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he linear map TpM → TpM : v → ∇vY (p). Thus we can think of the covariant derivative as a linear operator ∇ : Ω0(M, T M ) → Ω1(M, T M ). The equation (5.2.7) asserts that the operators X → ∇X indeed determine a linear operator from Ω0(M, T M ) to Ω1(M, T M ). Equation (5.2.8) asserts that this linear operator ∇ is a co... |
∗∇X, and if X, Y ∈ Vect(M ) are global vector fields, then ∇ φ∗X φ∗Y = φ∗(∇X Y ). (5.3.1) (5.3.2) (iii) If γ : I → M is a geodesic, then φ ◦ γ : I → M is a geodesic. (iv) If γ : I → M is a smooth curve, then for all s, t ∈ I, we have Φ φ◦γ(t, s)dφ(γ(s)) = dφ(γ(t))Φγ(t, s). (5.3.3) (v) φ∗R = R. 246 CHAPTER 5. CURVATURE P... |
0 252 CHAPTER 5. CURVATURE By the Gauß–Codazzi formula this implies Rp(u, v)w = hp(u)∗hp(v)w − hp(v)∗hp(u)w = dν(p)udν(p)vTw − dν(p)vdν(p)uTw = dν(p)v, wdν(p)u − dν(p)u, wdν(p)v and hence Rp(u, v)w, z = dν(p)u, zdν(p)v, w − dν(p)u, wdν(p)v, z. (5.3.8) Now fix four tangent vectors u, v, w, z ∈ TpM and consider the compos... |
1.8 may be dropped from (ii). Lemma 6.1.12. Let φ : M → M be a local isometry and let γ : I → M be a smooth curve on an interval I. Fix an element t0 ∈ I and define p0 := γ(t0), q0 := φ(p0), Φ0 := dφ(p0). (6.1.3) Then there exists a unique development (Φ, γ, γ) of M along M on the entire interval I satisfying the initia... |
atisfying the initial condition (6.1.2) as well as γ(I) ⊂ Ur and γ(I) ⊂ U r satisfying (6.1.1). r, then γ(1) = p0 =⇒ γ(1) = p 0, Φ(1) = Φ0. (iii) If (Φ0, γ0, γ 0) and (Φ1, γ1, γ γ0(1) = γ1(1) 1) are developments as in (ii), then 0(1) = γ γ 1(1). =⇒ (iv) If v ∈ Tp0M with |v| < r and γ(t) := exp p 0 γ(t) := expp0(tv), th... |
hile the former hypotheses are conditions on M and M respectively, namely, that the Riemann curvature tensor is invariant by parallel transport. It is rather amazing that this condition is equivalent to a simple geometric condition as we now show. 6.3. SYMMETRIC SPACES 273 6.3.1 Symmetric Spaces Definition 6.3.1. A Riem... |
metric (but not simply connected). This shows that the hypothesis of simple connectivity cannot be dropped in part (ii) of Corollary 6.3.5. Example 6.3.16. Below we define manifolds of constant curvature and show that they are locally symmetric. The simplest example, after a flat space, is the unit sphere Sm = x ∈ Rm+1 |... |
is also free. 6.4. CONSTANT CURVATURE 285 Examples and Exercises Example 6.4.14. Any flat Riemannian manifold has constant sectional curvature k = 0. Example 6.4.15. The manifold M = Rm with its standard metric is, up to isometry, the unique connected, simply connected, complete Riemannian m-manifold with constant sect... |
map is a diffeomorphism for every p ∈ Hm. Thus any two points in Hm are connected by a unique geodesic. Prove that the intrinsic distance function on hyperbolic space is given by d(p, q) = cosh−1 (Q(p, q)) for p, q ∈ Hm. Compare this with Example 4.3.11. (6.4.21) Exercise 6.4.27. In the case m = 2 the Poincar´e model of... |
be the unique geodesic with the endpoints γ(0, t) = γ0(t) and γ(1, t) = γ1(t). Then ρ(t) := d(γ0(t), γ1(t)) = 1 0 |∂sγ| ds = |∂sγ(s, t)| for all s and t and hence 1 ˙ρ(t) = ∂sγ, ∇t∂sγ |∂sγ| ds 0 ∂sγ(1, t), ∂tγ(1, t) − ∂sγ(0, t), ∂tγ(0, t) ρ(t) = (6.5.8) . Since d dt (ρ ˙ρ) = ρ¨ρ + ˙ρ2 and γ0 and γ1 are geodesics, this ... |
4 4 1 2 1 4 AP −1SP −1T − AP −1(∂tP )P −1S − 1 4 T P −1(∂sP )P −1A 1 2 T P −1AP −1S + 1 2 SP −1(∂tP )P −1A SP −1AP −1T − T P −1AP −1S − 1 4 SP −1AP −1T + 1 4 SP −1T P −1A T P −1SP −1A. 1 4 1 4 Insert ∂sP = S, ∂tP = T to obtain (6.5.19). This proves Lemma 6.5.19. 306 CHAPTER 6. GEOMETRY AND TOPOLOGY Proof of Theorem 6.5... |
) ˙γ(t), Xi(t) dt |∇tXi(t)|2 dt i=2 0 1 = (m − 1) 0 π2(m − 1) 2 . = π2 cos2(πt) dt Here the third step uses (6.6.5). Since m ≥ 2 it follows from this estimate that d(p, q)2 ≤ π2/δ and this proves Theorem 6.6.2. A direct consequence of Theorem 6.6.2 is that every compact manifold with positive Ricci curvature has a comp... |
