For older approaches regarding reality of sets see: Nature of sets at: http://zaljohar.tripod.com/natureofsets.pdf Nature of sets_2 at: http://zaljohar.tripod.com/realityofsets.pdf Reality of sets at: http://zaljohar.tripod.com/realityofsets.txt Simplified graphical representation of Sets: See: http://zaljohar.tripod.com/graphset.pdf Hierarchy Set Theory HST is just an alternative presentation of standard set theory, it interprets ZF in a nice manner, of course HST is at least equi-interpretable with Morse-Kelley set theory, it shows the model of ZF as it is built and therefore it is a more elegant presentation than the customary way of presenting ZF which is etiological and ad hoc looking. This representation shows that ZF is not arbitrary at all. HST is a pure class theory, no Ur-elements are encountered, just sets and proper classes. EXPOSITION HST: is the collection of all sentences entailed by first order logic with equality and membership from the list of axioms outlined below the following definition. Define (set): set(x) ⇔ ∃y. x ∈ y Axioms: I. Extensionality: ∀x.∀y. (∀z. z ∈ x ⇔ z ∈ y) ⇒ x=y II. Impredicative Class Comprehension: if ø is a formula in which x is not free then (∃x.∀y. y ∈ x ⇔ set(y)ø) is an axiom. Define ({|}): x={y|ø} ⇔ (∀y. y ∈ x ⇔ set(y)ø) III. Ordinals: Every accessible Von Neumann ordinal is a set. Define (Vi): V0 = 0 Vi = P(Vi-1) for every successor set ordinal i Vi=U(Vj) j Ex(x=X) BC_4: Ex(x=x) BC_5: X C Y ^ Y C X ->X=Y BC_6: If phi is a formula in which X is not free, then (EX(X={y|phi(y)})) is an axiom. Definitions in BC: Define: V={x|x=x}. Define: TC(X)={y|AZ(transitive(Z) & X C Z ->y e Z)} Define: |X|=<|Y| <-> (X C Y or EF(F:X-->Y, F is an injection)) Define: L"(A,B) <-> AxeB(Am(meTC({x}) ->|m|=<|A|)) In English B is hereditarily bounded by A iff every element of B is hereditarily subnumerous to A. All of the above is considered among the LOGICAL parcel. Statement of the full theory: MKZ is the theory written in the language of Classes with the following SINGLE extra-logical axiom. [1] Hereditary bound: L"(TC(X),Y) ^ |X|=|m| ->YeV / MKZ prove ZF- over sets of it. Proof: Theorem 1: L"(TC(x),Y) ->YeV Proof: L"(TC(X),Y) ^ |X|=|x| ->YeV [1] Let X=x, then L"(TC(x),Y) ^ |x|=|x| ->YeV [ID] which leads to L"(TC(x),Y) ->YeV. QED Theorem 2: X C TC(x) ->XeV Proof:Since X C TC(x) -> L"(TC(x),X) then XeV [th_1]. QED Theorem 3: U(x)eV Proof: Since U(x) C TC(x) then U(x)eV [th_2]. QED Theorem 4: X C x -> XeV Proof: Since X C x -> X C TC(x) then XeV [th_2]. QED Theorem 5: P(x)eV Proof: Since L"(TC(x),P(x)) Then P(x)eV[th_1].QED Theorem 6: {a,b}eV Since L"(TC(x),0) and since there exist a set [BC_4] then 0eV [th_1]. Now P(0)eV; P(P(0))eV [th_5] Define {0} as 1 Now for any a,b it is easy to prove that {{0},{0,a}}eV, {{1},{1,b}}eV, since both are subsets of P(P(a)) and P(TC(P(P(b)))) Denote those as: (0,a) and (1,b) So for any sets a,b we can prove the existence of a bijection from {0,1} to {TC(a),TC(b)} and since L"(TC({TC(a),TC(b)}),{a,b}) then {a,b}eV [1]. QED Theorem 7: {0,{0},{{0}},...}eV Proof: Since L"(TC({0}), {0,{0},{{0}},..}) and {0}eV; then {0,{0},{{0}},...}eV [th_1].QED Theorem 8: F:x-->Y ->YeV Proof: Since L"(TC(F),Y)^ |F|=|x| then YeV [1]. QED MKZ is stronger than MK-Reg. since it proves the existence of hereditary classes that the later can't. 24/1/2014, Notes on what mathematics is: The following notes are largely informal and so they are no that precise, the aim is to cast an imperssion as to what Mathematics is. I'll start with Logic which I'd define as the science of general inference. Logic is concerned with truth about statements of the most general kind, So logical tools can work over any subject, unlike non logical desciplines which are concered with inferences about a limited scope of knowledge. So the basic matter that discriminate logic from others is generality. However Logic as usually perceived doesn't by itself supply Ontology, in other words logic doesn't prove the existnece of multiple objects. A *logical Ontology* is what I call for extensions of logic that provide Ontology also in the most general manner, those extensions are writtin in a logical language extended with primitives that are so pervasive over human knowledge and often needed to explain the background of logic itself, examples of such primitives are 'set membership' and 'Identity', so trivial so pervasive relations over all fields of human knowledge, another example is 'part' relation which is as pervasive and also can be involved in explaining both set membership and identity themselves therefore indirectly also taking place in explaining the background (Meta_theory) of logic itself. So a logical Ontology is a theory that extends logic using pervasive primitives that are involved in explaining the meta_theory of logic, and of course that theory is axiomatized by motivations to capture trivial intuitions around those primitives. The aim of logical Ontology is primarily to provide Ontology over which the logical machinery operates in the most general manner. Mathematics can be seen as the field of knowledge that yields itself easily to