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Internal Set Theory and an Intuitive Development of the Calculus

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Internal Set Theory and an Intuitive Development of the Calculus INTERNAL SET THEORY AND AN INTUITIVE DEVELOPMENT OF THE CALCULUS SIMON LAZARUS Abstract. In this paper, we present Internal Set Theory (IST) as a means to more intuitively develop mathematics while still maintaining complete rigor. We begin by stating the axioms of IST and demonstrating several of their central consequences. We then use these results to rigorously introduce the notions of infinitesimal numbers and infinite closeness. Lastly, we develop the calculus in a nonstandard fashion based on these intuitive notions. Contents 1. Introduction 1 2. Fundamentals of Internal Set Theory 3 3. Standard Relations 7 4. Infinitesimal and Unlimited Numbers 9 5. The Calculus 12 Continuity 12 Di↵erentiatiation 14 Integration 19 6. Appendix 24 Acknowledgments 27 References 27 1. Introduction The vast majority of modern mathematics is based in principle upon Zermelo- Fraenkel Set Theory with the Axiom of Choice (ZFC). The axioms of ZFC stipulate the rules of set formation based upon the (undefined) binary predicate ,where x y intuitively states “x is an element of the set y.” Using and the2 logical symbols2 (for all), (there exists), (and), (or), (not), =2 (implies), and (is8 equivalent9 to; if and only if)^ under the_ rules¬ of first-order) logic and the axioms() of ZFC, one can create definitions, state theorems, and write proofs. Although these methods are quite powerful, in many cases the strict requirements of ZFC can lead to definitions and proofs that are less intuitive than we may wish them to be. For this reason, we introduce Internal Set Theory (IST) to allow for many more intuitive definitions and proofs to be perfectly rigorous. IST is an extension of ZFC: it introduces the (undefined) unary predicate “standard” (we say Date:September3,2012. 1 2 SIMON LAZARUS “standard(x)” to mean that the set x is standard) and adds to the axioms of ZFC three axiom schemata that allow us to work with formulas involving this predicate. We assume the reader is familiar with the basic structure of a well-formed formula in first-order logic1. We recall that a variable x is said to be free in a formula A if A does not quantify x with “ x” or “ x.” For example, the formula x R has x as 8 9 2 a free variable, while the formula x Q(x R) has no free variables; x is bound. A formula with no free variables is8 said2 to be2 a statement. Every theorem of ZFC is a statement formulated from (i) variables quantified by or and (ii) some or all of the symbols , , ,= , , and . Formulas of8 ZFC9 are created in a similar manner, but^ not_ ¬ all of) a formula’s() variables2 need to be bound. In much the same way, formulas of IST are created using free and bound vari- ables, some or all of the above symbols, as well as possibly the predicate “standard.” If A is a formula of IST in which “standard” does not appear, then A is clearly also a formula of ZFC. In this case, we say that A is an internal formula. If B is a formula of IST in which “standard” does appear, we say that B is external. IST merely adds new terminology to ZFC and gives axioms to explain how it can be used. In this way, any statement that is provable in ZFC will necessarily be provable in IST. However, the axioms of ZFC apply only to internal concepts. In particular, given any set x and any external formula with at least one free variable, the Axiom of Specification cannot be used to form a subset of x whose elements are exactly those of x which satisfy the given formula—it can only do so for internal formulas. We are therefore in need of new axiom schemata that allow us to form and work with sets based upon external formulas. These are the Principles of Transfer, Idealization, and Standardization. After stating these axiom schemata and a few of their important consequences, we will rigorously develop the notions of infinitesimal and unlimited numbers and infinite closeness, which we will use to develop the calculus in a fashion that we find much more intuitive than the usual presentation. One may wonder: even if IST does provide easier methods with which to state definitions and prove theorems, why should we believe anything IST begets? That is, just because we can show using the new methods of IST that some internal