Three Profound Theorems about Mathematical Logic

Transcription

1 Power & Limits of Logic Three Profound Theorems about Mathematical Logic Gödel's Completeness Theorem Thm 1, good news: only need to know* a few axioms & rules, to prove all validities. *Theoretically only: having more rules is convenient. lec 2F.1 Copyright lbert R. Meyer, lec 2F.2 xioms & Inference Rules Namely, starting from a few propositional & simple predicate validities, every valid assertion can be proved using just UG and modus ponens repeatedly! Cannot Determine Validity Thm 2, ad News: there is no procedure that determines when quantified assertions are valid (in contrast to propositional formulas). lec 2F.3 Copyright lbert R. Meyer, lec 2F.4 Gödel's Incompleteness Theorem for rithmetic Thm 3, Worse News: if we stick to domain, N, with predicates x + y = z, x y = z, then no proof system can prove all the true assertions! Three Profound Theorems We won't prove these Theorems. Their proofs usually require half a term in an intro logic course after lec 2F.5 Copyright lbert R. Meyer, lec 2F.6 1

2 Mathematics for Computer Science MIT 6.042J/18.062J Sets & Functions collection? Sets Informally, a set is a collection of mathematical objects with the collection treated as a single mathematical object. This is circular of course: what s a lec 2F.7 Copyright lbert R. Meyer, lec 2F.8 Some sets Real numbers, R complex numbers, C integers, Z the emptyset, the set, pow(z) of all sets of integers the power set lec 2F.9 Some sets {7, lbert R., π/2, Τ} set with 4 elements: two numbers, a string, and a oolean value. Same as { lbert R.,7,Τ, π/2} -- order doesn t matter Copyright lbert R. Meyer, lec 2F.10 Membership Membership x x is an element of π/2 {7, lbert R.,π/2, Τ} x x is a member of shorter: x is in π/3 {7, lbert R.,π/2, Τ} 14/2 {7, lbert R.,π/2, Τ} 7 Z 2/3 Z Z pow(r) lec 2F.11 lec 2F.12 2

3 In or Not In {7, π/2, 7} is same as {7, π/2} n element, like 7, is in a set, or not in the set. (No notion of being in the set more than once) is a subset of is contained in Every element of is also an element of. Z R, R C, {3} {5,7,3} every set Containment lec 2F.13 lec 2F.14 Team Problem Problem 1 Defining Sets The set of elements, x, in such that P(x) is true. { x P( x )} lec 2F.15.. Copyright lbert R. Meyer, 2005 lec 2F.16 Copyright lbert R. Meyer, 2005 Defining Sets The set of even integers: { n Z n is even } { n Z m Z. n = 2m} New sets from old Venn Diagram for and lec 2F.17 lec 2F.19 3

4 union intersection :: = { x ( x ) ( x )} :: = { x x x } lec 2F.20 lec 2F.21 set difference complement = D known domain D :: = { x ( x ) ( x )} :: = { x D x } = D lec 2F.22 lec 2F.23 power set pw o ( ):: = { S S } { } { ab} { ab} { a} { b} pow(, ) =,,,, Quickie What is Pow( )? ns: { } What is Pow(Pow( ))? ns: {{ }, } lec 2F.24 lec 2F.25 4

5 Russell s Paradox Let W :: = { S Sets S S } so S W S S Let S be W and reach a contradiction: W W W W Russell s Paradox So the collection, Sets, of all sets, cannot be considered a set! lec 2F.26 lec 2F.27 Mathematics for Computer Science MIT 6.042J/18.062J Functions f : function, f, from set to set associates an element, f ( a ) with an element a. Example: f is the string-length function: f( aabd )=4 lec 2F.31 lec 2F.32 f : f is the string-length function., the domain of f, is the set of strings., the codomain of f, is N 2 g : 1 R R gxy (, ) = x y domain(g) = all pairs of reals codomain(g)= all reals ut g is partial: not defined on all pairs of reals lec 2F.33 lec 2F.34 5

6 g : D R g ( x, y ) = 1 x y { } 2 where D = R ( xy, ) y = x Then g is total: defined on all pairs in domain D. f : Total functions is total iff every element of is assigned a -value by f a b. f( a ) = b lec 2F.35 lec 2F.36 f : Onto functions is onto iff every element of is f of something b a. f( a ) = b a surjection f : 1-1 functions is 1-1 iff every element of is f of at most 1 thing aa,.( f( a ) = f( a ')) ( a = a ') an injection lec 2F.37 lec 2F.38 ijections f : is a bijection iff it is all those good things: total, onto, and 1-1 ijections f : is a bijection iff it is an exact correspondence: f( )= lec 2F.39 lec 2F.40 6

7 ijections Team Problem exactly one arrow out f( )= exactly one arrow in Problem 2 lec 2F.41 lec 2F.42 7