Foundations of Mathematics

Transcription

1 Foundations of Mathematics Andrew Monnot 1 Construction of the Language Loop We must yield to a cyclic approach in the foundations of mathematics. In this respect we begin with some assumptions of language and a minimal set of axioms within that language and use the results to establish the existence of algebraic objects with which another language can be constructed. What separates this approach from that of circular reasoning is that the secondary language created is sort of bounded above by the initial metalanguage. One simply starts at an abritrary level of abstraction and creates lower levels by travelling forwards in the meta loop. meta loop mathematics 1. There exists a set called a language, belonging to which is a set of letters, a set of terms, and a set of grammar/morphological operations. That this paper has any meaning could itself be construed as a special case of this axiom. For convenience we use that special case: the existence of the English language, in which case the above axiom is well-defined. We now adopt set theoretic notation to simplify the English sentences, including the operations,, :=, and quantifier on propositions together with: x y := ( x y) xφ(x) := x φ(x) x y := x y x y := (x y) (y x). Note that for the quantifiers and, one must specify a domain of discourse. That is, the basic formulas involving quantifiers are ( x X)(φ(x)) or ( x X)(φ(x)). 2. ( of Pairing) x y z w(w z w = x w = y). By this axiom, if we have two sets x and y, we can define their pair set {x, y}. We can thus define a singleton of x as {x} := {x, x} and set {x} = {x, x}. Moreover we can define the notion of an ordered pair: (x, y) := {{x}, {x, y}} and inductively by (x 1,..., x n ) := ((x 1,..., x n 1 ), x n ). 3. (Schema of Separation) x a y b φ(b, a)(b y b x φ(b, a)). 1

2 This allows us to define subsets of some set whose elements b satisfy a condition φ(b, a). In particular we have the intersection and difference of two sets, as well as the empty set: x y = {a x : a y} x y = {a x : a / y} = {x : x x}. We say sets x and y are disjoint if x y =. 4. ( of Union) x y a(a y b(b x a b)). This yields the existence of a union set over a family of sets: X = {a x : x X}. We can now define and x y = {x, y} x y = (x y) (y x). 5. ( of Power Set) x y a(a y a x). The set y above is called the power set of x, which can be formally written as P (x) = {y : y x}. We are now in a position to define an important set. Definition as 1.1. Let X and Y be sets. We define the cartesian product of X and Y X Y = {(x, y) : x X and y Y }. Note that the cartesian product is a set since X Y P (P (X Y )). A relation between X and Y is a subset R X Y. An n-ary relation on a set X is a subset of X n = n 1X. A relational structure is a set X together win an n-ary relation. Definition 1.2. A function from X to Y is a relation f between X and Y such that (i) ( x X)( y Y )((x, y) f) (right-extendable) (ii) ( (x, y) f)( (x, z) f)(y = z) (functional). We may write f : X Y. If (x, y) f, we may write y = f(x). We further define domain, range, and preimage: dom f = {x X : ( y Y )(y = f(x))} ran f = {y Y : ( x X)(y = f(x))} f 1 (A) = {x X : f(x) A Y }. 2