is problem was formulated in 1960 by Hidehiko Yamabe and was eventually settled in the affirmative in 1984 by the combined work of Hidehiko Yamabe [78], Thierry Aubin [5], Neil Trudinger [75], and Richard Schoen [69]. The proof for a compact manifold M of dimension m > 2 relies on finding a positive function f : M → R and... |
. . . , em is an orthonormal basis of TpM . The right hand side of equation (6.8.6) is independent of the choice of this orthonormal basis and is a 2-form by Lemma 6.8.2. 6.8. THE WEYL TENSOR* 323 Now let M be an oriented Riemannian 4-manifold. Then a 2-form ω on M is called self-dual iff it satisfies the condition ωp(e... |
(7.1.3) Thus mγ(τ ) = 0 whenever τ is not a conjugate point of γ. 7.1. CONJUGATE POINTS AND THE MORSE INDEX* 329 In §4.4 we have addressed the question when a geodesic γ : [0, 1] → M with the endpoints γ(0) = p and γ(1) = q minimizes the lengths of curves with the same endpoints globally, i.e. when it satisfies L(γ) = d... |
isely k more negative eigenvalues than for τ0 − δ < τ < τ0. Hence the number of the negative eigenvalues of A1 with multiplicities is 0<τ <1 dim(ker(Aτ )). This proves Theorem 7.1.6. For more detailed explanations and other closely related applications of the Kato Selection Theorem the reader is referred to [61, 76]. 3... |
TpiM → Texppi (vi)M is not injective. Passing to a subsequence, if necessary, we may assume that vi converges to a vector v0 ∈ Tp0M . Then, by smoothness of the exponential map, the derivative d expp0(v0) : Tp0M → Texpp0 (v0)M is not injective. Since |v0| ≤ r, this contradicts the fact that r < inj(p0; M ). This prove... |
fields (defined below). Proof. See §7.3.4. 344 CHAPTER 7. TOPICS IN GEOMETRY 7.3.2 The Topology on the Space of Isometries The next lemma shows that for each p ∈ M the set Ip in (7.3.4) is a closed subset of Gp and that the map ιp : I(M ) → Ip in (7.3.3) is a homeomorphism with respect to the C∞ topology on I(M ). Lemma... |
(p, γ(t)) ≤ t 0 | ˙γ(s)| ds = t|X(p)| for all t ∈ I. Since M is complete, the closed ball of radius R about p is compact for every R > 0 (Theorem 4.6.5). Hence the restriction of γ to any bounded subinterval of I is contained in a compact subset of M amd by Corollary 2.4.15 this implies I = R. This proves (iii) and Lem... |
mi + 1)τi. Then Step 2 asserts that φT (p) = limi→∞ φmi i (p). Since φmi : M → M is an isometry, it follows that T |X(p)| < inj(φmi i i (p); M ) for all i ∈ N and hence T |X(p)| < injφT (p); M . By Step 1 this implies [0, T ] ⊂ IφT (p). Hence [0, 2T ] ⊂ Ip and, by Step 2, φ2T (p) = lim i→∞ φmi i φT (p). Continue by ind... |
n of §7.3.1. By Corollary 6.4.13 a complete, connected, simply connected manifold M satisfies Ip = Gp if and only if it has constant sectional curvature. Exercise 7.3.17. Consider the incomplete 2-manifolds M0 := R2 \ {(0, 0)}, M1 := R2 \ Z2. Prove that for i = 0, 1 every isometry of Mi extends to an isometry of R2 and ... |
spans Z(g). Part (a) was noted above, part (b) follows from Exercise 2.5.39 because the Lie algebra Z(g) is commutative, and part (c) follows from the Hopf–Rinow Theorem 4.6.6. That the set Λ is an additive subgroup of Z(g) follows directly from (b). Moreover, by (a) and (b) the exponential map restricts to a local diff... |
= R and d(q, γ(R)) = |w(R)|, the triangle inequality yields R − d(p, q) ≤ |w(R)| ≤ R + d(p, q). (7.5.10) Combinig the inequalities (7.5.7), (7.5.8), (7.5.9), and (7.5.10) we find that ∂ ∂R FR,q,p(v) ≤ | ˙w(R) − w(R)/R| |w(R)| ≤ |vp − vq| |w(R)| ≤ d(p, q) R(R − d(p, q)) . This proves the estimate (7.5.6). 7.5. CONVEX FU... |
s and is a real number (and not −∞). Since d dt f (γ(t)) = −|∇f (γ(t))|2 and the function t → f (γ(t)) is bounded below by c, there must exist a sequence ti → ∞ such that lim i→∞ ∇f (γ(ti)) = 0. Since γ(ti) ∈ B for all i, we may also assume that the limit p∞ := lim i→∞ γ(ti) (7.5.18) (7.5.19) exists (after passing to a... |
tion ρ : [0, ∞) → R in (7.5.31) is bounded. This proves Claim 1 and Lemma 7.5.14. 7.5. CONVEX FUNCTIONS ON HADAMARD MANIFOLDS* 381 Proof of Theorem 7.5.12. Let G ⊂ SL(V ) be a Lie subgroup which acts irreducibly on V . Then G acts on P0(V ) by isometries. By Lemma 7.5.14 the induced action on the sphere at infinity S∞(P... |
