Table of Contents
- 1. Logic Hierarchy
- 2. Fisrt-Order Logic Language
- 3. First-Order Theory
- 3.1. ZF Set Theory
- 3.2. Magma(Groupoid) Theory = (Set, closure under an operation •)
- 3.3. Semigroup Theory = (Groupoid, Associativity)
- 3.4. Monoid Theory = (Semigroup, Identity) (Natural Number with addition as a Model)
- 3.5. Group Theory = (Monoid, Inversibility)
- 3.6. Abelian Group Theory = (Group, Commutativity) (Integer with addition as a Model)
- 3.7. Ring with Identity = (Set, +, *, 0, 1) = (Abelian Group, *, 1)
- 3.8. Commutative Ring with Identity = (Ring with Identity, commutativity of *) (Integer with addition and multiplication as a Model)
- 3.9. Field = (Commutative Ring with Identity, ) (Rational with addition and multiplication as a Model)
- 4. Structures of Fisrt-Order Languages
- 5. Models of First-Order Theory (As Types in Type Theory)
- 6. Elementary Class. (As Type class in Type Theory)
- 7. Embedings and Isomorphisms.
1 Logic Hierarchy
- Propositional Logic (propositional variables, logical connectives)
First Order Logic: (Propositional Logic, quantifiers)
quantifies only variables that range over individuals (elements of the domain of discourse).
- Second Order Logic (Fisrt Order Logic, quantifiers over relations) (relations are sets of elements in the domain).
Higher Order Logic
First-order logic quantifies only variables that range over individuals;
second-order logic, in addition, also quantifies over sets;
third-order logic also quantifies over sets of sets, and so on.
Higher-order logic is the union of first-, second-, third-, …, nth-order logic;
i.e., higher-order logic admits quantification over sets that are nested arbitrarily deeply.
2 Fisrt-Order Logic Language
- A language is countable if it has only countable many nonlogical symbols.
- A language is finite if it has finitely many nonlogical symbols.
2.1 A first Order Logic language L consists of two types of symbols:
- Logical Symbols: (Same for all fisrt-order logic language)
- a sequence of variables, V. (x, y, z, u, v, w, …)
- logical connectives. (not ¬, disjunction ∨)
- logical quantifiers. (Exists ∃, For-all ∀)
- equality symbols. (=)
- Non-logical Symbols: (Signature of the specified first-order logic language)
- a set of constant symbols, C, { ci, i ∈ I }.
- a set of n-ary function symbols, F, n ≥ 0, { fj, j ∈ Jn }. (return values) (0-ary function symbols are constants.)
- a set of n-ary relation symbols, R, n ≥ 0, { Pk, k ∈ Kn }. (return true of false) (1-ary relation symbols are predicates on a varaible; 0-ary relation symbols are propositions.)
2.2 Terms of a language: (No Relation Symbols)
The set of all terms of a language L is the smallest set T of expression of L that contains all variables and constant symbols and is closed under operation f ∈ F.
- the rank of a term t is the smallest natural number k such that t is of rank ≤ k, shows how many nested functions inside that term.
- all variables and constant symbols are terms of rank 0.
- (t1, t2, …, tn) ∈ T and the ranks of ti ≤ k ⇒ f(t1, t2, …, tn) is a term of rank at most k + 1.
The set of all varaible-free terms of lanugage is the smallest set T' is constructed as T without variables. And they are called closed terms.
2.3 Formulas of a language: (invoking Relation Symbols)
The set of all formulas of L is the smallest set of all expressions of L that contains all the atomic formulas and that is closed under negation, disjunction, and existential quantification
- atomic formula:
- t, s ∈ T ⇒ t=s is atomic formula.
- t1, t2, …, tn ∈ T and P ∈ R with n-ary ⇒ P(t1, t2, …, tn) is atomic formula.
- fomula:
- atomic formula is formula of rank 0.
- A, B ∈ Formula of rank ≤ k, v ∈ V ⇒ A, B, or v, connected with connetives ∈ Formula of rank ≤ k+1.
3 First-Order Theory
- A first order theory, or simply a theory, T consists of a first-order language L and a set of formulas of L as nonlogical axioms of T.
- The language of T is denoted by L(T).
- A theory is called countable if its language is countable.
- A theory is finite if its language is finite.
3.1 ZF Set Theory
- Nonlogical Axioms:
- Set existence: There exists a set. ∃ x (x = x)
- Extensionality: Two sets are the same if they contain the same sets. ∀ x ∀ y (∀ z (z ∈ x ↔ z ∈ y) → x = y)
- Comprehension (subset) schema: given any property of sets to define a subset of the set contains those sets. ∀ φ [x, w1, …, wn] ∀ z ∀ w1 … ∀ wn (∃ y ∀ x (x ∈ y ↔ x ∈ z ∧ φ) )
- Replacement schema.
- Pairing. ∀ x ∀ y ∃ z (x ∈ z ∧ y ∈ z)
- Union.
- PowerSet.
- Infinite.
- Foundation.