3 Hence we have f 1 (ran f) = dom f. These are all sets since dom f, ran f, f 1 (A) f. We let Y X = {f : X Y } denote all functions from X to Y, which is also a set since Y X P (X Y ). A function f : X Y is surjective if ran f = Y. f is injective if ((x, y) f (z, y) f) x = z. An n-ary operation on X is a function f : X n X. A structure is a set together with an n-ary operation. The signature of a structure is a sequence σ(x) = (n 1,..., n k,...) where n k is the number of k-ary operations. Definition 1.3. Let X and Y be structures with the same signature such that each k-ary operation of X is assigned to a k-ary operation of Y (i.e. f(o i ) = O i where O i is the ith k-ary operation of X). A homomorphism between structures X and Y is a map ϕ : X Y such that ϕ(o i (x 1,..., x n )) = O i (ϕ(x 1 ),..., ϕ(x n )). Note that a nullary operation on X is a map O : X. That is, it is simply an element of X. Now let A be a set which we will call an alphabet, and its elements will be called letters. A monoid X has a nullary operation, 1 X called a space (or empty letter), and a binary operation, which will simply be denoted by concatenation. We define the free monoid on A as the monoid A consisting of all strings of elements in A. We now have two definitions of a language, of which the first is traditional and the second is mine: Definition 1.4. A language is a subset of A. Alt. Definition 1.4. Let W A, T be a relational structure (a set together with an n-ary relation), and F be a structure. The language L F,T,W is defined as F [T [W ]]/T [W ] where X[Y ] is the free X-structure on Y. In particular elements of W are called words, elements of T [W ] are called terms, and elements of L F,T,W are called formulas. The definition of the factorization depends on whether or not F has an identity. If so, we simply set the terms equal to an identity formula. Otherwise we can interpret the factorization as set subtraction. Definition 1.5. A theory of L F,T,W is a subset X L F,T,W. Elements of a theory are called axioms. Elements of F [X] are called propositions. A theory X of L F,T,W is called a reduced theory if for all φ, ψ X, ψ O(φ, x 1,..., x n 1 ) for all n-ary operations of F and all placements of φ in evaluation of the operation. (That is, the theory is reduced if no axiom is in the orbit of another). For example, the theory L F,T,W is called the trivial theory. The theory is called the empty (or agnostic) theory. Definition 1.6. An n-ary logic system on a theory X is a homomorphism ϕ : F [X] V where V has cardinality n and F and V have the same signature. We further say that the theory is normal if ϕ(φ) = ϕ(ψ) for all φ, ψ X. In traditional logic V is a two element boolean algebra. Traditional logic also has a special kind of function on its language. Definition 1.7. A quantifier on L F,T,W is a function : T [W ] L F,T,W L F,T,W. We may write: (x X, φ) = ( x X)φ. 3

4 In particular it is a pseudo operation, and gives the language a pseudo structure. This is similar to modules, where in this case a product of a term and a formula are sent to a formula. Hence our initial assumption of four axioms (as well as the ability to understand the English language), have in turn given us the ability to create a notion of a language of which a degenerate English can be construed as a special case. This is certainly circular in some sense, but in foundations we must appeal to some cyclic process. One subtlety worth noting is that the secondary language created will always be strictly bounded above by the initial language; they aren t truly equivalent. (In fact this last statement is similar to the antecedent of Godel s Second Incompleteness theorem). 2 Absorption of Mathematical Logic The next task is to absorb the traditional area of mathematical logic. One key missing ingredient is a model. Let us recall the traditional setup (taken from [1]). Definition 2.1. Let S be a set (of symbols). An S-structure is a pair A = (A, a) where A is a nonempty set, called a universe, and a is a map sending symbols to elements, functions, and relations of A. An assignment of an S-structure (A, a) is a map β : S A. An S-interpretation is a pair I = (A, β) where A is an S-structure and β is an assignment in A. For shorthand notation, the convention (with some of my modifications) is to write: c A = β(c), (f(t 1,..., t n )) A = a(f)(β(t 1 ),..., β(t n )), and (xry) A = β(x)a(r)β(y). These are the terms. Formulas are then built from the terms using traditional (although this can be generalized) logical connectives. The notion of a model is then defined via induction on formulas. Definition 2.2. Let I = (A, β) be an S-interpretation. We say that I satisfies a formula φ (or is a model of φ), denoted I φ, if φ A holds, where φ A is defined via its components and β and a where necessary. Formal languages in convention are built up from the formulas mentioned above, which are nothing more than special cases of Alt Definition 1.4. A model for a language is hence nothing more than an A-interpretation into a structure, where A is an alphabet (provided it is equipped with a logic system). This is precisely what I have constructed in section 1; the symbols of W A are mapped to the universe L F,T,W. The next thing to establish is that every model is a language model. This is trivial since a model by definition satisfies a set of formulas as well as compounds of them (i.e. it must satisfy a language). Hence we have no need to trouble ourselves with interpretations and may simply stick to the algebra of section 1. While we have absorbed model theory, there are a few more critical topics to absorb from mathematical logic. We return to the terminology of section 1. Let X be a theory of L F,T,W and ϕ : F [X] V be a binary logic system. A formula φ L F,T,W is derivable in X if it is a proposition (i.e. is in F [X]). We may write X φ. This definition is in complete agreement with the traditional definition (namely, there being a derivation, or finite number of steps, that begin with axioms and use inference rules); it is nothing more than saying it is in F [X]. Similarly φ L F,T,W is valid if φ, or equivalently, it is 4