1/2 P P −1/2. Then, by Lemma 6.5.18, γ(t) = P 1/2 exp(tS)P 1/2 and hence fw(γ(t)) = ρ(P −1/2)w, exp(−t ˙ρ(S))ρ(P −1/2)wW . This implies d2 dt2 fw(γ(t)) = ˙ρ(S)ρ(P −1/2)w, exp(−t ˙ρ(S)) ˙ρ(S)ρ(P −1/2)wW ≥ 0 for all t and hence fw is convex. Hence Mw = Crit(fw) by Lemma 7.5.9. That fw is Gw-invariant follows directly fro... |
c vector space and ω is called a symplectic form on V . The isotropy subgroup of ω in GL(V ) is called the symplectic linear group. Denote this group and its Lie algebra by ω(g·, g·) = ω , Sp(V, ω) := g ∈ GL(V ) sp(V, ω) := Lie(Sp(V, ω)) = A ∈ End(V ) ω(A·, ·) + ω(·, A·) = 0 . The group Sp(V, ω) is connected and contai... |
algebra. Lemma 7.6.9. Let g be a finite-dimensional simple real Lie algebra. Then the center of g is trivial, the adjoint representation ad : g → Der(g) is injective, the commutant is [g, g] = g, and trace(ad(ξ)) = 0 for all ξ ∈ g. Proof. The center Z(g) is an ideal in g. It is not equal to g because g is not abelian, ... |
form follows from Lemma 7.6.8. The remaining assertions of Theorem 7.6.10 are direct consequences of the nondegeneracy of the Killing form. In particular, the adjoint representation ad : g → Der(g) is bijective and Der(g) ⊂ sl(g) by Lemma 7.4.3. Hence Aut0(g) ⊂ SL(g) and so Gw = G = Aut0(g) in the notation of §7.5.3. ... |
ugate to K. (ii) The map K × Der+(g) → Aut0(g) : (u, δ) → exp(δ)u is a diffeomorphism. (iii) If there exists a c > 0 such that κ(ξ∗, η) = cξ, η for all ξ, η ∈ g, then Mg := Crit(fg) = P0(g) ∩ Aut(g) = {exp(δ) | δ ∈ Der+(g)} is a totally geodesic and geodesically convex submanifold of P0(g) and so is a Hadamard manifold ... |
δ2 = 0. (g) The space of oriented Lagrangian Lie subalgebras l ⊂ gc isomorphic to g (that can be joined to g by a path of such subspaces) is diffeomorphic to the quotient space Lg := Kc/K, where Kc := Aut0(gc) ∩ SU(gc) and K := Aut0(g) ∩ SO(g). Hint: Choose the embedding Φ + iΨ in (f) such that Φ∗Φ + Ψ∗Ψ = 1l, Φ∗Ψ − Ψ∗Φ... |
ce of o ∈ En. Note that En contains the “preferred” point 0 while En has no preferred point. Such spaces En and En would arise in linear algebra by taking En to be the space of solutions of k − n independent inhomogeneous linear equations in k unknowns while En is the space of solutions of the corresponding homogeneous... |
ings of the American Mathematical Society 20 (1969), 603. [63] Dietmar Arno Salamon, Analysis I. Lecture Course, ETHZ, HS 2016. [64] Dietmar Arno Salamon, Analysis II. Lecture Course, ETHZ, FS 2017. https://people.math.ethz.ch/~salamon/PREPRINTS/analysis2.pdf [65] Dietmar Arno Salamon, Spin Geometry and Seiberg–Witten... |
refinement, 101 regular value, 21, 32 relative topology, 9 relatively closed, 9 relatively open, 9 Ricci tensor, 309 in local coordinates, 309 positive, 310 Riemann curvature tensor, 232 covariant derivative, 275 in local coordinates, 278 first Bianchi identity, 239 in local coordinates, 254 second Bianchi identity, 276... |
= ˙x = m k g + g e−kt/m. Since v = u at t = 0, we get c = u − m Integrating once gives x = m k gt − u − m k g m k e−kt/m + d. Since x = 0 at t = 0. So So x = m k gt + m k u − g m k (1 − e−kt/m). In component form, let x = (x, y), u = (u cos θ, u sin θ), g = (0, −g). So x = cos θ(1 − e−kt/m) mu k mgt k y = − + m k u si... |
)2 a2 + y2 b2 = 1, where a = 1 − e2 and b = √ 1 − e2 ≤ a. b O a ae a and b are the semi-major and semi-minor axis. is the semi-latus rectum. One focus of the ellipse is at the origin. If e = 0, then a = b = and the ellipse is a circle. – Hyperbola: (e > 1). For e > 1, r → ∞ as θ → ±α, where α = cos−1(1/e) ∈ (π/2, π). T... |
ct vanishes at the equator. Note that only the horizontal effect of horizontal motion vanishes at the equator. The vertical effects or those caused by vertical motion still exist. Example. Suppose a ball is dropped from a tower of height h at the equator. Where does it land? In the rotating frame, ¨r = g − 2ω × ˙r − ω × ... |
