First Order Logic and Nonstandard Analysis Julian Hartman September 4, 2010 Abstract This paper is intended as an exploration of nonstandard analysis, and the rigorous use of infinitesimals and infinite elements to explore properties of the real numbers. I first define and explore first order logic, and model theory. Then, I prove the compact- ness theorem, and use this to form a nonstandard structure of the real numbers. Using this nonstandard structure, it it easy to to various proofs without the use of limits that would otherwise require their use. Contents 1 Introduction 2 2 An Introduction to First Order Logic 2 2.1 Propositional Logic . 2 2.2 Logical Symbols . 2 2.3 Predicates, Constants and Functions . 2 2.4 Well-Formed Formulas . 3 3 Models 3 3.1 Structure . 3 3.2 Truth . 4 3.2.1 Satisfaction . 5 4 The Compactness Theorem 6 4.1 Soundness and Completeness . 6 5 Nonstandard Analysis 7 5.1 Making a Nonstandard Structure . 7 5.2 Applications of a Nonstandard Structure . 9 6 Sources 10 1 1 Introduction The founders of modern calculus had a less than perfect understanding of the nuts and bolts of what made it work. Both Newton and Leibniz used the notion of infinitesimal, without a rigorous understanding of what they were. Infinitely small real numbers that were still not zero was a hard thing for mathematicians to accept, and with the rigorous development of limits by the likes of Cauchy and Weierstrass, the discussion of infinitesimals subsided. Now, using first order logic for nonstandard analysis, it is possible to create a model of the real numbers that has the same properties as the traditional conception of the real numbers, but also has rigorously defined infinite and infinitesimal elements. 2 An Introduction to First Order Logic 2.1 Propositional Logic The most basic form of logic is propositional logic, an it serves a building block for first order logic. Propositional logic takes statements, and links them together with "and" or "or". Basic sentence symbols are regarded as well-formed formulas, and combine with each other in specific ways to make larger well-formed formulas. However, there are no quantifiers in this basic form of logic. First order logic remedies this lack. 2.2 Logical Symbols Predicate logic serves as a refinement of basic propositional calculus, or sentential logic. First order logic allows for the expression of more complicated conditions such as "for all" or "for some"; it allows for the quantification of the basic propositions first seen in propositional logic. There are a few basic symbols used in first order logical systems that come from propositional logic: the implication arrow ()), the negation symbol (:), the 'and' symbol (^), the inclusive 'or' symbol (_), and the inference symbol (`). In addition to these symbols from propositional logic, we introduce the quantifier symbols 8, the universal quantifier, and 9, the existential quantifier. 2.3 Predicates, Constants and Functions In first order logic, the equivalent of an english noun or noun phrase is what is called a term. Definition 1. A term is as follows: 1. A variable, or individual constant is a term. n 2. If t1; t2; :::; tn are all terms, fj (t1; t2; :::; tn) is a term, where f is a function letter. 3. An expression is a term if an only if it is a term as defined in 1 or 2. It is easiest to think of terms in the intuitive sense mentioned above as nouns that are then combined with predicate letters to form atomic formulas. Predicates define relations between different terms. 2 n Definition 2. Pj (t1; t2; :::; tn) is an atomic formula, when P is a predicate letter, and ti is a term 8 i. These atomic phrases further combine to form well-formed formulas, which are the es- sential building blocks for a first order logic. An example of an atomic formula is < xy, or x is less than y, where x and y are both terms. The symbol < is a two-place predicate symbol, which means that it defines a relation between the two components x and y. 2.4 Well-Formed Formulas Well-formed formulas, or ’wffs’, are expressions made from atomic formulas, and various quantifiers. Definition 3. The classification of expressions as a Well-Formed Formula is dependent on three rules: 1. Every atomic formula is a wff. 2. If A and B are wfs, and x is a variable, then (:A), A)B, and ((8y)A) are wffs. 3. An expression is a wff if an only if it is a wff as defined in 1 or 2. Though by the definition of a well-formed formula, the above example of an atomic for- mula also serves as an example of a wff, you can make more complicated wffs out of the atomic phrases. For example, the phrase < xy ) : ≈ xy is a wff because "x is less than y" and "x equals y" are both atomic formulas, and the implication and negation are allowed under section two of the definition. Note: At first, it would seem that the quantifier 9 cannot be in a wff, due to the defini- tion's lack of provision for this quantifier. However, upon further examination we see that 9 is logically equivalent to :((8y):X). That is, "there exists y such that y has property X" is logically equivalent to the statement "it is not true that for all y, y does not have the property X". Therefore, wffs can use the existential quantifier. Using well formed phrases and quantifier logic, it becomes much easier to translate various phrases from english to mathematical language that it was using propositional logic. For example, the sentence, "Every person has a mother." can be translated as "8y, 9x such that xMy, with aMb indicating that a is the mother of b. 3 Models 3.1 Structure A structure of a first order logical language is roughly defined as the universe in which a first order logic operates. All of the parameters of a first order language must be accounted for in the structure. 3 Definition 4. A structure, Φ, of a first order language consists of a set and functions assigning the set of parameters of the first order language to specific aspects of the structure's set. 1. The structure, Φ assigns the universal quantifier symbol (8) some nonempty set, φ. n 2. For each n-place predicate symbol with n different components, Pj , Φ assigns an n-place relation in φ n n Φ 3. For each n-place function with n inputs, fj , Φ assigns an n-place operation, (fj ) , from φn to φ. 4. For each constant c, Φ assigns a constant cΦ in the universe φ For an example of a structure, let us consider a language with only the 2-place predicate relation aDb meaning a is more delicious than b. Now, we want to assign a structure to this language. If we use a structure with the universe, φ being all the different types of fruit, and DΦ being the set of pairs of fruit such that one fruit is more delicious than the other. This satisfies parts one and two of the definition of structure, and as this particular language has no functions, part three is also satisfied. For an example of the structure of a language with an operation, let us turn to the lan- guage of ordered abelian groups. Ordered abelian groups have one 2-place predicate relation, ≤, where a ≤ b means that a is less than or equal to b. An ordered abelian group has several axioms that it must satisfy, including having a binary operation, and inverses. The operation must be associative, and communative. The group must be closed, and there must be an identity element, such that the binary operation between this element and any other element does not change the latter element. A structure which satisfies this language is the integers, with the operation addition. The ≤ relation is preserved in the integers, and addition a the binary, communicative, associative operation as specified. Negative and positive numbers are inverses of each-other, and 0 is the identity element. The integers are also closed under addition. In the language of well-formed formulas, the integers, variables, and expressions such as x + y, or −y, or z + 5 are considered terms. Expressions such as 5 ≤ 4 or x + y ≤ −6 are atomic phrases, the most basic well-formulated formulas. These can be combined into more complex wffs, such as 8x , x + 5 ≤ x + 10 3.2 Truth A well-formulated formula is not always true, it is just generated in a specific way outlined above. For example, in the model and structure of abelian groups outlined above, the statement 3 + 5 ≤ 1 is a wff, although it is clearly not true. Similarly, the statement: there exists an integer, x, such that for every integer y, x is less than or equal to y, is a wff, but is not true. 4 If a sentence is true in a structure Φ, we say that that structure is a model of that sentence. Extending this, a structure Φ is a model for a set of wffs if and only if every wff in the set is true for Φ. However, to make a statement about truth, it is necessary to give a rigorous definition of truth. To this end, we introduce the concept of satisfaction. 3.2.1 Satisfaction Let Φ be a structure of a language β. φ is the domain, or universe, of Φ. Σ is the set of all the sequences in φ, the domain of the structure Φ.