are well-formed, provided Φ ( X, x)

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1 (October 27) 1 We deine an axiomatic system, called the First-Order Theory o Abstract Sets (FOTAS) Its syntax will be completely speciied Certain axioms will be iven; but these may be extended by additional ones at a later time (as it is expected rom the experience with the orse-kelly system) In FOTAS, we only have sets; no classes However, we also have unctions as a primitive notion We use: set variables:, YZ,, [thus: capital letters are sets now; not classes ; in act they are abstract sets ]; element variables: x, x, y, z, Each element-variable must be declared to be typed by a set-variable, thus: x : The intuitive meanin o x : is that x is an element o ; however, x : is not a poposition, and, or instance, one cannot write ( x : ) This, o course, is amiliar rom ordinary type theory The only dierence is that in x :, itsel is a variable The combinations x : Φ (, x), x: Φ (, x), x: Φ (, x) are well-ormed, provided Φ (, x) is a well-ormed proposition (ormula), with possibly other ree variables The irst quantiied ormula says, o course, that or all x in, Φ (, x) holds ; the third that there exists such that or all x in, Φ (, x) holds Note however that Φ (, x) (with x : as beore) is not meaninul and, correspondinly, it will be declared illeal (not well-ormed) syntactically Supposin that the ree variable x actually occurs in Φ (, x) (we ll see examples below), in Φ (, x), we have lost its roundin as x : : we have no particular any more to relate x to: we cannot evaluate it at values o x We have unction variables,, Each unction variable has to be typed thus: : Y, with and Y set-variables, possibly the same O course, the meanin o : Y is that is a unction whose domain is, and whose codomain is Y (in particular, the rane o is included in Y ) The type o is thus a dependent type: Arr(, Y ) ( arrow rom to Y ), and we have the variable declaration : Arr(, Y ) We write : Y synonymously to : Arr(, Y ) The upshot is that it is not possible to talk about unctions in eneral; only ones with pre-assined domain and codomain We can use a quantiier over the collection o arrows rom a iven to a iven Y, but not over unctions in eneral (directly, at least) Suppose we have a well-ormed proposition (ormula) Φ (, Y, ), with possibly other ree variables Then the ormula Y : Arr( Y, ) Φ ( Y,, ) is meaninul 1

2 and well-ormed For instance, i Φ (, Y, ) is T ( identically true ), then Y : Arr(, Y) T says that rom every object (set), there is an arrow (unction) to at least one object The statement : Arr(, ) T is also meaninul It is not all-riht to say, however, that Y : Arr(, Y) T, which would, apparently, say that is an arrow rom every set to every set at the same time The latter statement, even i it looks meaninul to you, is not expressible in our lanuae (it is not simply alse) Let us note that, once the variable is declared thus: : Arr(, Y ) (equivalently: : Y ), we may abbreviate Y : Arr(, Y) T as Y T, althouh this now looks ambiuous We will have variable declarations and judments separately, or a ull description o a situation:, thus: : Set; Y : Set; : Y :: Y T For each set variable, we have the equality predicate = : equality or elements o The rammar o = is this: or variables x :, : 1 x2, the atomic ormula x1 = x2 is well-ormed, with ree variables x1 and x2 (o course) For each pair o set variables and Y, we have the equality predicate or arrows : Y The rammar o = Y, is this: or variables : Y, : Y, the atomic ormula = Y, is well-ormed, with ree variables and (o course) = Y, We have an operation symbol App(lication) which works like this: : Set; Y : Set; : Y; x : :: App(, x): Y We mean that in case the variables, Y,, x are declared as shown, we have the wellormed term App(, x) typed as a term o type Y : App(, x) : Y We will abbreviate the term App(, x ) as x, or even ( x), or x O course, App(, x ) siniies the (unique) value o the unction at the arument x With the usual iterated term-ormation rules, as an example, we can then orm the ollowin typed term : Y Z, x: :: ( ( x)): Z 2

