Formal Systems : follow a procedure to enlarge a formal system Axioms Rules of inference Definition: Set Theory: most mathematicians use ZFC ; vocabulary, grammar; - statements about sets; $S$ - member of set $s$ - $s \in S$ - $f : S_1 \rightarrow S_2$ - map between sets = function Cartesian Product: Example: $R X R$ = $R^2$ = $\{ (a,b)| a,b \in R\}$ True vs Valid element = member function = map bigger formal system: Fields - Definition: A field is a set $F$ equipped with two functions: $F$ x $F\rightarrow F$ - set {a,b} where a,b $\in$ $F$ - addition: $+$ : $F$ x $F\rightarrow F$ - multiplication: $\cdot$ : $F$ x $F\rightarrow F$ - binary operation on $F$ - Axioms: gave last time: - Other fields: rational numbers - Finite fields - Vector Spaces subsume fields - field is special case of set - Definition of Vector Space is just enlarging fields - by adjoining new words and new axioms - State as Definition: Vector Space is over a Field - fix the field - for each $F$ there is a separate theory - A Vector Space over $F$ is a set $V$ equipped with two functions: - + is $V X V \rightarrow V$; given two elements u, v $\rightarrow$ u + v - scalar multiplication: $F X V \rightarrow V$ - use $\lambda , \mu$ - Axioms of Vector Space in book 1B - 1. $\forall$ u, v $\in$ V, u + v = v + u : commutativity - 2. a. ( u + v) + w = u + ( v + w) - 2.b. (ab) v = a (bv), a,b $\in$ $F$, v $\in$ $V$ like associativity but not, since not in same set - 3. Additive identity: $\exists$ 0 $\in$ $V$ such that st v + 0 = v - 4. $\forall$ v $\in$ V, $\exists$ w $\in$ V st v+w = 0 - 5. what is 1? look at $F$; stipulated $\exists$ 0 $\in$ F; $\exists$ 1 in $F$; additive identity and multiplicative identity; structure that exists in F, so can use it; - 1 $\cdot$ v = v - 6a. Distributive law: a ( u + v) = au + av, $\forall$ a $\in$ F, $\forall$ u, v $\in$ V - 6b. (a + b)v = av + bv; $\forall$ a,b $\in$ F, $\forall$ v $\in$ V - Gödel's Second Incompleteness Theorem - in formal system should not be able to prove every statement - propositional logic underlies ZFC; negation - $\forall$ statements, can make opposite statement; both cannot be true - intuitionist mathematics; law of excluded middle; allow possibility that both are true - formal system that does not produce a statement and its negation is called consistent - purpose of a formal system is to prove, but not all statements; if can prove both ways, can prove any statement; - how do we know a given formal system is consistent? for most we don't; second incompleteness says in a formal system cannot prove its own consistency - First Incompleteness theorem is already shocking: in natural numbers, can have a statement that is true, but which cannot be proved; - Wiggle out by adding axioms to ZFC - large cardinals - step out of ZFC, then prove consistency - ZFC', ZVC''' but better than that; confidence; haven't found inconsistency - easy to find instance of proof of s and proof of its negation; - easy in sense of counterexample; Euclidian geometry, simpler system can be proved. - theory of Turing machines; halting problem; culprit is infinity - Can add axioms; can remove axioms - fields: existence of multiplicative inverse is axiom ; 1 is there; - you can remove axiom of multiplicative inverse; get something smaller; get theory of rings; ring of integers; but can adjoin something to axiom of ring; to integers; - Tree, not russian doll, of systems - could build formal system for which there are no examples; - in physics, quantum mechanics, quantum systems are vectors over the complex numbers - infinite dimensional vector sispace - that ability to discern what is interesting ; feed computer program a string of symbols; how to discern , how will they discern that something is more interesting than something else? - Examples: - Simplest Vector Space; - simple set: empty set; - simple field: no empty set, since has 0, 1: Boolean addition: 1 + 1 = 0 - simplest Vector space; no 1, that is in field; in VS, one element: $\{0\}$ - has 0; addition is simple; 0+0= 0; - notation; fixed F; comes with 0,1; but different notation; 0 in V not same as 0 in F; underline; - simple example $R^n$, n from 1 to infinity; n=0 means $R^0$ - examples; - columns: - 1 is additive inverse: 1 + 1 = 0 in simple; - $R^2$ parallelogram; - Proof: uniqueness of additive identity - u,v,w $\in$ V; v,w are additive identity; - u+v = u; u+w = u ; v = (u+v)+ v