be interpreted in a logical Ontology, so in other words Mathematics is "Applied logical Ontology", and if we consider, roughly speaking, any logical Ontology as being a varient of actuation of logic, and thus a variant of logic, then in that sense mathemtics would be defined as "Applied Logic". A simple example of a logical Ontology is the following theory extending first order logic, by adding symbols of set membership "e" and Identity "=" and adding the axioms of identity theory and the following axioms: [1] For all x,y(for all z(z e x <-> z e y) -> x=y) [2] If phi is a formula in which x is not free, then (Exist x(for all y(y e x <-> phi(y) ^ Exist z(y e z)))) is an axiom. [3] For all a,b,x(for all y(y e x ->y=a V y=b) -> Ez(x e z)) / This theory is motivated solely by the consideration of pervasively extending FOL with identity over a wider Ontology, it has nothing to do with any particular mathematical motivation whatsoever, the first axiom which is Extensionality asserts a common intuitive aspect about classes and sets, the second which is a comprehension scheme aims at building object extensions of predicates which is solely an Ontological extension of logic matter, the last axiom aims at providing a sufficient media for speaking about Relations and thus providing Ontology withing which those relations thrives, and so it is also a general kind of motivation to extend logic Ontologically since logic contains relation symbols and we need to understand those in the object world. What I'm saying here is that the above theory has actually a pure logical motivation. So it is verified to call it a "LOGICAL ONTOLOGY". Now Second order arithemtic is faithfully interpretable in the above theory, and so MOST of ordinary mathematics is interpretable in the above theory, and this addresses my point, that of mathematics just being "Applied Logic" or more precisely an "application of some logical Ontology". It might of course be possible for Non mathematical stuff to be interpretable also in the above theory but those interpretations are either non pervasive or otherwise very complex ones that one can hardly view them as applications of the above theory, one can say that they are not faithfull interpretations. Set theory in general is a Logical Ontology, and it appears as if all of mathematic is an application of Set theory. However some thoughts about Mereology and perhaps Category theory might not be seen as being application of set theory, still the later ones must yield themselves to some logical Ontology. Mereology by itself is a logical Ontology, however I'm not sure if Category theory is a logical Ontology, it might be so, otherwise it'll need to be encoded in some logical Ontology to secure it. Zuhair 15/2/2014, Simpler definitions of Finite x Infinite x is infinite if and only if there exist a non empty set of subsets of x that is closed under existence of proper superset relation. formally this is: Inf(x) <-> Ey(Ekey^AueyEwey(wCx^uCw^~u=w)) On the other hand finiteness is defined as: x is finite if and only if every non empty set of subsets of x that is 'non x' closed under existence of proper superset relation, must contain x as an element. formally this is: Fin(x) <-> Ay(Ekey^Auey(~u=x->Ewey(wCx^uCw^~u=w)) ->xey) Those definitions are formally simpler than the customary definitions. It is noticeable that the definition of finite is more extensive than that of infinite, this is expected since finiteness is a more strict condition than infiniteness. Zuhair 13/2/2014, My notes on possiblity of having a TOPOS on top of most of NFU EXPOSITION to NFU+primitive type level ordered pairs "<>", add the primitive binary relation symbol F Axioms about <> = -> (a=c <-> b=d) empty(a) ^ empty(b) -> empty () Axioms about F: 1.F(a,b) ^ F(c,d) ->(a=c<->b=d) 2.F(x,y) -> empty(y) 3.set(x)-> Exist y. F(x,y) 4.F(a,b) ^ F(c,d) -> Exist y. F(,y) We'll use F symbol as a function symbol for simplicity. Define [AxB]_F= {| aeA,beB} [AxB]_F is read as the F_Cartesian product of A and B. F_relations from A to B are just subsets of the F_Cartesian product of A and B. F_functions from A to B are just F_relations from the F_domain of A to the F_codomain of B. It is understood that the F_domains and F_codomains are of course those that have objects the F_images of which are the first and second projections of elements of the F_functions respectively. Of course all F_relations are nothing but sets of empty objects!!! Define (F_object): F_object(x) <-> set(x) ^ for all m in x (Exist n. F(m,n)) this is read as: x is an F_object iff every member of set x has an image under relation F. 5.Axiom: A,B are F_objects -> Exist [A,B]_F Define (F_morphism), An F_morphism is just an F_function. Definition: any formula phi is said to be an extended stratified formula if it can be meet the stratification requirements around all of its primitives, those around membership and equality are the convensional ones; the primitive ordered pairs are assigned exactly the same type as there projections; all variables appearing on the right of the symbol F (or the image of F when using F in a function style symbolism) will receive the same assignment throughtout the fromula while variables appearing on the left of F (or arguments of F when using F in a function style symbolism) are free to have any type assignments as far as those doesn't conflict with stratification requirements around the other primitives. 