretic function is representable in a theory K iff it is strongly representable in K. Proof. Strong representability implies representability via (A4), MP, and Gen. If f is representable via formula B, then f is strongly representable via formula ( ) ( )! z B(x 1,, x n, z) B(x 1,, x n, y)! z B(x 1,, x n, z) y = 0 Prop If K (0 = 0 ) then a number-theoretic relation is expressible in K iff its characteristic function is (strongly) representable in K. 3. Primitive Recursive and Recursive Functions Initial functions (unary) zero, (unary) successor, (n-ary) projection functions.
8 8 Rules: substitution (composition) f(x 1,, x n ) = g(h 1 (x 1,, x n ),, h m (x 1,, x n )) recursion (definability via recursive equations) (restricted) µ-operator f(x 1,, x n, 0) = g(x 1,, x n ) f(x 1,, x n, y + 1) = h(x 1,, x n, y, f(x 1,, x n, y)) f(x 1,, x n ) = µy(g(x 1,, x n, y) = 0) = the least y such that g(x 1,, x n, y) = 0 assuming (the restriction) that x 1 x n y(g(x 1,, x n, y) = 0). primitive recursive = obtainable from initial functions using substitution and recursion. recursive = obtainable from initial functions using substitution, recursion, and the restricted µ-operator. All primitive recursive functions are recursive. Prop The set of primitive recursive (recursive) functions is closed under permutation of variables, also under adding dummy variables and identifying variables. Cor More primitive recursive functions: n-ary zero function, n-ary constant functions, extended substitution rule. Prop More primitive recursive functions: addition x + y, multiplication x y, exponentiation x y, predecessor function δ(x) = x 1 if x > 0 and δ(0) = 0, x y, factorial, { 0 x = 0 sg(x) = 1 x = 1, { 1 x = 0 sg(x) = 0 x = 1, min, max, remainder, quotient, etc. Define bounded sums and bounded products. Prop The set of primitive recursive (recursive) functions is closed under bounded sums and products. Example: τ(x) is the number of divisors of x if x > 0 and τ(0) = 1. τ is primitive recursive because τ(x) = y x sg(rm(y, x)) Define the bounded µ-operator. Relations are primitive recursive (recursive) iff their characteristic functions are primitive recursive (recursive). Equality, less-than, divisibility, and primality are primitive recursive relations. Prop The set of primitive recursive (recursive) relations is closed under intersection, union, complementation, and bounded µ-operators.
9 Examples of primitive recursive functions: the nth prime p n, the exponents of the prime factorization of a number, the concatenation function, etc. Prop The set of primitive recursive (recursive) relations is closed under definitions by cases. Define primitive recursive encodings of pairs, triplets, etc., of natural numbers. 9 f#(x 1,..., x n, y) := u<y p f(x 1,...,x n,u) u Prop The set of primitive recursive (recursive) functions is closed under courseof-values recursion. Cor The set of primitive recursive (recursive) relations is closed under course-ofvalues recursion. Lemma 3.22 (Gödel s β-function) β is primitive recursive, where β(z, c, i) := rm((i + 1)c + 1, z). Gödel s β-function can be used to code finite sequences of arbitrary length; it is needed for representability of functions obtained using the restricted µ-operator. p.190, 3.30(a) Given a, b Z. Let d be the least positive element of I := {au + bv : u, v Z}. Such a d exists unless a = b = 0. Since d I, there are u, v Z such that d = au+bv. Divide a by d, obtaining a quotient q and a remainder r such that 0 r < d and a = qd + r. Then r = a qd = a q(au + bv) = (1 qu)a + bv I. If 0 < r then r is a positive element of I that is smaller than d, contradicting the choice of d, so r = 0, hence d divides a. Similarly, d divides b. If e divides both a and b then there are k and m such that a = ke and b = me, hence d = au + bv = keu + mev, hence e divides d. Thus d is the greatest common divisor of a and b. Lemma 3.23 k ω, y 1,, y k ω, z, c ω, i {1,..., k}, β(z, c, i) = y i. Proof. (3) (4) (5) (6) k, y 1,..., y k ω given c := max(k, y 1,..., y k )! def x i := c(i + 1) + 1 def y i < x i (3), (4) Claim: i j (x i, x j ) = 1 Proof. Assume p is prime. Then p x i, p x j p x i x j p (i j)c (4) p i j p c i j p c p i j (3) p c p 1 p c, p x i (4) p 1
10 10 (7) (8) (9) (10) x i := x j j i def (x i, x i ) = 1 (7), (6) u i, z i Z, x i z i + x i u i = 1 (8), 3.30(a) z := x 1 z 1 y x k z k y k def Let i denote congruence modulo x i for 1 i k (11) (12) z i x i z i y i (8), (10) z i y i, i = 1,..., k (9)(11) (12) holds for all z ± mx where x := x 1 x 1, so assume z > 0 (13) (14) β(z, c, i) := rm(c(i + 1) + 1, z) def of β β(z, c, i) = rm(x i, z) (4) β(z, c, i) i z def of rm β(z, c, i) i y i (12) β(z, c, i) = y i (13) (5) Prop Every recursive (or primitive recursive) function is representable in S. Proof. The zero, successor, and projection functions are (strongly) representable in S. If g, h 1,, h n are (strongly) representable, then g(h 1,, h n ) is (strongly) representable. Use Gödel s β-function to represent recursion. Say g is represented by G, h by H. Define f from g, h by f(x 1,, x n, 0) = g(x 1,, x n ) f(x 1,, x n, y + 1) = h(x 1,, x n, y, f(x 1,, x n, y)) Then there is a finite sequence of intermediate results from the computation of any particular value of f. Gödel s β is used to express this. f(k 1,, k n, l) = m iff there are y 1,..., y l such that y 1 = g(k 1,, k n ) y 2 = h(k 1,, k n, 0, y 1 ) y 3 = h(k 1,, k n, 1, y 2 ) y 4 = h(k 1,, k n, 2, y 3 ). y l = h(k 1,, k n, l 2, y l 1 ) m = h(k 1,, k n, l 1, y l )