3 O course, ( ( x )) abbreviates App(, App(, x)) The above essentially completes the speciication o the lanuae o FOTAS We have explained the (restricted) use o the quantiiers We have the connectives T (true), ( alse), ( and), ( or), ( implies) used as usual (no restriction) Φ abbreviates Φ I should add that in the ormula x : x= x, the variable is a ree variable For clarity s sake, let s review the natural eneral (ensemblist) semantics o the lanuae o FOTAS ( ala ' Tarski) We are usin set theory, in act orse-kelly, in this speciication A structure or FOTAS consists o: a collection (a set, or a class) S (= structure ) (the collection o sets ); Set, the interpretation o the sort Set in the or each set, or object S, a set El ( )(= El ( ) ), the collection o elements o (here we insist that El ( ) be a set); or each pair o objects, Y S, a set Arr (, Y ) (or: hom (, Y )) o all arrows rom to Y ; or each object S, the binary relation = on the set El ( ) is taken to be real equality (This is the standard interpretation, within the ramework o Freean absolute equality In another version, a non-standard one, = is an arbitrary binary relation = on the collection El ( )); similar specs or = Y, ; a ternary operation App = ( App ) that applies to any quadruple (, Y,, x ) where S, Y S, Arr(, Y), and x El ( ), and ives, as the output, a value denoted (naturally) as ( x ) in El ( Y ) In other words, or any, Y,, x as stated we have an actual unction (carelessly denoted as) : El ( ) El( Y ) Under the semantics, we have, by an obvious (implied-by-the-above) Tarskian truthdeinition or ormulas o FOTAS in eneral, resultin in a truth-value o any ormula, at any admissible evaluation o its ree variables Admissible here means, or instance, 3

4 that i we had the ree variables x and, with x : declared, then in evaluatin x and, we must have observed the condition that x El ( ) (Note also that there is a correspondin E -valued semantics o FOTAS, or any Setlike universe (cateory) E ; or instance, or any topos E ) The two main axioms (the second bein an axiom scheme, in act) are Function extensionality: : Y, : Y, x : :: ( = Y, x( ( x) = Y ( x))) Several abbreviatin devices may be used Firstly, in an axiom, the initial universal quantiiers may be omitted (and considered bein there, ater all) Secondly, the subscripts o the equality sins may be omitted, since they can be uniquely restored rom the context We obtain the simpliied statement = x( ( x) = ( x)) Function comprehension: Given any ormula Φ (, Y, x, y) with the ree variables x : and y : Y, and possibly other ree variables, the ollowin is an axiom: x! y Φ(, Y, x, y) ( : Y) x y( ( x) = y Φ (, Y, x, y)) (! y Φ ( y) abbreviates y y ( Φ( y ) y = y) as usual; o course, y : Y ) Uniqueness o in comprehension is assured by extensionality The eect o the axioms is this Suppose we have any model o the axioms so ar Suppose urther that we have a ormula Φ(, Y, x, y; a), with x : and y : Y, and with the ree variables denoted a are all iven (admissible!) values (parameters) in, also denoted by a The ormula Φ(, Y, x, y; a) provides or a deinable (with parameters) relation R El( ) El( Y) I this relation R is unctional, that is, R is a unction with domain El( ) in the usual set-theoretic sense, then there is a unique arrow, an element o hom (, Y ), : Y, that denotes R, ie, such that ( x ) = y i xry ( x El( ), y El( Y) ) (remember that ( x) = y abbreviates App (, x) = y ) As an example, let Y Z, x:, z: Z, and consider the ormula ( ( x)) = z (with ree variables all the displayed variables) Suppose all the ive 4

5 variables in Y Z are iven (appropriate) values (simply: assume that we have objects and arrows Y Z in ) I claim that it is obvious that ( ( x)) deines a unctional relation rom to Z : or all x El( ), there is a unique z El( Z ) such that ( ( x )) = z Thereore, there is a well-deined arrow such that hx ( ) = zi ( ( x)) = z, or all x El( ), z El( Z) ; that is, hx ( ) = ( ( x)) or all x El( ) We denote h as, and call it the composite o and = z h: Z The ormula x = y ( x :, y: ) deines, or any iven object ( set ), the identity unction id : Extensionality shows that the associative law or composition: in case Y Z W, with all entities in, then h( ) ( h) : both sides deine the unction F: El( ) El( W) or which F( x ) = h( ( ( x)) ) or all x El( ) = We have shown that every model o FOTAS ives rise to what is called a concrete cateory The cateory has objects the elements o Set, ie, what we called objects above; and arrows what we called arrows above Denotin this cateory by too, we have the aithul unctor F: Set, with Set the cateory o sets and unctions, where F( ) = El ( ), and, or : Y in, F( ) the unction that we wrote as : El ( ) El( Y ) It is ar rom true, however, that every concrete cateory appears as a model o FOTAS The axiom scheme o unction comprehension will ive rise to arrows that would not be there without it Remark o apoloy: there are some inconsistencies in the onts used (and I was too lazy to correct it) For instance, I write sometimes App, and sometimes App or the same thin There is no daner o conusion thouh There are only two meanins o App : the syntactical one (a symbol), and its interpreted version, App, reardless o onts used h 5