lid object that cannot deform. We call these rigid bodies. Definition (Rigid body). A rigid body is an extended object, consisting of N particles that are constrained such that the distance between any pair of particles, |ri − rj|, is fixed. The possible motions of a rigid body are the continuous isometries of Euclidean ... |
of motion is So ˙L = G. I ¨θ = −M g 2 sin θ, or 3g 2 which is exactly equivalent to a simple pendulum of length 2/3, with angular ¨θ = − sin θ. frequency 3g 2 . This can also be obtained from an energy argument: E = T + V = I ˙θ2 − M g 1 2 2 cos θ. We differentiate to obtain dE dt = ˙θ(I ¨θ + M g 2 sin θ) = 0. I ¨θ = −... |
e on. 8.3 Relativistic physics Now we can look at all sorts of relativistic weirdness! 65 8 Special relativity IA Dynamics and Relativity Simultaneity The first relativistic weirdness is that different frames disagree on whether two evens are simultaneous Definition (Simultaneous events). We say two events P1 and P2 are s... |
mes, the events lie on the boundary of each other’s light cones. e.g. different points in the trajectory of a photon are lightlike separated, hence the name. Note that ∆s2 = 0 does not imply that P and Q are the same event. The Lorentz group The coordinates of an event P in frame S can be written as a 4-vector (i.e. 4-c... |
m2 kg s−1 is Planck’s constant. For massless particles, this is consistent with Planck’s relation: E = hc λ = hν, where ν = c λ is the wave frequency. Newton’s second law in special relativity Definition (4-force). The 4-force is F = dP dτ This equation is the relativistic counterpart to Newton’s second law. It is relat... |
the following arc true for all sets A, B, and C? (a) IfA V Band B(ZC, thenA IZC. (b) If A B and B C, then A Pd- C. (c) If A C B and / C, then A 1Z C. (d) If A C Band BCC, then C CT- A. (e) If A C Band B C C, then A VC. 3.5. Show that for every set A, A C 0 if A = 0. 3.6. Let A,, A2, . , A. be n sets. Show that A,CA2C .... |
of their intersection. If xCBnC., then xC BandxEC.IlencexCA U B and x E A U C, so again x is a member of their intersection. (b) Proof that (A U B) n (A U C) C A U (BnC). Let x C (AUB)n(AUC).Then xEAUBand xEAUC.Hence x C A, or x E B and x C C. These imply that x C A U (B n C). Identities 1 and 1' are referred to as th... |
trate that a proof of the dual results. following equations is an identity. 5.3. Using only the identities in Theorems 5.1 and 5.2, show that each of the (a) (AnBfX)U(Af BnCfXfY)U(AnXnA) =AfBnX. (b) (A n RnC)U (1nBnC)UPUC= U. (c) (AntinCn )u(AnC)u(7nc)u(CnX)=C. (d) [(Af B)U(AnC)U(AnXnY)] n[(Ar) 1nc)U(Ai1X(1Y)u(AnBnY)] ... |
et. Intermediate is the identity relation in X, symbolized by t. or tx, which is {(x, x)lx C X}. For x, y in X, clearly, xtxy if x = y. If p is a relation and A is a set, then p[A] is defined by p[A] = {yj forsome xinA,xpy}. 1.6 1 Relations 27 This set is suggestively called the set of p-relatives of elements of A. Cle... |
d call it the quotient set of X by p. The significance of the partition of a set X accompanying an arbitrary equivalence relation p on X is best realized by comparing p with the extreme equivalence relation on X of identity. We classify identity on X as an extreme equivalence relation because the only element equal to ... |
available the following terminology. A function f is into Y if the range of f is a subset of Y, and f is onto Y if Rf = Y. For corresponding notation for the domain of a function we shall say that f is on X when the domain off is X. The symbols f: X-*- Y and X-f3.- Y are commonly used to signify that f is a function on... |
t be shown that if [x] _ [Y] then f(x) = f(y). But [x] _ [y] if xpy if f(x) = f(y); so g is a function. Finally, we let i be the injection of f[X] into Y. Collectively, we have defined three functions j, g, i where j: X-' X/p with J(x) = [x], g: X/p-} f[X] with g([x]) = I(x), i: f[X] -} Y with i(y) = y. Clearly, j is o... |
ion 0, (of subsets of U) is defined to be (x C Ulx C .I for all X in Ct}. For a nonempty collection, the new definition agrees with the old. The cliffercncc is the way in which they treat the empty collection; according to the new definition, n xcox = U, which seems to be a more reasonable result. Algebraic properties ... |