11 11 iff there are z, c such that for all i {2,..., l 1}, y 1 = g(k 1,, k n ) y i = h(k 1,, k n, i 2, y i 1 ) m = h(k 1,, k n, l 1, y l ) This last condition can be expressed by a formula using the β-function. Cor All recursive relations are expressible in S. The converses of 3.25 and 3.24 are also true; see Prop. 3.29, Cor. 3.30, and Cor Consequently a relation on numbers is expressible in Peano arithmetic if and only if it is recursive, and a function on numbers is representable in Peano arithmetic if and only if it is recursive. There are finitely axiomatizable fragments of S, such as Robinson s Q, called RR in the text, which have the same key properties as Peano arithmetic. Lemma 3.32 lists specific sentences provable in RR, from which one gets Prop. 3.33, that all recursive functions are representable of RR. Prob 3.33 (Mendelson, 3rd Edition). There is a recursive function that is not primitive recursive, defined by A(n, 0) = n + 1 Rewrite these equations as follows. A(0, m + 1) = A(1, m) A(n + 1, m + 1) = A(A(n, m + 1), m) A 0 (n) = n + 1 A m+1 (0) = A m (1) A m+1 (n + 1) = A m (A m+1 (n)) n+1 {}}{ Show that A m+1 (n + 1) = A n+1 m (1) = A m (A m (...(1))). Then A 0 is the function +1, A 1 is the function +2, A 2 (n) = 2n + 3, etc. (Try calculating A 3 (n) and A 4 (n).) It can be shown that the function defined by f(n) = A n (n) grows faster than any primitive recursive function, and hence is not primitive recursive. 4. Arithmetization, Godel numbers Arithmetization of syntax treat numbers as symbols and formulas and proofs associate symbols with numbers via a function g : parts of L A ω, where g(a) = Gödel number of A. n {}}{ Recall that n = 0 for all n ω. Let (15) A = g(a). All syntactical notions (such as substitution, free variables, etc.) can be defined as primitive recursive number-theoretic functions and relations. The proof consists of several pages of definitions; see Propositions 3.26, 3.27, and 3.28.
12 12 Two important examples: Gödel s β-function β(x, y, z) = rm(1+(z +1)y, x) = remainder when x is divided by 1 + (z + 1)y, and the diagonal function D, which is a primitive recursive function D : ω ω such that, for every formula A(x) with exactly one free variable x, D(g(A(x))) = g(a( A )) = g(sub x A (A)). 5. The Fixed-Point Theorem. Gödel s Incompleteness Theorem Prop Diagonalization Lemma, 3.35 Fixed-Point Theorem Suppose D is representable in K via the formula D(x, y), and E is any formula with one free variable. Let (16) F (x) := y(d(x, y) E(y)). Then K F ( F ) E( F ( F ) ). Proof. By the definition of D, we have (17) D(g(F )) = g(f ( F ), so, since D represents D, (18) K D( F, F ( F ) ). Let s = F and t = F ( F ), so we have (19) K D(s, t). By A5 and (16), (20) F (s) (D(s, t) E(t), so, by PC and (19) we get half of the desired conclusion, namely (21) K F (s) E(t). Since D represents D, we know that K can prove D is functional, i.e., (22) K D(s, z) (D(s, y) z = y). By Gen, A5, obtain the following instance of (22). (23) K D(s, t) (D(s, y) t = y), so, by MP, (19), and (23), (24) K D(s, y) t = y. Since K is a theory with equality, (25) K E(t) (t = y E(y)) so by PC, (24), and (25), we get (26) K E(t) (D(s, y) E(y)). From this it follows, by Gen and A3, that (27) K E(t) y (D(s, y) E(y)), i.e., by (16) (the definition of F ), (28) K E(t) F (s).
13 K is ω-consistent iff, for every formula F (x) with one free variable x, it is not the case that K xf (x) and K F (n) for every n ω. Prop For a theory with equality, ω-consistency implies consistency. Proof. Assume K is ω-consistent. Since K is a theory with equality, we have K x(x = x). By the ω-consistency of K applied to the formula x = x, there must be some n ω such that K (n = n). Since there is a formula that K cannot prove, it must be consistent. For all x, y, ω, Let Pf be the binary number-theoretic predicate such that y, x Pf iff y is the Gödel number of a proof in K of the formula with Gödel number x. Assume Pf is represented in K by Pf. Let E(x) be a formula that says x has no proof in K, that is, let (29) E(x) := ypf(y, x). Define F (x) by (30) F (x) := y(d(x, y) E(y)). Assume the primitive recursive diagonal function D is represented in K by D. Then (31) K D( F, F ( F ) ). By the Fixed-Point Theorem, (32) K F ( F ) E( F ( F ) ). Let (33) G := F ( F ), so (34) K G E( G ), K G ypf(y, G ). Prop Godel s Incompleteness theorem (i) If K is consistent, then K G. (ii) If K is ω-consistent, then K G. 13 Proof. (i): 1. K G Hyp. 2. n ω, Pf(n, g(g)) K Pf(n, G ) K ypf(y, G ) 3., (A4), PC 5. K E( G ) 1., (34) 6. K ypf(y, G ) 5., def of E(x), (34) 7. K is inconsistent 4., 6.
