| Abstract |
In the introduction of this article, we make a brief his¬torical retrospection of Mathematical Logic, from Ancient Greece and Aristotle to the modern era and the great evolu¬tion of Logic. At the beginning of the 20th century, three schools of mathematical philosophy, each opposite to the other, appeared: Formalism, Intuitionism and Logicism.
According to Logicism, logic is the basis for any kind of argument. And since Mathematics are considered as an ex-tension of logic, most or all mathematics are reducible to logic. Russell and Whitehead championed this theory, fa¬thered by Gottlob Frege.
On the other hand, Intuitionists such as G.Boole, J.Wenger and E.Schroder, were mostly interested in studying the rela-tionship between numerical and logical operations.
Regarding formalism, one of its first supporters was David Hilbert, whose theory was intended to be a complete and consistent axiomatization of all mathematics. His main ob¬jective was to prove that these axiomatic systems do not lead to contradictions.
In the second chapter, we study various concepts from the field of Topology. Our ultimate goal is to present an alternative proof of the Compactness' Theorem via Topology techniques.
Definition 1 A topological space X is called compact iff for each class {Oa : a £ I} of open sets of X, foo which X = Uiei Oi, there is a finite I0 C I such that X = Ui£lo Oi.
Definition 2 A topological space X is called compact if for each class of {Ca : a £ I} of closed sets of X, which are the complement of open sets (Ca = X — Oa), for which Πi£l Ci = 0, there is a finite number of I0 C I such that Ci£lo Ci = 0.
The alternative formulation of the above definition will be more useful for the following proofs.
Remark 1 Let X1,..., Xn,... topological spaces. We recall the definition of the Product Topology, which we symbolize:
oo oo
Π Xn = {f : N - (J Xn : Vn £ N f (n) £ Xn] (1)
n=l n=l
In order to prove the Compactness' Theorem, we assume that, for the Topology Spaces X1,..., Xn,... of the remark (1), we can accept that X1 = ... = Xn = {0,1}. We shall denote this space by 2ω = {0,1}ω = {0,1} χ {θ, 1} χ ..., where each of Xi = {0,1} is trivially compact. Moreover, all subsets of 2ω are both open and close, using the discrete topology.
Definition 3 Let {xi : i £ I} be a family of topological spaces and X = ni£j Xi their product. The open sets in product topology are unions (finite or infinite) of sets of the form: X = Πi£i Ui, where every Ui is an open subset of Xi and Xi = Ui only finitely many i.
The following theorem of Topology plays an important role towards the proof of the Compactness theorem.
Theorem 0.1 (Tychonoff) Let {Xi : i £ I}, a family of compact topological spaces. Then the product X = i£i Xi is a compact topological space.
According to the theorem, the space 2ω is compact and therefore, its open subsets are, by definition, sets of the form:
Οι χ O2 χ•••χΟρ χ {0,1} χ {0,1} χ • • • (2) where Oi are open subsets of {0,1},for i £ {1,...,p}.
Lemma 0.2 Given a formula „, we denote the set F„ = {u : u(„) = 1}. Then, both of the following statements are valid:
1. F„ is an open set, for every formula „.
2. F„ is a closed set, for every formula „.
Proof: We will prove the lemma in two steps. First, we will prove the 2, assuming that 1 is true. The proof is then completed by showing the validity of 1.
Proof of 2. For each formula „ the following applies:
F„ = Fc„, (3)
where denotes the complement of set F-,„. The set F-,„ is an open set and therefore F„, for each formula „, is closed.
Proof of 1. First, suppose that the random formula „ is just a variable, xn. We define the set:
FXn = {u : u(xn) = 1}, (4)
where u(xn) is the valuation of variable xn.
The elements of this set are sequences of 0 and 1, whose nth term is equal to 1 and all others are either 0 or 1. Notice that the set FXn is an element of the base of open subsets, as defined above, therefore it is an open set.
We will show that the same principle also applies to the
case where „ contains more than one variable. Suppose that
the variables appearing in the „ formula are between the
variables Obviously, the remaining variables from
xn+i do not affect the valuation of „.
We consider a valuation u and let every i £ {1,...,n} have u(x^)= -u^.For each i £ {1,...,n} we define:
xi,wi — 1 /r
(5)
We then define the set W = (wi,w2,...,wn) £ {0, 1}n which contains all η-groups of valuations of variables appear¬ing in the formula φ, for which φ is true.