relation p that is reflexive and transitive is a. preordering. A potential shortcoming of such a relation, in connection with establishing an order of precedence in a set X, is the possibility of p being "indifferent" to some distinct pair x, y of objects in the. sense that both xpy and ypx. For example, in some popul... |
to a collection consisting of all subsets of some set. Such those of the form (4'(A), q), do not partially ordered sets, that is, typify partially ordered sets in general, since they have special features. For example, each contains an clement (namely, 0) less than every other element and an element (namely, A) greate... |
a one-to-one mapping on N into N - (0}. If M is a subset of N, such that 0 C M and m' C M whenever m C M, then h1 = N. Property N2 (which has its origin in the assertion that the succession of discrete steps -consisting of starting with 0 and repeatedly passing from a number to its successor-yields all of the natural n... |
g(n', g(fu)) = g(n', k(n')). Hence, k(0) = c and k(a) _ g(n, k(n)) for n C N. 't'hat k is unique is shown by a straightforward induction proof, which is left as an exercise. We turn now to the definition of an ordering relation for N. The basis for the intuitive ordering of the natural numbers is the order in which the... |
m) = nm', where we have used the preliminary result n'O = n and the one distributive law already proved. Hence M2 follows by the principle of induction. M,. For fixed n and p consider {m C NIm(np) = (mn)p}. This set contains 0, and if it contains m then it contains in', since m'(np) = m(np) + np = (mn)p + np = (mn + n)... |
ation of n which contains p as a factor. If the theorem is false for n then it has a second representation. If q is the smallest prime present in this second representation, then q > p, since this other representation of n does not involve p and p is the smallest divisor (> 1) of n. Let n = qn2 and q = p + d. Then n = ... |
ve for exercises similar derivations for irmultiplication, the exponential function, and time factorial function. 2.4. The predecessor of x, pd(x), is defined by pd(0) = 0 and pd(x') = x. It is prirnitivc-recursive by virtue of the primitive-recursive derivation: Cot (x) - 0, U; (x, y) = x, Jpd(O) = r,,,, ' pd(x') = Ui... |
dinal number. It can be successfully argued that is immaterial as to what cardinal numbers are, for mathematics it explicitly, so long as they have the property A = 13 iff A ti B. Indeed, by virtue of this property, we shall find that all questions regarding equality and inequality of cardinals can be reduced to questi... |
t q < c p. We turn our attention next to the nonfinite cardinals. A nonfinite cardinal is an infinite or transfinite cardinal. If the cardinal number of a set is infinite the set is called infinite. The cardinal number of the set of natural numbers is symbolized by bto. TI!EOREM 3.5. If n is a finite cardinal, then n <... |
theorem, the proposition now at hand can be reduced to the case of a function whose domain is a subset of N. So consider f: A -. 13 where A _C N and B is the range off. It is to be shown that B is countable. Let C be the set of all memf(y). That is, C bers x of A such that if y E A and y < x, then f(x) consists of the... |
ion of a denumerable family of denumerable sets is denumerable, prove'I'heorem 4.5. 4.6. Show that N can be represented as a union of a denumerable family of denumerable disjoint sets. 4.7. Give an intuitive proof that any infinite set includes a denumerable subset. From this deduce that (1) No is the least infinite ca... |
ermines a single order type. This is not true of an infinite set, which admits of a simple ordering; relative to various orderings there will correspond different order types. For the purpose of illustrating this remark, as well as for later examples, it is convenient to employ our notation (...) for an ordered n-tulle... |
we have a > f(a). I;et B be the subset of A of all such elements and b its least rtrenther. Since b > f(b) it follows chat f(b) > f(f(b)). Thus f(b) E B, which is a contradiction. f The hypothesis that xo have property P is redundant, for it is that instance of the second hypothesis which results upon choosing z as xo... |
