Well let s see The first subchapter is 8 5 The Problem of Division by Zero This he follows up with 8 7 Five approaches to Division by Zero which I will repeat in its entirety This will take more than one sitting to type in The following is from Suppes 1957 163 169 1 The first approach differs from the others in that it recommends a change in the baseic logic to deny meaning to expressions like 1 0 1 0 Without attempting to characterize the basic changes necessary we may still offer some general objections to this approach more to follow tomorrow 2 The second approach is to let x 0 be a real number but to define division by the conditional definition stated in the last section IF y 0 Then x y z if and only if x y z In this case for every number x x 0 is a real number but we are not able to prove what number it is 2 1 0 2 0 The inability to prove or disprove 2 is an argument against the second approach since we want our axioms to be as complete as possible 3 The third approach agrees with the second in making x 0 a real number but it differes in making x 0 0 for all x This eliminates the undecidability of statements like 2 For the third approach the appropriate definition of division is a proper definition similar to the axiom free definition discussed at the end of 8 5 x y z y 0 x y z y 0 z 0 An advantage of the third approach is that it permits the definition of zero by a straight forward proper definition fully satisfying the criteria of elminability and non creativity The main disadvante of this appraoch is the one mentioned in 8 5 many mathematicians feel uneasy with the identity x 0 0 4 The forth approach agrees with the second and third in requiring no basic change of logic it differes in not making it possible to prove that x 0 is a real number The basic idea is to introduce a predicate R which means is a real number Hypotheses using this predicate must be added to all the axioms stated in Chapter 7 Furthermore to guarantee that addition and multiplication of numbers yield numbers we must add the two closure axioms R x R y R x y R x R y R x y The introduction of the predicate R widens our domain of individuals for now it does not follow that everything in the domain is a real number in particular the 1 0 is a real number The indroduction of R has the further consequence of making it natural to make all our definitions of arithmetical relations and operations conditional There is no point in defining the realition of equal to or less than for instance for things which are not numbers Thus we would ahve R x R y x y x y V x y The definition of division would be 3 R x R y R z y 0 x y z x y z If x is a real number with 3 at hand we can not prove x 0 is a real number And we can not prove x 0 is not a real number But we are not faced with the counterintuitive situation of being forced to call x 0 a real number The situation can be improved by introducing into our system a primitive symbol for some object which is not a real number Without saying what the object is let us designate it by v They we have the axiom R v here is or bent bar i e NOT that is the assertion that v is not a real number We may now definie division by 4 R x R y x y z y 0 R z x y z y 0 z v The virtue of 4 is that it definitely places x 0 outside the domain of real numbers for any number x Such a consequence would seem to be in closest accord with ordinary mathematical usage Definition 4 also has the virtue of making x 0 y 0 Where x and y are real numbers thus eliminating a vast proliferation of odd mathematical entities On the other hand to see once for all that all is not for the best in this best of all possible worlds notice that if we adopt 4 we can not decide whether or not v v is a real number If we use conditioncal dfinitions and insist on not tampering with the basic law of identity for therms then we must be prepared for the undecidability of the statue so fentities like v v We make thake the attitude that everything is in good order in the domain where we intend to use the division operation and we really do not care what is going on elsewhere if not inconsistencies can creap in 5 A fifth approach will have to wait until tomorrow I will finish up the 1st one also I ve basically typed in the entire section Bill