As in the case of one variable, we will define the sets
Fxt = {u : u(xi) = 1}, (6)
for i £ {1,...,η}, which are both open and closed. Obvi¬ously φ is logicably equivalent to the formula:
φ = V X • x2 • ... • xn (7)
and therefore F„ is:
F„ = U (Fx! Π FX2 fl ... fl FXn). (8)
W £w
Therefore, the set F„ is a finite union of finitely many intersections of closed sets and as such, it is a closed set.
□
Theorem 0.3 (Compactness' Theorem in Logic). Let Σ be a set of formulas. If Σ is finitely satisfiable, then Σ is satis-fiable.
Proof: To prove that the set Σ = {φ,φ2,...} is satisfiable, we need to show that
Π F„k = 0, (9)
k£N
where F„k as in Lemma (an/kl).
We shall prove this by contradiction, therefore we sup¬pose that the intersection of F„k is the empty set. Applying Tychonoff's theorem, we know that the space 2ω is compact thus we get, from the equivalent definition of compactness in Topology, that there is a finite number of sets F„k whose intersection is the empty set. But this leads to a contradic¬tion, because it is known that the set Σ is finitely satisfiable, so for every finite subset of Σ is satisfiable. Therefore, for every n £ N
Π F„k = 0. (10)
k£{1,...,n}
□
In the third chapter, we make a brief introduction to quan-tum mechanics and study the experiment of the two slits and the conlusions derived by it.
The double slit experiment, first performed by Thomas Young in 1801 is considered one of the most famous exper¬iments in quantum physics. The layout of the experiment is quite simple. We have a wall with two very small slots in close proximity and on the opposite side a screen that records the result of the experiment.
In front of the wall we place a source. In the first version of the experiment we consider a source that sends out particles of macroscopic dimension, for example small balls. Some will bounce off the wall, but some of them will travel through the slits and hit the screen behind. The screen marks all the spots where a ball has hit and the pattern that will be formed on the screen will be in the form of two lines of marks roughly the same shape as the slits.
The second time we run the experiment with a source that emits (classic) waves. As the wave passes though both slits, it essentially splits into two new waves, each spreading out from one of the slits. These two waves interfere with each other. For some of these waves, the crest of one will meet the trough of the other, and they will cancel each other out (destructive interference). For some other waves, their crests will meet resulting in a wave of double the amplitude (constructive interference). When the waves meet the screen placed behind the wall, we will see a stripy pattern, called an interference pattern.
Lastly, we run the experiment with a source that sends out particles of the microcosm, for example electrons. Imagine firing electrons at our wall with the two slits, but block one of tho slits for the moment. You'll find that some of the electrons will pass through the open slit and strike the screen just as small balls did at the first experiment. Something unexpected occurs when we repeat the experiment with both slits open. You would expect two rectangular strips on the second wall, as with the small balls, but what is actually observed is very different: the spots where electrons hit build up to replicate the interference pattern from a wave.
There is a possibility that the electrons might somehow interfere with each other, so they don't arrive in the same places they would if they were alone. However, the interfer¬ence pattern remains even when we fire electrons one by one, so that they have no chance of interfering. Strangely, each individual electron contributes one dot to an overall pattern that looks like the interference pattern of a wave. his means that somehow each electron splits, passes through both slits at once, interferes with itself, and then recombines to meet the second screen as a single particle.
To find out what exactly happens, we place a detector by the slits, to see from which slit each electron passes through. If you do that, then the pattern on the screen turns into the particle pattern of two strips, as seen in the first experiment. The interference pattern disappears. Somehow, the act of looking makes sure that the electrons travel like well-behaved little balls.
This experiment suggests that what we call "particles", such as electrons, somehow combine characteristics of par¬ticles and characteristics of waves. That's the famous wave particle duality of quantum mechanics. It also suggests that the act of observing, of measuring, a quantum system has a profound effect on the system. The question of exactly how that happens constitutes the measurement problem of quantum mechanics.
In the last chapter, we present the proof of the Kochen-Specker Theorem and examine its importance for quantum mechanics. The Kochen-Specker theorem was suggested by S.Kochen and E.Specker in 1967. It is also known as the Bell-KS theorem, because in 1966 Bell had proposed an al¬ternative formulation that led to similar conclusions.