or example, w > 1and1+w<w-I-Iwhile w-1.2>w+I and (w -I- 1) -}- (w -I- 2) > (w + 21 - (w A- 1). 7.10. According to '['heorcrn 7.5, if the well-ordered set A is of order type a, .c(a), the set of ordinals less than a. Ilence this set of ordinals may be then A used to index the members of A. That is, we may describe A as ... |
it should be named an axiom is simply an indication that no one has been able to infer the existence of such a set, in general (other than from an equivalent property of sets). 1 he axiom of choice has been the subject of serious controversy among totally on such grounds as the utter mathematicians. Some reject it imp... |
The Axio,n of Choice and Zorn's Lemma 117 (111) implies (IV). Let X be any set. We consider ordered pairs (A, p) where A C X and p simply orders A. Let S be the set of all (A, p) such that p well-orders A. If (A,, pl) and (A2, p2) are members of g, define (A,, p,) < (A2, P2) ill (a) A. S A2, (b) P, C P2, and (c) if a, ... |
of the idcmpotcncy of multiplication is due to Zorn (1944). To facilitate the exposition we introduce a temporary definition: To say that a set A has property 9, symbolized i(A), shall mean that A has at least two distinct members and A2 = 1. Additional properties of such sets, as well as some properties of related se... |
ssible in view of Theorem 10.1. Hence c < 1!I(c), from which it follows that every transfinite cardinal is an aleph. This, in turn, yields the axiom of choice. 126 The Natural Number Sequence and its Generalizations I C H A P P. 2 'I' I I E 0 R E M 10. 5 (Tarski). The axiom of choice is equivalent to the assertion that... |
h respect to multiplication--that is, the minimum enlargement of 'Z to attain the solvability of all equations of the form. xb = a with a, b C Z and b s 0. The resulting set is the set Q of rational numbers. The third extension amounts to the completion ofO with respect to order---that is, the minimmum enlargement of a... |
e latter, consider any integer [(p, q)Jj. Exactly one of p > q and p < q holds. In the former case, p = q + it with it C N, and hence [(p, q)]i = [(n, 0)] C Z°. In the latter case, q = p -l- in with in C N - {0) and [(p, q)]i = [(0, rn)J which completes the proof. It is a straightforward exercise to demonstrate that th... |
rlier proofs carry over without change. We are now in a position to simplify the notation for rationals. The string of identities CaavJA tiiJ= l 8 = [1J]e L1 tie' = aiba 3.4 I Rational Numbers 141 shows that each rational can he written in terms of integral rational numbers. We shall drop the subscript "s" from now on ... |
, = IXnyn - unyn + unyn - unvnl < lynl Ixn - u,, + 111"1 lyn -- and < < E. 52(e/252) That if x is positive and x -, u, then u is positive is shown as follows. By assumption, there exists a positive rational number 2E and an integer N, such that for every n > N, x > 2E, and there exists an N2 such that for every n > N2 ... |
ue of (1) arid, in turn, (2), for every n (3) (4) bn-2-n <u, u < bn. f We shall now prove that u is an upper bound of A. Assume to the contrary that a > u for some a in A. Then there exists an n such that 2" > (a - u)--l or 2-n<a-u Addition of this to (3) yields the inequality bn < a, a contradiction of the fact that b... |
e of B. in SR. Suppose now that X is any superreal number. Let x E X, which means that x is a Cauchy sequence of real numbers. According to Theorem 7.1, x has a limit y, whence x is -.,equivalent to (y, y, ), which implies that X = yer. This means that the image of B. in SR exhausts SR or, in other words, that SR is an... |
,P--3Q,PHQ is a statement. Let us elaborate. On the basis of the usual meaning of "not," if a statement is true, its negation is false, and vice versa. For example, if S is the true stateinent (has truth value T) "The moon is a satellite of the earth," then --S is false (has truth value F). By convention, the conjuncti... |