14 14 (ii): 1. K G Hyp. 2. K E( G ) 1., (34) 3. K ypf(y, G ) 2., def of E(x), PC 4. n ω, K Pf(n, G ) 3., K is ω-consistent 5. n ω, Pf(n, g(g)) Pf represents Pf 6. K G K is inconsistent 1., 6. The inconsistency of K contradicts Prop Assume K is a recursively axiomatized theory in a language containing the constant 0, the unary function symbol, and the binary function symbol +. Let Neg be the binary number-theoretic predicate such that x, y Neg iff x is the Gödel number of a formula whose negation has Gödel number y. Thus Neg(g(A), g( A)) Assume Neg is represented in K by formula N. Let x y be the formula z(x + z = y), where z is the first variable distinct form x and y. Let E(x) be a formula that says if y is a proof of x, then there is a proof of the negation of x that is smaller than y. For example, we may let (35) E(x) := z(n(x, z) y(pf(y, x) w(w y Pf(w, z)))). Also let (36) (37) F := y(d(x, y) E(x)), R := F ( F ). So R is a formula that says, every proof of me has a shorter proof of my negation. By the Fixed-Point Theorem, (38) K R E( R ). Prop Gödel-Rosser Theorem (i) If K is consistent, then K R. (ii) If K is consistent, then K R. Proof. (i): FOLE means by some laws concerning equality that are provable in First- Order Logic with Equality. 1. K R Hyp. 2. K E( R ) 1., (38) 3. n ω, Pf(n, g(r)) 1. def of Pf 4. K Pf(n, R ) 3., Pf represents Pf in K 5. K N( R, R ) Neg(g(R), g( R)), N represents Neg in K 6. K w(w n Pf(w, R )) 2., 4., 5., (35), (A4), MP
15 15 7. K w n w = 0 w = 1 w = n Hyp. on K 8. K Pf(0, R ) Pf(n, R ) 6., 7., FOLE 9. K R 1., K is consistent 10. 0, g( R) / Pf,..., n, g( R) / Pf K Pf(0, R ),..., K Pf(n, R ) 10., Pf represent Pf 12. K is inconsistent 8., 11. (ii): 1. K R Hyp. 2. K E( R ) 1., (38) 3. K z(n( R, z) y(pf(y, R ) w(w y Pf(w, z)))) 2., (35) Neg(g(R), g( R)) 4. K N( R, R ) N represents Neg in K 5. K y(pf(y, R ) w(w y Pf(w, R ))) 3., 4., K Neg is functional 6. K y(pf(y, R ) w(pf(w, R ) w y)) FOL 7. n ω, Pf(n, g( R)) K Pf(n, R ) 7., Pf represents Pf in K 9. K n y y n Hyp. on K 10. K y(pf(y, R ) y n) 6., 8., K y n y = 0 y = 1 y = n Hyp. on K 12. K Pf(0, R ) Pf(n, R ) 10., 11., FOLE 13. K R 12., FOLE 14. K is inconsistent 1., 13. Recall that the standard model N of Peano arithmetic has domain ω, interprets the function symbol as the successor function, the function symbol as multiplication, the symbol + as addition, and the relation symbol = as the equality relation between numbers. By the existence of the standard model, the theory S (Peano arithmetic) is consistent. (Any formal version of this consistency proof would require enough axioms to prove the existence of the standard model and any such theory happens to be more powerful than S itself.) The sentences G and R were specially constructed in order to prove Propositions 3.37 and 3.38 for S. We know S is consistent (and ω-consistent because S has the model N), so K G, K R, K R, and K G. Are G and R true in the standard model? Since G is provably equivalent to the statement that G is not provable in Peano arithmetic, and K G (i.e. G is not provable in Peano arithmetic), what G says is true, so N = G. Answer the same question for R in HW Problem Additional HW Problems
First-Order Logic. 1 Syntax. Domain of Discourse. FO Vocabulary. Terms
First-Order Logic 1 Syntax Domain of Discourse The domain of discourse for first order logic is FO structures or models. A FO structure contains Relations Functions Constants (functions of arity 0) FO
More informationInformal Statement Calculus
FOUNDATIONS OF MATHEMATICS Branches of Logic 1. Theory of Computations (i.e. Recursion Theory). 2. Proof Theory. 3. Model Theory. 4. Set Theory. Informal Statement Calculus STATEMENTS AND CONNECTIVES Example
More informationPeano Arithmetic. CSC 438F/2404F Notes (S. Cook) Fall, Goals Now
CSC 438F/2404F Notes (S. Cook) Fall, 2008 Peano Arithmetic Goals Now 1) We will introduce a standard set of axioms for the language L A. The theory generated by these axioms is denoted PA and called Peano
More informationHerbrand Theorem, Equality, and Compactness
CSC 438F/2404F Notes (S. Cook and T. Pitassi) Fall, 2014 Herbrand Theorem, Equality, and Compactness The Herbrand Theorem We now consider a complete method for proving the unsatisfiability of sets of first-order
More information03 Review of First-Order Logic
CAS 734 Winter 2014 03 Review of First-Order Logic William M. Farmer Department of Computing and Software McMaster University 18 January 2014 What is First-Order Logic? First-order logic is the study of
More informationPropositional and Predicate Logic - XIII
Propositional and Predicate Logic - XIII Petr Gregor KTIML MFF UK WS 2016/2017 Petr Gregor (KTIML MFF UK) Propositional and Predicate Logic - XIII WS 2016/2017 1 / 22 Undecidability Introduction Recursive
More information2.2 Lowenheim-Skolem-Tarski theorems
Logic SEP: Day 1 July 15, 2013 1 Some references Syllabus: http://www.math.wisc.edu/graduate/guide-qe Previous years qualifying exams: http://www.math.wisc.edu/ miller/old/qual/index.html Miller s Moore
More informationThis is logically equivalent to the conjunction of the positive assertion Minimal Arithmetic and Representability
16.2. MINIMAL ARITHMETIC AND REPRESENTABILITY 207 If T is a consistent theory in the language of arithmetic, we say a set S is defined in T by D(x) if for all n, if n is in S, then D(n) is a theorem of
More informationPart II Logic and Set Theory
Part II Logic and Set Theory Theorems Based on lectures by I. B. Leader Notes taken by Dexter Chua Lent 2015 These notes are not endorsed by the lecturers, and I have modified them (often significantly)
More informationPart II. Logic and Set Theory. Year
Part II Year 2018 2017 2016 2015 2014 2013 2012 2011 2010 2009 2008 2007 2006 2005 2018 60 Paper 4, Section II 16G State and prove the ǫ-recursion Theorem. [You may assume the Principle of ǫ- Induction.]
