Infinity and Beyond

2. Infinity and Beyond

3. Lecture 1 Infinity in Real Arithmetic

4. What is infinity()? • It’s a “number”  that is larger than all natural numbers, i.e. for all n{0,1,2,3,…}, n < . • Questions: • 1) Are there more than one infinity? • 2) Can we perform algebraic operations with ? • 3) Is  + 1 = ? Is  +  = ? Is  = ? • 4) How about   , /, 0? • 5) Can we write 1/ = 0? How about 1/0? • Answer: This depends on our definitions.

5. Is  (or ) a real number? • We defined: For all n{0,1,2,3,…}, n < . • Likewise: For all n{0,1,2,3,…}, n > . • Question: Is  (or ) a real number? • Answer: This depends on our definition of real numbers. • Usually we define real numbers in a way that excludes both  and . • Question: What are real numbers?

6. The Set of Real Numbers R: • Definition 1: This is the set of all points on the real line. • This definition is intuitive and visual, but faces the following problems: • What does a line mean? • Is a line physical? In this case are there infinitely many real numbers? • Does the line have a beginning and an end? I.e. do  and  belong to it?

7. The Set of Real Numbers R: • Definition 2: This is the set of all decimal infinite sequences of digits (including a decimal point). • E.g.  = 3.14159265358979323846… • Also 1/3 = 0.3333333333333333333… • Caution: 1 = 1.000000… = 0.9999999… • This set does not contain a sequence like …9999.9999…. (infinite in both directions). • Thus, both ,R.

8. The Set of Real Numbers R: • Definition 3: Instead of defining the set R, we define the structure (R,+,,<) by the following axioms: • 1) (R,+,) is a field, i.e. + and  satisfy the usual properties, e.g. x(y + z) = xy + xz. • 2) (R,<) is a linear order, i.e. for any x and y, either x < y or x = y or x > y, and the relation < is transitive, i.e. for all x,y, and z; x < y < z  x < z. • 3) < is congruent with respect to + and , i.e. for all x,y, and z; x < y  x + z < y + z. Also x < y and z > 0  xz < yz. • 4) Every nonempty subset of R that is bounded above, has a least upper bound.

9. Do  and  belong to R? • Theorem: No. There is no real number that is larger than all natural numbers. • Proof: If there were such a number (called ), then the set N of all natural numbers is bounded above by . • Thus, using Axiom 4, we can get a least upper bound . • From 1 < 0, it follows that   1 < . • Since  is a least upper bound of N, it follows that   1 is NOT an upper bound of N. • Thus, there is an nN, such that  1  n. • It follows that  n + 1 < n + 2, contradicting the fact that  is an upper bound of N.

10. The extended real numbers R* • In real arithmetic, we can choose to extend the set of real numbers R with the two new elements  and , i.e. we define: R* = R{,}. • Note: We have only one  and one . • (R*,<) is an extension of (R,<) by defining: • For all real numbers r,  < r <  , and  < . • Note: (R*,<) is a linear order as before, i.e. for any x and y, either x < y or x = y or x > y, and the relation < is transitive, i.e. for all x,y, and z; x < y < z  x < z.

11. Algebraic Properties of  and  • We extend the algebraic operations by the following definitions: For all rR, pR+, nR, • r +  =  + r =  +  =  • r + () = () + r = () + () =  • p = p = n() = ()n =  = ()() =  • n = n = p() = ()p = () = () =  • 1/ = 1/() = 0 • Note: Since we can define x  y = x + (1)y, and also x/y = x(1/y), subtraction and division involving  and  can be defined.

12. Example: Show that, for all rR, pR+, nR, • r   = ()  r = ()   =  •   r = r  () =   () =  • r/ = r/() = 0 • /p = ()/n =  • /n = ()/p = 

13. Undefined Quantities involving  and  • The following quantities are left undefined: •  + (), () + , • and consequently   , ()  (), • 0, 0, 0(), ()0, • and consequently /, /(), ()/, ()/() • Also, 1/0 is still not defined (informally, we can not choose between + and ), • and consequently /0, ()/0 • Question: Why don’t we just define them as we like?

14. Answer: • If we tried defining the previous undefined quantities, we will ruin the following theorem. • Theorem: Whenever defined, arithmetic expressions involving  and  obey Axioms 1 and 3 of Slide 8. • E.g., for all x,y,zR*=R{,}, the equation x + (y + z) = (x + y) + z holds, if both sides are defined. • Also, x < y and z > 0  xz < yz, etc.. • Proof: An exhaustive check of all properties.

15. Defining the Undefined • What goes wrong if we defined  + () = 0, say? • Answer: The required properties will not hold anymore. If they did, we can derive the contradiction 0 = 1 as follows: • 0 =  + () = (1 + ) + () = 1 + ( + ()) = 1 + 0 = 1. • Exercise: Show that we can not define 0, without ruining Axiom 1. Hint: Use 0 = 1 + (1) • What about 1/0?

16. Challenge! Give a definition of a set R** that contains all real numbers and (possibly many) infinities, with the operations of + and  defined for ALL elements, and obeying all of Axioms 1,2,3.

17. Thank you for listening. Wafik