es and that we extend this set by adjoining precisely all of those sentences which can be formed by using, repeatedly and in all possible ways, the various sentential connectives. Then the extended set has the following property. If A and B are members, then so are each of -i A, A V B, A A B, A --* B, and A-+ B. We sha... |
prime component in at least one of A and B. In terms of truth tables, the definition amounts to this. Suppose that . -, P,4 is the union of the sets of prime components contained I P,, P2, in A and B, respectively, and that we compute the truth tables of A and of B as if both contained P,, P2, , P,,, as prime component... |
1 + A) B) B, which, as one sees immediately, is identically 0. In the algebra at hand, 2x = 0, x(x + 1) = 0, and x2 = x for all x. These facts make the simplification of long expressions an easy matter. Prove some of the tautologies in Theorem 3.4 by this method. 3.8. (a) With Exercise 3.7 in mind, show that the functi... |
derivation, incorporating some practical abbreviations. In this form the reader is called on to supply the tautologies employed. {1} {1) (3) {3} (1) A -> C (2) A V B--3CV B (3) B -- D (4) CV {6} {1,3,6) B (6) A V B (7) CV D D p It p 2,41 p 5,61 4.3. As a more elaborate illustration we prove that WVP->I,I-CVS,S-U,-,CA-... |
y the final line is any contradiction. For example, suppose that it is a question of the satisfiability of a set of statements which may be symbolized as AFiB, B --*C, - - 1 We adopt these as a set of premises and investigate what inferences can be made. {1} {2) {3} {4} {5} (4,5) {4, 5} (1,2) 11, 2, 4, 5} {3, 51 {1, 2 ... |
." In logic the word "predicate" has a broader role than it has in grammar. The basis for this is the observation that if a predicate is supplemented by including a variable as a placeholder for the intended subject (for example, "x is a real number"), the result behaves as a "statement function" in the sense that for ... |
e old are lawyers. Some women are both lawyers and Congressmen. (Wx) No woman is both a politician and a housewife. (lIx) There are some women lawyers who are housewives. All women who are lawyers admire some judge. (Axy) Some lawyers admire only judges. Some lawyers admire women. Some shysters admire no lawyer. Judge ... |
bound and, clearly, x serves as a variable. That x in the formula (3x) (y ; x) 204 Logic I CHAP. 4 serves as a variable is made more plausible on recalling that this formula has the same meaning as (x) --I (Y 7,!5 X) In conclusion, we note that it is now possible to give a precise definition of the word "statement." A ... |
e predicate calculus. To proceed with our first illustration, let us call a formula of the predicate calculus prime for the statement calculus if no sentential connectives appear in it. In terms of the composition of a formula from such prime formulas we can introduce the notion of tautology into the predicate calculus... |
ic. That is, we take as the domain D the set of natural numbers. Further, let < and + have their familiar meanings; thus <(x, y) is a 2-place predicate letter, and + (x, y, z) is a 3-place one. The (true) statement "There does not exist a greatest natural number" may be symbolized by Its negation, (x)(3y)(x < y)- -, (x... |
g them. In the following simple example illustrating them we employ a lower-case Greek letter to designate an object which is involved in the "act of choice" accompanying an instance of the rule es. Every member of the committee is wealthy and a Republican. Some committee members are old. Therefore, there are some old ... |
successful attempts to substantiate the belief that the parallel postulate could be derived from Euclid's remaining axioms. Bolyai and Lobachevsky dispelled this belief by developing a geometry in which the parallel postulate was replaced by the statement "In a plane, if the point A is not on line 1, then there exists ... |
everal theories is the possibility of enriching and extending given theories in an inexpensive way. For example, a 5.2 I Informal Theories 227 theorem in one theory may be the origin of a theorem in the derived theory and it, in turn, may yield a new result in another parent theory. In addition to the possible enrichme... |