3.2 Magma(Groupoid) Theory = (Set, closure under an operation •)
- a binary function: •
- Nonlogical axioms:
- closure: ∀ x ∀ y ∃ z (z = x • y)
3.3 Semigroup Theory = (Groupoid, Associativity)
- a binary function: •
- Nonlogical axioms:
- closure: ∀ x ∀ y ∃ z (z = x • y)
- Associativity: ∀ x ∀ y ∀ z (x • (y • z) = (x • y) • z)
3.4 Monoid Theory = (Semigroup, Identity) (Natural Number with addition as a Model)
- a constant symbol: e
- a binary function: •
- Nonlogical axioms:
- closure: ∀ x ∀ y ∃ z (z = x • y)
- Associativity: ∀ x ∀ y ∀ z (x • (y • z) = (x • y) • z)
- Identity: ∀ x (x • e = x ∧ e • x = x )
3.5 Group Theory = (Monoid, Inversibility)
- a constant symbol: e
- a binary function: •
- Nonlogical axioms:
- closure: ∀ x ∀ y ∃ z (z = x • y)
- Associativity: ∀ x ∀ y ∀ z (x • (y • z) = (x • y) • z)
- Identity: ∀ x (x • e = x ∧ e • x = x )
- Inversibility: ∀ x ∃ y (x • y = e ∧ y • x = e)
3.6 Abelian Group Theory = (Group, Commutativity) (Integer with addition as a Model)
- a constant symbol: e
- a binary function: •
- Nonlogical axioms:
- closure: ∀ x ∀ y ∃ z (z = x • y)
- Associativity: ∀ x ∀ y ∀ z (x • (y • z) = (x • y) • z)
- Identity: ∀ x (x • e = x ∧ e • x = x )
- Inversibility: ∀ x ∃ y (x • y = e ∧ y • x = e)
- Commutativity: ∀ x ∀ y (x • y = y • x)
3.7 Ring with Identity = (Set, +, *, 0, 1) = (Abelian Group, *, 1)
- constant symbols: 0, 1
- binary function symbols: +, *
- Nonlogical axioms:
- closure of +: ∀ x ∀ y ∃ z (z = x + y)
- Associativity of +: ∀ x ∀ y ∀ z (x + (y + z) = (x + y) + z)
- Identity for +: ∀ x (x + e = x ∧ e + x = x )
- Inversibility of +: ∀ x ∃ y (x + y = e ∧ y + x = e)
- Commutativity of +: ∀ x ∀ y (x + y = y + x)
- closure of *: ∀ x ∀ y ∃ z (z = x * y)
- Associativity of *: ∀ x ∀ y ∀ z (x * (y * z) = (x * y) * z)
- Identity of *: ∀ x (x * e = x ∧ e * x = x )
- Left Distributability: ∀ x ∀ y ∀ z (x * (y + z) = x * y + x * z)
- Right Distributability: ∀ x ∀ y ∀ z ((y + z) * x = y * x + z * x)
3.8 Commutative Ring with Identity = (Ring with Identity, commutativity of *) (Integer with addition and multiplication as a Model)
- Commutativity: ∀ x ∀ y (x * y = y * z)
3.9 Field = (Commutative Ring with Identity, ) (Rational with addition and multiplication as a Model)
- constant symbols: 0, 1
- binary function symbols: +, *
- Nonlogical axioms:
- closure of +: ∀ x ∀ y ∃ z (z = x + y)
- Associativity of +: ∀ x ∀ y ∀ z (x + (y + z) = (x + y) + z)
- Identity for +: ∀ x (x + e = x ∧ e + x = x )
- Inversibility of +: ∀ x ∃ y (x + y = e ∧ y + x = e)
- Commutativity of +: ∀ x ∀ y (x + y = y + x)
- closure of *: ∀ x ∀ y ∃ z (z = x * y)
- Associativity of *: ∀ x ∀ y ∀ z (x * (y * z) = (x * y) * z)
- Identity of *: ∀ x (x * e = x ∧ e * x = x )
- Left Distributability: ∀ x ∀ y ∀ z (x * (y + z) = x * y + x * z)
- Right Distributability: ∀ x ∀ y ∀ z ((y + z) * x = y * x + z * x)
- Inversibility of *: ∀ x (¬ (x = 0) → ∃ y (x * y = 1 ∧ y * x = 1) )
4 Structures of Fisrt-Order Languages
Define the interpretation or the structure of a language L as a nonempty set A together with the interpretations or meanings of all nonlogical symbols of L.
Note that which statement is true in a structure and which is not is irrelevant in the definition of a structure.
A structure or an interpretation of a first-order language L(T) consists of:
- a nonempty set M. (the universe of the structure)
- for each constant symbol c of L(T), a fixed element cM ∈ M.
- for each n-ary function symbol f of L(T), an n-ary map fM: Mn → M.
- for each n-ary relation symbol P of L(T), an n-ary relation PM ⊂ Mn.
- The interpretation of
=
is always taken to be the equality relation in M.
5 Models of First-Order Theory (As Types in Type Theory)
A model M of a first-order theory T is a structure of L(T) with universe M in which all nonlogical axioms of T are valid. M |= A. E.g. Any group is a model of group theory, but the set N of natural numbers, together with the usual 0 and + as the interpretations of e and · respectively, is definitely a structure for the language of group theory but not a model of group theory.
6 Elementary Class. (As Type class in Type Theory)
A class M of structures of a language L is called elementary if there is a theory T with language L such that elements of M are precisely the models of T.
7 Embedings and Isomorphisms.
M and N denote structures of a fixed first-order language L. An embedings of N into M, N -> M, is a one-to-one map α : N -> M, where:
- For all constant symbol c of language L, α(cN) = cM.
- For every n-ary function symbol f of language L, and every β ∈ Nn, α(fN(β)) = fM(β).
- For every n-ary relation symbol P of language L, and every β ∈ Nn, α(PN(β)) ⇔ PM(β).
if α is surjective, then α is an isomorphism. And M and N are isomorphic structures. An automorphism of M is an isomorphism from M onto itself.
N is a substructure of M if
- N ⊂ M.
- α : N -> M is an embeding.