More informationChapter 1 : The language of mathematics.
MAT 200, Logic, Language and Proof, Fall 2015 Summary Chapter 1 : The language of mathematics. Definition. A proposition is a sentence which is either true or false. Truth table for the connective or :
More informationby Yurii Khomskii There is a weaker notion called semi-representability:
Gödel s Incompleteness Theorem by Yurii Khomskii We give three different proofs of Gödel s First Incompleteness Theorem. All three proofs are essentially variations of one another, but some people may
More informationCS156: The Calculus of Computation
Page 1 of 31 CS156: The Calculus of Computation Zohar Manna Winter 2010 Chapter 3: First-Order Theories Page 2 of 31 First-Order Theories I First-order theory T consists of Signature Σ T - set of constant,
More informationMotivation. CS389L: Automated Logical Reasoning. Lecture 10: Overview of First-Order Theories. Signature and Axioms of First-Order Theory
Motivation CS389L: Automated Logical Reasoning Lecture 10: Overview of First-Order Theories Işıl Dillig Last few lectures: Full first-order logic In FOL, functions/predicates are uninterpreted (i.e., structure
More information5. Peano arithmetic and Gödel s incompleteness theorem
5. Peano arithmetic and Gödel s incompleteness theorem In this chapter we give the proof of Gödel s incompleteness theorem, modulo technical details treated in subsequent chapters. The incompleteness theorem
More informationCOMP 409: Logic Homework 5
COMP 409: Logic Homework 5 Note: The pages below refer to the text from the book by Enderton. 1. Exercises 1-6 on p. 78. 1. Translate into this language the English sentences listed below. If the English
More informationChapter 0. Introduction: Prerequisites and Preliminaries
Chapter 0. Sections 0.1 to 0.5 1 Chapter 0. Introduction: Prerequisites and Preliminaries Note. The content of Sections 0.1 through 0.6 should be very familiar to you. However, in order to keep these notes
More informationArithmetic and Incompleteness. Will Gunther. Goals. Coding with Naturals. Logic and Incompleteness. Will Gunther. February 6, 2013
Logic February 6, 2013 Logic 1 2 3 Logic About Talk Logic Things talk Will approach from angle of computation. Will not assume very much knowledge. Will prove Gödel s Incompleteness Theorem. Will not talk
More informationGödel s Incompleteness Theorems by Sally Cockburn (2016)
Gödel s Incompleteness Theorems by Sally Cockburn (2016) 1 Gödel Numbering We begin with Peano s axioms for the arithmetic of the natural numbers (ie number theory): (1) Zero is a natural number (2) Every
More informationChapter 1. Sets and Mappings
Chapter 1. Sets and Mappings 1. Sets A set is considered to be a collection of objects (elements). If A is a set and x is an element of the set A, we say x is a member of A or x belongs to A, and we write
More informationMATH 3330 ABSTRACT ALGEBRA SPRING Definition. A statement is a declarative sentence that is either true or false.
MATH 3330 ABSTRACT ALGEBRA SPRING 2014 TANYA CHEN Dr. Gordon Heier Tuesday January 14, 2014 The Basics of Logic (Appendix) Definition. A statement is a declarative sentence that is either true or false.
More informationLogic: The Big Picture
Logic: The Big Picture A typical logic is described in terms of syntax: what are the legitimate formulas semantics: under what circumstances is a formula true proof theory/ axiomatization: rules for proving
More informationNotes for Math 601, Fall based on Introduction to Mathematical Logic by Elliott Mendelson Fifth edition, 2010, Chapman & Hall
Notes for Math 601, Fall 2010 based on Introduction to Mathematical Logic by Elliott Mendelson Fifth edition, 2010, Chapman & Hall All first-order languages contain the variables: v 0, v 1, v 2,... the
More informationOn some Metatheorems about FOL
On some Metatheorems about FOL February 25, 2014 Here I sketch a number of results and their proofs as a kind of abstract of the same items that are scattered in chapters 5 and 6 in the textbook. You notice
More informationProofs. Chapter 2 P P Q Q
Chapter Proofs In this chapter we develop three methods for proving a statement. To start let s suppose the statement is of the form P Q or if P, then Q. Direct: This method typically starts with P. Then,
More informationSyntax. Notation Throughout, and when not otherwise said, we assume a vocabulary V = C F P.