5 derivable in any theory. In our setup this would imply φ F [ ] =. Hence no formula is valid. Let F have a unary operation and ϕ : F [X] V be a logic system on a theory X. If we assume to be idempotent ( φ = φ), then since ϕ is a homomorphism, we have ϕ(φ) = ϕ( φ) = ϕ(φ). That is, the corresponding unary operation in V must also be idempotent on ran(ϕ). Definition 2.3. A unary operation (not necessarily idempotent) is consistent in ϕ if for all φ F [X], ϕ(φ) ϕ( φ). If we assume is consistent in ϕ and that ϕ is a binary logic system, then the corresponding in V is idempotent since ϕ(φ) = 0 ϕ( φ) = 1 ϕ( φ) = ϕ(φ) = 0. Again, proofs in a binary system are independent of the choice of valence. If we assume consistency and idempotency, then we have a nonidentity negation which is idempotent on the range. The case for assuming binary system and idempotency yields either a trivial mapping of propositions (all to 0 or all to 1), or that is consistent and idempotent on V. And lastly if we assume all three (idempotency and consistency of together on a binary system), we obtain a surjective assignment with idempotent negation in V. Let ϕ : F [X] V be a binary logic system where V is a boolean algebra. Then the completeness and compactness theorems are trivial. Recall these statements: Completeness Theorem. For all formulas φ and models I, I φ X φ where I X. Compactness Theorem. For all formulas φ and models I, X φ I φ where I X. Traditionally these apply to, what we would call, a binary logic system ϕ : F [X] V where V is a boolean algebra (hence F has a consistent, idempotent negation) under traditional operations, and in particular this fixes the operational/relational structures of F, T, and W, but X is arbitrary. In this setup, all formulas (or what we would hence call propositions since they are generated by a theory) are trivially satisfiable since they have a language model. Hence Compactness is true. Moreover since they are propositions in a binary logic system, they are in some F [X] for a theory X and are hence derivable; so we have Completeness. Lastly we wish to address Godel s Second Incompleteness Theorem; recall its statement: Godel s Second Incompleteness Theorem. A theory contains a statement of its own consistency if and only if it is inconsistent. We have only defined what it means for a unary operation in a logical system to be consistent. Hence we can say that a binary logic system with a unary operation is consistent if its unary operation is consistent. But all of these traditional theorems of mathematical logic are assuming a binary logic system where V is a boolean algebra, is idempotent, and the map ϕ : F [X] V is surjective. Hence is consistent (from above 5

6 discussion), and the right-hand side of the theorem is false. The simplest example of the antecedent of Godel s theorem is to use a structure to create itself (i.e. a self-swallowing structure), which makes no sense, let alone using it to create a larger structure within which is a statement about the initial structure. That a binary logic system with unary operation contains a statement of its own consistency is itself a contradiction, since such a statement is in the complement of the logic system within the metalanguage. Hence the left-hand side is also false. So both implications are true. 3 Absorption of Category Theory Category theory can easily be interpreted as a language. The alphabet consists of all the symbols we need to represent categories, objects, morphisms, membership, products, coproducts, logical connectives for our statements...etc. The words in turn consist the concatenations we want. The relational structure is defined on our category and object words: we define a reflexive and transitive relation on {Cat} 2 and Ob(Cat) 2. This gives us the identity and composition rules for functors and morphisms. The formulas then consist of statements involving categories/objects. 6

7 References [1] Ebbinghaus, H.-D., J. Flum, and W. Thomas. Mathematical Logic. Second Edition. Undergraduate Texts in Mathematics. New York: Springer-Verlag