Theorem A holds true, does that necessarily mean Theorem A follows from ZFC alone? The answer to this question is a resounding yes. In particular, we have the following result. Conservation Theorem (Powell). Let A be an internal theorem. If A is provable in IST, then A is provable in ZFC. A proof of the Conservation Theorem is omitted in this paper. Within the proof given in [1], Nelson presents a method of reducing any proof of A using IST to a proof of A using ZFC alone. Therefore, we can have full confidence in the fruits of IST: any proof of an internal theorem that employs IST immediately implies a proof of that same theorem that employs only ZFC. Additionally, this theorem guarantees that IST is consistent if ZFC is consistent. With this in mind, we begin our formal discussion of Internal Set Theory. 1Essentially, this means the reader will neither accept nor produce combinations of symbols that amount only to nonsense. INTERNAL SET THEORY AND AN INTUITIVE DEVELOPMENT OF THE CALCULUS 3 2. Fundamentals of Internal Set Theory Before stating the three axiom schemata mentioned in the previous section, we introduce some notation. First, if y is any set, we say finite(y)ifthereisno bijection between y and any proper subset of y. Now let A be any formula with free variable(s) including x. Then we introduce the following notation. Notation. (i) We use finxA(x) to mean x(finite(x)= A(x)) and we use finxA(x) to mean8 x(finite(x) A(x)).8 ) 9 (ii) We use 9stxA(x) to mean^ x(standard(x)= A(x)) and we use stxA(x) to mean8 x(standard(x) 8A(x)). ) 9 (iii) We use 9st finxA(x) to^ mean st(x)(finite(x)= A(x)) and we use st finxA(8x) to mean stx(finite(8x) A(x)). ) 9 9 ^ We can now state the three axiom schemata of Internal Set Theory. Transfer Principle. Let A(x, t1,...,tn) be an internal formula with free variables x, t1,...,tn and no other free variables. Then (T) stt ... stt ( stxA(x, t ,...,t )= xA(x, t ,...,t )). 8 1 8 n 8 1 n )8 1 n Idealization Principle. Let A(x, y) be an internal formula with free variables including x and y.Then (I) st finz x y zA(x, y) x styA(x, y). 8 9 8 2 () 9 8 Standardization Principle. Let A(z) be a (not necessarily internal) formula with free variable(s) including z.Then (S) stx sty stz(z y z x A(z)). 8 9 8 2 () 2 ^ For the remainder of this section, we develop the intuitive meaning of these axiom schemata as we employ them. By replacing A with A and employing contraposition in (T) and (I) above, we can arrive at the logical¬duals of (T) and (I): (T1) stt ... stt ( xA(x, t ,...,t )= stxA(x, t ,...,t )). 8 1 8 n 9 1 n )9 1 n (I1) st finz x y zA(x, y) x styA(x, y). 9 8 9 2 () 8 9 We now discuss some important consequences of (T) and (T1). First, we observe that if A is any internal formula whose only free variable is x and there exists a unique x such that A(x) holds true, then that x must be standard, as (T1) asserts the existence of a standard x satisfying A. Thus, one way of thinking about the standard sets is that any set that can be uniquely described within ZFC is standard. For example, the set of natural numbers N, the function sin, and the real number e are all standard sets since they are all sets which are uniquely described within ZFC. (The reader should recall that every entity in ZFC is a set, including numbers and functions.) More generally, (T1) tells us that any set which is uniquely determined by some finite number of other standard sets will itself be standard. We o↵er a few examples of this below. 4 SIMON LAZARUS Examples 2.1. (i) If A is a standard set, then the power set of A (the set of all subsets of A) is standard since it is uniquely determined by A. (ii) If x and y are standard sets, then the ordered pair (x, y) is standard since it is uniquely determined by x and y. (iii) If f is a standard function and a is a standard element of the domain of f, then f(a) is standard since it is uniquely determined by f and a. (iv) If f and g are standard functions such that f g is a function, then f g is standard since it is uniquely determined by f◦ and g. ◦ Remark 2.2. We note that restricted quantifiers maintain their restrictions through applications of (T) and (T1). For example, if X and Y are sets and B is an internal formula whose only free variables are x, t1,...,tn, then for all standard t1,...,tn we have stx XB(x)= x XB(x), and letting ! denote unique existence, we also have8 2!xB(x)= )8!stxB2(x).
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