First-Order Logic Syntax The alphabet of a first-order language is organised into the following categories. Logical connectives:,,,,, and. Auxiliary symbols:.,,, ( and ). Variables: we assume a countable
More information07 Equational Logic and Algebraic Reasoning
CAS 701 Fall 2004 07 Equational Logic and Algebraic Reasoning Instructor: W. M. Farmer Revised: 17 November 2004 1 What is Equational Logic? Equational logic is first-order logic restricted to languages
More informationGödel s Incompleteness Theorems
Seminar Report Gödel s Incompleteness Theorems Ahmet Aspir Mark Nardi 28.02.2018 Supervisor: Dr. Georg Moser Abstract Gödel s incompleteness theorems are very fundamental for mathematics and computational
More information23.1 Gödel Numberings and Diagonalization
Applied Logic Lecture 23: Unsolvable Problems in Logic CS 4860 Spring 2009 Tuesday, April 14, 2009 The fact that Peano Arithmetic is expressive enough to represent all computable functions means that some
More informationGödel s Incompleteness Theorem. Part II: Arithmetical Definability. Computability and Logic
Gödel s Incompleteness Theorem Part II: Arithmetical Definability Computability and Logic The Language of Arithmetic The language of arithmetic L A contains the following four non-logical symbols: 0: constant
More information1 Predicates and Quantifiers
1 Predicates and Quantifiers We have seen how to represent properties of objects. For example, B(x) may represent that x is a student at Bryn Mawr College. Here B stands for is a student at Bryn Mawr College
More informationVAUGHT S THEOREM: THE FINITE SPECTRUM OF COMPLETE THEORIES IN ℵ 0. Contents
VAUGHT S THEOREM: THE FINITE SPECTRUM OF COMPLETE THEORIES IN ℵ 0 BENJAMIN LEDEAUX Abstract. This expository paper introduces model theory with a focus on countable models of complete theories. Vaught
More informationProofs. Chapter 2 P P Q Q
Chapter Proofs In this chapter we develop three methods for proving a statement. To start let s suppose the statement is of the form P Q or if P, then Q. Direct: This method typically starts with P. Then,
More informationMathematical Logic. Reasoning in First Order Logic. Chiara Ghidini. FBK-IRST, Trento, Italy
Reasoning in First Order Logic FBK-IRST, Trento, Italy April 12, 2013 Reasoning tasks in FOL Model checking Question: Is φ true in the interpretation I with the assignment a? Answer: Yes if I = φ[a]. No
More information2 Lecture 2: Logical statements and proof by contradiction Lecture 10: More on Permutations, Group Homomorphisms 31
Contents 1 Lecture 1: Introduction 2 2 Lecture 2: Logical statements and proof by contradiction 7 3 Lecture 3: Induction and Well-Ordering Principle 11 4 Lecture 4: Definition of a Group and examples 15
More informationLecture Notes 1 Basic Concepts of Mathematics MATH 352
Lecture Notes 1 Basic Concepts of Mathematics MATH 352 Ivan Avramidi New Mexico Institute of Mining and Technology Socorro, NM 87801 June 3, 2004 Author: Ivan Avramidi; File: absmath.tex; Date: June 11,
More informationMore Model Theory Notes
More Model Theory Notes Miscellaneous information, loosely organized. 1. Kinds of Models A countable homogeneous model M is one such that, for any partial elementary map f : A M with A M finite, and any
More informationMODEL THEORY FOR ALGEBRAIC GEOMETRY
MODEL THEORY FOR ALGEBRAIC GEOMETRY VICTOR ZHANG Abstract. We demonstrate how several problems of algebraic geometry, i.e. Ax-Grothendieck, Hilbert s Nullstellensatz, Noether- Ostrowski, and Hilbert s
More informationGödel s Completeness Theorem
A.Miller M571 Spring 2002 Gödel s Completeness Theorem We only consider countable languages L for first order logic with equality which have only predicate symbols and constant symbols. We regard the symbols
More informationIntroduction to Model Theory
Introduction to Model Theory Jouko Väänänen 1,2 1 Department of Mathematics and Statistics, University of Helsinki 2 Institute for Logic, Language and Computation, University of Amsterdam Beijing, June
More informationConstructions of Models in Fuzzy Logic with Evaluated Syntax
Constructions of Models in Fuzzy Logic with Evaluated Syntax Petra Murinová University of Ostrava IRAFM 30. dubna 22 701 03 Ostrava Czech Republic petra.murinova@osu.cz Abstract This paper is a contribution
More informationMathematics 114L Spring 2018 D.A. Martin. Mathematical Logic
Mathematics 114L Spring 2018 D.A. Martin Mathematical Logic 1 First-Order Languages. Symbols. All first-order languages we consider will have the following symbols: (i) variables v 1, v 2, v 3,... ; (ii)
More information185.A09 Advanced Mathematical Logic
185.A09 Advanced Mathematical Logic www.volny.cz/behounek/logic/teaching/mathlog13 Libor Běhounek, behounek@cs.cas.cz Lecture #1, October 15, 2013 Organizational matters Study materials will be posted
More informationProperties of Relational Logic
Computational Logic Lecture 8 Properties of Relational Logic Michael Genesereth Autumn 2011 Programme Expressiveness What we can say in First-Order Logic And what we cannot Semidecidability and Decidability
More informationFoundations of Mathematics MATH 220 FALL 2017 Lecture Notes
Foundations of Mathematics MATH 220 FALL 2017 Lecture Notes These notes form a brief summary of what has been covered during the lectures. All the definitions must be memorized and understood. Statements
More informationArithmetical classification of the set of all provably recursive functions
Arithmetical classification of the set of all provably recursive functions Vítězslav Švejdar April 12, 1999 The original publication is available at CMUC. Abstract The set of all indices of all functions
More informationPart I: Propositional Calculus
Logic Part I: Propositional Calculus Statements Undefined Terms True, T, #t, 1 False, F, #f, 0 Statement, Proposition Statement/Proposition -- Informal Definition Statement = anything that can meaningfully
More informationFirst Order Logic (FOL) 1 znj/dm2017
First Order Logic (FOL) 1 http://lcs.ios.ac.cn/ znj/dm2017 Naijun Zhan March 19, 2017 1 Special thanks to Profs Hanpin Wang (PKU) and Lijun Zhang (ISCAS) for their courtesy of the slides on this course.