shall be called a-statements. (Parenthetically we remark that Z--sentences, are usually 5.2 I Informal Theories 231 written using a combination of words and symbols, as in the foregoing examples, instead of the purely symbolic style of Examples 4.7.1 -4.7.3.) An interpretation of `;" consists of selecting a particular ... |
or all a (b c) _ (a b) c, in X there exists an a' in X such that a- a' = a theory Z is axiomatized by defining a set-theoretical predicate, what we have called up to this point the primitive symbols (or terms) of the theory appear in the running text immediately preceding the axioms. Also in this circumstance models of... |
be derived from I'; that is Z is negation complete. We may loosely relate consistency and completeness in the following way. An axiomatic theory is consistent if it does not have too many theorems and it is complete if it does not have too few. If an axiomatic theory is both consistent and negation complete, then all q... |
r with a relation that is it-reflexive and transitive on X (see Exercise 1.11.3). Another example is implicit in a remark made in Section 1; rephrased, it amounts to the assertion that Hilbert and Pieri gave different formulations of a theory which axiomatizes intuitive plane geometry. Different formulations of a theor... |
GRAPHICAL NOTE Discussions of axiomatic theories and the axiomatic method, pitched at about the same level as ours, appear in R. L. Wilder (1952), E. It. Stabler (1953), and A. Tarski (1941). CHAPTER 6 Boolean Algebras T; r. THEORY or Boolean algebras has historical as well as present-day practical importance. For the ... |
to the meet operation, since, just as for the algebra of sets, anb=a if aUb=b. The proof of this as well as the proofs of such related facts as a < b iff a nb' = 0 and a < b if b' < a' are left as exercises. Important features of the new relation are stated in the next theorem. 6.2 I Some Basic Properties of a Boolean ... |
the axioms except B i = 1, 2, 3, 4. Below arc defined four systems which demonstrate the independence of the axiom with the corresponding label. c a b c (B,) B = (a, bB2) B = (a, bB,) B = (aB4) B = (A C v'(Z+) I7+ - A is a finite set). n is set intersection. ' is defined as follows. We note that for each A in B there e... |
of "the Boolean algebra (B, n, ')." Let us consider now the relationship of a Boolean algebra B/0 to the algebra B from which B/0 is derived using a proper congruence relation. Let p be the natural mapping (see Section 1.9) on the set B onto the set B/0, that is, the mapping p: 13 -*- B/0, where p(b) = b. Since anb =a... |
st we note that x n y is the meet of two elements in B, and A(x) n A(y) is the set of those elements common to A(x) and A(y). Now, assume that a C A(x n y). Then a< x n y, and hence a< x and a< y. Thus a E A(x) n A(y). hence A(x n y) 9A(x) n A(y). Reversing the foregoing steps establishes the reverse inequality, and he... |
that B be complete and atomic. In this event, B is isomorphic to the algebra of all subsets of its set of atoms. Proof. Since the necessity of these conditions has already been observed, we turn to a proof of their sufficiency. Suppose, therefore, that B is complete and atomic and let 7' be the set of all atoms of B. A... |
gebra are as unrestricted as is possible if they are to have the structure of a Boolean algebra. Intuitively this is clear, since the only relations which have been imposed upon them are a necessary and sufficient set to insure that they do have that structure. There are alternative definitions of a free Boolean algebr... |
B, then Ix C Bjx' E I} is a filter; if F is a filter, then (x C Bjx' C F} is an ideal, as is easily proved. 'T'his pairing provides a bridge for transferring observations about ideals to filters. For example, both B and (1) are filters of B. Again, if a C B, then {x E Bjx > a} is a filter; this is the principal filter ... |
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