Theorem 0.4 (KS Theorem) There is no function S2 — {0,1}, where S2 is the unit sphere, such that for each system {a,a2,a3} this representation takes the value 0 in just one direction.
In order to understand the significance of the KS theo¬rem, we must make reference to some well known facts and definitions of the Quantum mechanics field.
Quantum mechanics based on experimental data, makes the following assumptions regarding the electron spin on the orthhohelium atom:
I. The projection s(a, t) of the spin in the direction a £ S2,
where S2 is the unit sphere, at the moment t is measurable.
Also, s(a, t) randomly takes only one of the values -1,0,1 with
some probability each.
II. The lengths |s(a^,t)|2,for i = 1, 2,3, of the three pro-
jections in an orthonormal basis {a,a2,a3} C S2 are mea-
surable. Each such orthonormal basis will be called system.
The sum of the lengths of the projections of the spin onto
the vectors of every system is always equal to 2.
From now on, we will refer to the above properties cis ax¬ioms I and II. According to classical mechanics, the quantity spin had to be a function of different variables. The key question we consider is this: Can the spins be considered as a function of natural variables and at the same time satisfy axioms I and II?
The answer to the above question is negative due to the KS theorem. In particular, we begin by assuming that there exists such a function s(a,t) that satisfy both axioms.
From axiom I, we get that the value s(a,t) is either 0 or 1, whereas from axiom II, we derive that, for each system {ai,a2,a3}, only one of s(a^) equals 0. We shall also refer to this property of s(ai,t) as basic property. Finally, we derive a contradiction using the basic property of s(ai5t) and the KS theorem.
Lastly and in order to prove the KS theorem, we present the proof of the following Technical Theorem.
Technical Theorem. It is possible to construct a finite set Γ c S2 which has 117 vertices and the following property: "For any function k: Γ — {0,1}; there is a sys¬tem {a1,a2,a3} c Γ in which k will not take value 0 exactly once, or there will be a pair of {ai,a2}c Γ perpendicular directions, in which k is equal to 1."
A direct consequence of this theorem is the KS theorem. To prove the Technical Theorem, we first need the following two intermediate lemmas.
Lemma 0.5 Let a and b vertices on S2 such that: sinB £ [0,1/3], where θ is the angle formed between a and b. Then, the following graph can be constructed, for which a0 — a and a9 — b.
Lemma 0.6 The vertices a0 and a9 of the graph (1) will always be either both 0 or both 1.
Proof: Suppose that in the graph we assign different values at the vertices a0 and a9. We select the value 1 for a0 and 0
for ag. According to the basic property, we will give values to the other vertices as shown in the figure (2):
Figure 2: G2
This requires that vertices a5 and a6 are orthogonal and both take the value 1, which is forbidden. Hence, two vertices closer than arcsin(1/3) cannot have different values.
□
Technical Theorem's Proof: We have to construct another KS graph in the following way. Consider a realization of the graph (1), where, according to Lemma(0.5), the angle between vertices a0 and a9 is θ = π/10 6 sin-1 (1/3).
Now we choose three orthogonal vertices p0,q0,r0 and space interlocking copies of (1) between them such that every in¬stance of vertex a9 of one copy of (1) is identified with the instance of a0 of the next copy. In this way five interlock¬ing copies of (1) are spaced between p0 and q0 and all five instances of a8 are identified with r0. Working in the same way, five interlocking copies are spaced between q0 and r0, identifying all copies of a8 with p0, and between p0 and r0, identifying all copies of a8 with q0.
Figure 3: G3
That graph is constructible is born out directly by the construction itself. If from the 15 copies of (1) used in the
process of constructing (3) we subtract those vertices that were identified with each other, we end up with the requested set of 117 vertices.
However, although (3) is constructible, no values can be attributed to its vertices according to the basic property. We know from Lemma (0.5) that a copy of (1) with θ = π/10 6 sin-1 (1/3), requires that vertices a0 and a9 have the same value. Now, since a9 in one copy of (1) is identical to a0 in the next copy, a9 in the second copy must have the same value as a0 in the first. Indeed, by repetition of this argument all instances of a0 must have the same value.
Vertices p0,q0,r0 are identified with points a0, so they must have the same value, which are inconsistent with the constraint that exactly one of them must get the value 0 and the other two must get the value 1.
|
Collections