More informationCS 514, Mathematics for Computer Science Mid-semester Exam, Autumn 2017 Department of Computer Science and Engineering IIT Guwahati
CS 514, Mathematics for Computer Science Mid-semester Exam, Autumn 2017 Department of Computer Science and Engineering IIT Guwahati Important 1. No questions about the paper will be entertained during
More informationMoreover this binary operation satisfies the following properties
Contents 1 Algebraic structures 1 1.1 Group........................................... 1 1.1.1 Definitions and examples............................. 1 1.1.2 Subgroup.....................................
More informationSet Theory and the Foundation of Mathematics. June 19, 2018
1 Set Theory and the Foundation of Mathematics June 19, 2018 Basics Numbers 2 We have: Relations (subsets on their domain) Ordered pairs: The ordered pair x, y is the set {{x, y}, {x}}. Cartesian products
More informationModel Theory MARIA MANZANO. University of Salamanca, Spain. Translated by RUY J. G. B. DE QUEIROZ
Model Theory MARIA MANZANO University of Salamanca, Spain Translated by RUY J. G. B. DE QUEIROZ CLARENDON PRESS OXFORD 1999 Contents Glossary of symbols and abbreviations General introduction 1 xix 1 1.0
More informationLecture 14 Rosser s Theorem, the length of proofs, Robinson s Arithmetic, and Church s theorem. Michael Beeson
Lecture 14 Rosser s Theorem, the length of proofs, Robinson s Arithmetic, and Church s theorem Michael Beeson The hypotheses needed to prove incompleteness The question immediate arises whether the incompleteness
More information06 Recursive Definition and Inductive Proof
CAS 701 Fall 2002 06 Recursive Definition and Inductive Proof Instructor: W. M. Farmer Revised: 30 November 2002 1 What is Recursion? Recursion is a method of defining a structure or operation in terms
More informationGödel s Incompleteness Theorem. Overview. Computability and Logic
Gödel s Incompleteness Theorem Overview Computability and Logic Recap Remember what we set out to do in this course: Trying to find a systematic method (algorithm, procedure) which we can use to decide,
More informationCS156: The Calculus of Computation Zohar Manna Winter 2010
Page 3 of 31 Page 4 of 31 CS156: The Calculus of Computation Zohar Manna Winter 2010 First-Order Theories I First-order theory T consists of Signature ΣT - set of constant, function, and predicate symbols
More informationA Model Theoretic Proof of Completeness of an Axiomatization of Monadic Second-Order Logic on Infinite Words
A Model Theoretic Proof of Completeness of an Axiomatization of Monadic Second-Order Logic on Infinite Words Colin Riba ENS de Lyon, Université de Lyon, LIP colin.riba@ens-lyon.fr http://perso.ens-lyon.fr/colin.riba/
More informationGödel s Incompleteness Theorem. Overview. Computability and Logic
Gödel s Incompleteness Theorem Overview Computability and Logic Recap Remember what we set out to do in this course: Trying to find a systematic method (algorithm, procedure) which we can use to decide,
More informationExamples: P: it is not the case that P. P Q: P or Q P Q: P implies Q (if P then Q) Typical formula:
Logic: The Big Picture Logic is a tool for formalizing reasoning. There are lots of different logics: probabilistic logic: for reasoning about probability temporal logic: for reasoning about time (and
More informationA Simple Proof of Gödel s Incompleteness Theorems
A Simple Proof of Gödel s Incompleteness Theorems Arindama Singh, Department of Mathematics, IIT Madras, Chennai-600036 Email: asingh@iitm.ac.in 1 Introduction Gödel s incompleteness theorems are considered
More informationKreisel s Conjecture with minimality principle
Kreisel s Conjecture with minimality principle Pavel Hrubeš November 9, 2008 Abstract We prove that Kreisel s Conjecture is true, if Peano arithmetic is axiomatised using minimality principle and axioms
More informationUndecidable propositions with Diophantine form arisen from. every axiom and every theorem in Peano Arithmetic. T. Mei
Undecidable propositions with Diophantine form arisen from every axiom and every theorem in Peano Arithmetic T. Mei (Department of Journal, Central China Normal University, Wuhan, Hubei PRO, People s Republic
More informationAlgebraizing Hybrid Logic. Evangelos Tzanis University of Amsterdam Institute of Logic, Language and Computation
Algebraizing Hybrid Logic Evangelos Tzanis University of Amsterdam Institute of Logic, Language and Computation etzanis@science.uva.nl May 1, 2005 2 Contents 1 Introduction 5 1.1 A guide to this thesis..........................
More informationLogic Synthesis and Verification
Logic Synthesis and Verification Boolean Algebra Jie-Hong Roland Jiang 江介宏 Department of Electrical Engineering National Taiwan University Fall 2014 1 2 Boolean Algebra Reading F. M. Brown. Boolean Reasoning:
More informationPREDICATE LOGIC: UNDECIDABILITY AND INCOMPLETENESS HUTH AND RYAN 2.5, SUPPLEMENTARY NOTES 2
PREDICATE LOGIC: UNDECIDABILITY AND INCOMPLETENESS HUTH AND RYAN 2.5, SUPPLEMENTARY NOTES 2 Neil D. Jones DIKU 2005 14 September, 2005 Some slides today new, some based on logic 2004 (Nils Andersen) OUTLINE,
More informationPropositional Logic: Syntax
Logic Logic is a tool for formalizing reasoning. There are lots of different logics: probabilistic logic: for reasoning about probability temporal logic: for reasoning about time (and programs) epistemic
More informationNONSTANDARD ARITHMETIC AND RECURSIVE COMPREHENSION
NONSTANDARD ARITHMETIC AND RECURSIVE COMPREHENSION H. JEROME KEISLER Abstract. First order reasoning about hyperintegers can prove things about sets of integers. In the author s paper Nonstandard Arithmetic
More informationDiscrete Mathematics. W. Ethan Duckworth. Fall 2017, Loyola University Maryland
Discrete Mathematics W. Ethan Duckworth Fall 2017, Loyola University Maryland Contents 1 Introduction 4 1.1 Statements......................................... 4 1.2 Constructing Direct Proofs................................
More informationMAT115A-21 COMPLETE LECTURE NOTES
MAT115A-21 COMPLETE LECTURE NOTES NATHANIEL GALLUP 1. Introduction Number theory begins as the study of the natural numbers the integers N = {1, 2, 3,...}, Z = { 3, 2, 1, 0, 1, 2, 3,...}, and sometimes
More informationOctober 12, Complexity and Absoluteness in L ω1,ω. John T. Baldwin. Measuring complexity. Complexity of. concepts. to first order.
October 12, 2010 Sacks Dicta... the central notions of model theory are absolute absoluteness, unlike cardinality, is a logical concept. That is why model theory does not founder on that rock of undecidability,
More informationBound and Free Variables. Theorems and Proofs. More valid formulas involving quantifiers:
Bound and Free Variables More valid formulas involving quantifiers: xp(x) x P(x) Replacing P by P, we get: x P(x) x P(x) Therefore x P(x) xp(x) Similarly, we have xp(x) x P(x) x P(x) xp(x) i(i 2 > i) is
More informationThe Mother of All Paradoxes
The Mother of All Paradoxes Volker Halbach Truth and Intensionality Amsterdam 3rd December 2016 A theory of expressions The symbols of L are: 1. infinitely many variable symbols v 0, v 1, v 2, v 3,...
More informationA field F is a set of numbers that includes the two numbers 0 and 1 and satisfies the properties:
Byte multiplication 1 Field arithmetic A field F is a set of numbers that includes the two numbers 0 and 1 and satisfies the properties: F is an abelian group under addition, meaning - F is closed under
More informationWhat are the recursion theoretic properties of a set of axioms? Understanding a paper by William Craig Armando B. Matos
What are the recursion theoretic properties of a set of axioms? Understanding a paper by William Craig Armando B. Matos armandobcm@yahoo.com February 5, 2014 Abstract This note is for personal use. It
More informationNotes on ordinals and cardinals
Notes on ordinals and cardinals Reed Solomon 1 Background Terminology We will use the following notation for the common number systems: N = {0, 1, 2,...} = the natural numbers Z = {..., 2, 1, 0, 1, 2,...}
More informationAutomata Theory and Formal Grammars: Lecture 1
Automata Theory and Formal Grammars: Lecture 1 Sets, Languages, Logic Automata Theory and Formal Grammars: Lecture 1 p.1/72 Sets, Languages, Logic Today Course Overview Administrivia Sets Theory (Review?)
More informationHandbook of Logic and Proof Techniques for Computer Science
Steven G. Krantz Handbook of Logic and Proof Techniques for Computer Science With 16 Figures BIRKHAUSER SPRINGER BOSTON * NEW YORK Preface xvii 1 Notation and First-Order Logic 1 1.1 The Use of Connectives
More informationLecture Notes on DISCRETE MATHEMATICS. Eusebius Doedel
Lecture Notes on DISCRETE MATHEMATICS Eusebius Doedel c Eusebius J. Doedel, 009 Contents Logic. Introduction............................................................................... Basic logical
More information07 Practical Application of The Axiomatic Method
CAS 701 Fall 2002 07 Practical Application of The Axiomatic Method Instructor: W. M. Farmer Revised: 28 November 2002 1 What is the Axiomatic Method? 1. A mathematical model is expressed as a set of axioms
More informationContribution of Problems
Exam topics 1. Basic structures: sets, lists, functions (a) Sets { }: write all elements, or define by condition (b) Set operations: A B, A B, A\B, A c (c) Lists ( ): Cartesian product A B (d) Functions
More informationArithmetical Hierarchy
Arithmetical Hierarchy 1 The Turing Jump Klaus Sutner Carnegie Mellon University Arithmetical Hierarchy 60-arith-hier 2017/12/15 23:18 Definability Formal Systems Recall: Oracles 3 The Use Principle 4
More informationMath 4606, Summer 2004: Inductive sets, N, the Peano Axioms, Recursive Sequences Page 1 of 10
Math 4606, Summer 2004: Inductive sets, N, the Peano Axioms, Recursive Sequences Page 1 of 10 Inductive sets (used to define the natural numbers as a subset of R) (1) Definition: A set S R is an inductive
More informationPart IA Numbers and Sets
Part IA Numbers and Sets Definitions Based on lectures by A. G. Thomason Notes taken by Dexter Chua Michaelmas 2014 These notes are not endorsed by the lecturers, and I have modified them (often significantly)
More informationOutline. We will now investigate the structure of this important set.
The Reals Outline As we have seen, the set of real numbers, R, has cardinality c. This doesn't tell us very much about the reals, since there are many sets with this cardinality and cardinality doesn't
More informationArithmetical Hierarchy
Arithmetical Hierarchy Klaus Sutner Carnegie Mellon University 60-arith-hier 2017/12/15 23:18 1 The Turing Jump Arithmetical Hierarchy Definability Formal Systems Recall: Oracles 3 We can attach an orcale
More informationThis paper is also taken by Combined Studies Students. Optional Subject (i): Set Theory and Further Logic
UNIVERSITY OF LONDON BA EXAMINATION for Internal Students This paper is also taken by Combined Studies Students PHILOSOPHY Optional Subject (i): Set Theory and Further Logic Answer THREE questions, at
More informationThe Syntax of First-Order Logic. Marc Hoyois
The Syntax of First-Order Logic Marc Hoyois Table of Contents Introduction 3 I First-Order Theories 5 1 Formal systems............................................. 5 2 First-order languages and theories..................................
More informationReminder of Notation. For a variable-free term t, we let t N N stand for the interpretation of t in N. (For example, (SSS0 SS0) N equals 6.
Reminder of Notation Language is always L NT = (0, S, +,, E,
More informationIntroduction to Model Theory
Introduction to Model Theory Charles Steinhorn, Vassar College Katrin Tent, University of Münster CIRM, January 8, 2018 The three lectures Introduction to basic model theory Focus on Definability More
More informationGödel s First Incompleteness Theorem
Gödel s First Incompleteness Theorem Alex Edmonds MAT 477 January 2014 Alex Edmonds (2014) Gödel s First Incompleteness Theorem January 2014 1 / 29 Incompleteness of Peano Arithmetic (Main Theorem): If
More information0 Sets and Induction. Sets
0 Sets and Induction Sets A set is an unordered collection of objects, called elements or members of the set. A set is said to contain its elements. We write a A to denote that a is an element of the set
More informationComputably Enumerable Algebras, Their Expansions, and Isomorphisms
Computably Enumerable Algebras, Their Expansions, and Isomorphisms Bakhadyr Khoussainov 1 and Steffen Lempp 2 and Theodore A. Slaman 3 1 Computer Science Department, The University of Auckland, New Zealand
More informationRepresentability of Recursive Functions
16 Representability of Recursive Functions In the preceding chapter we connected our work on recursion with our work on formulas and proofs in one way, by showing that various functions associated with
More informationChapter 2 Axiomatic Set Theory
Chapter 2 Axiomatic Set Theory Ernst Zermelo (1871 1953) was the first to find an axiomatization of set theory, and it was later expanded by Abraham Fraenkel (1891 1965). 2.1 Zermelo Fraenkel Set Theory
More information06 From Propositional to Predicate Logic
Martin Henz February 19, 2014 Generated on Wednesday 19 th February, 2014, 09:48 1 Syntax of Predicate Logic 2 3 4 5 6 Need for Richer Language Predicates Variables Functions 1 Syntax of Predicate Logic
More informationShort Introduction to Admissible Recursion Theory
Short Introduction to Admissible Recursion Theory Rachael Alvir November 2016 1 Axioms of KP and Admissible Sets An admissible set is a transitive set A satisfying the axioms of Kripke-Platek Set Theory
More information1 FUNDAMENTALS OF LOGIC NO.10 HERBRAND THEOREM Tatsuya Hagino hagino@sfc.keio.ac.jp lecture URL https://vu5.sfc.keio.ac.jp/slide/ 2 So Far Propositional Logic Logical connectives (,,, ) Truth table Tautology
More informationCMPSCI 601: Tarski s Truth Definition Lecture 15. where
@ CMPSCI 601: Tarski s Truth Definition Lecture 15! "$#&%(') *+,-!".#/%0'!12 43 5 6 7 8:9 4; 9 9 < = 9 = or 5 6?>A@B!9 2 D for all C @B 9 CFE where ) CGE @B-HI LJKK MKK )HG if H ; C if H @ 1 > > > Fitch
More informationContribution of Problems
Exam topics 1. Basic structures: sets, lists, functions (a) Sets { }: write all elements, or define by condition (b) Set operations: A B, A B, A\B, A c (c) Lists ( ): Cartesian product A B (d) Functions
More informationMAGIC Set theory. lecture 2
MAGIC Set theory lecture 2 David Asperó University of East Anglia 22 October 2014 Recall from last time: Syntactical vs. semantical logical consequence Given a set T of formulas and a formula ', we write
More information