6.Axiom scheme of Stratified Cartesian Separation: if phi is an extended stratified formula then: (For all F_objects A,B: {y|y in [AxB]_F ^ phi(y)} exists) is an axiom. 7.Axioms of NFU+ primtive type level ordered pairs. / Perhaps we need to add another separation axiom using extended stratified formulas on the set of all F_functions between two F_objects, however I think it is provable from Cartesian sepration on the identity functions on sets of F_functions, then extracting there domains, not so sure? Now the set of All F_objects and F_morphisms would most possibly be Cartesian closed "CC" in which all *sets of sets* of NFU thrives, so for example the set of all sets in NFU is an object of that Cartesian closed Category. And it might be possible for that Category to be a TOPOS. The reason why I would be CC is because the main obstacle against CC imposed by stratification in the ordinary sense are removed and there is no F_object towards which every F_object would have a monic, so both T.Forester's and McLarty's argument are bypassed! What remains is to: [1] Prove that the F_Category (Category of all F_objects) is CC. [2] Prove the consistency of this system relative to ZFC, preferably without using any large cardinal axiom. [3] See if the system is extendable in the usual manner NFU is. [4] See if F_Category is a TOPOS [5] Compare it to known Topoi. Zuhair Al-Johar Feb. 15/2014 Logicism Again: There had been many formal trials to identify whether a symbol represent a logical or extra-logical concept. Here is my take on that subject. I'll consider First order logic to be logic and so all of the connectives and the quantifiers in it are logical symbols. Now if we add any new predicate symbol to FOL, then this mere addition is still logical! In other words even if we have INFINITELY many such added predicate symbols of ALL n_arity functions and relations, then if NO axiom is added to the list of axioms of FOL, then the resulting system which is FOL+these new primitives, *is* a logical system! The reason is because the added symbols are zero axiomatized, i.e. not axiomatized extralogically, so the only machinery at effect is actually the logical machinery of first order logic. So my stand on this subject is that the mere addition of a symbol to FOL is itself logical, however if we add axioms about that symbol then if those axioms carry concepts external to those after which logic is motivated, then the symbol become an "extra-logical" symbol, while if those axioms are motivated solely by the same grounds that motivated FOL in the first place, then that symbol would be "logical". FOL is motivated by aiming at having rules of general inference over the object world. FOL is normative for that, it just stipulate those rules. Anyhow extending those rules Ontologically, i.e. adding existential claims that are so general reflecting the bits and pieces present in FOL, like extending predicate symbols and function and relation symbols into the object world as well as the connectives and quantifiers into the object world is a kind of seeing the Actuation of Logic Ontologically, of course the result is stronger than the mere potential of having rules. Now that motivation of extending FOL by objects is so general motivation that cannot be said to fit any limited scope of human knowledge, it directly reflect logic into the object world, such a motivation is actually nearer to logic than anything else, therefore it can be considered as logical also. So a theory that is merely motivated by Ontologically extending logic would be a logical theory and all symbols in it are logical symbols. Under the above rationale I consider the following theory as a theory of logic. FOL + Extensionality: (Az. z e x <-> z e y) -> (Az. x e z <-> y e z) Impredicative Class comprehension: as in MK Pairing: (Ay. y in x ->y=a or y=b)-> Ez. x in z. / The above theory is solely motivated towards having unique object extensions per equivalent predicates that have the capacity to express relations and functions in the object world. A pure motivation of Ontologically extending logic. So it is a logical theory and the membership relation symbol which is the only primitive symbol added to FOL in this thoery is also to be considered here as a Logical symbol. This is what I call as Relational Class logic. It is nice to know that almost all of ordinary mathematics is interpretable in such a purely logically motivated theory. This strongly support the case for logicism over ordinary mathematics. Zuhair Al-Johar Feb. 16 / 2014 Very Old trials that contains naive premature ideas of mine before I was aware of formal approaches to math are: The Infinite Calculus: at http://zaljohar.tripod.com/the_infinite_calculus.doc The sub\supra_fintie numbers: at http://zaljohar.tripod.com/suprafinites.pdf Zuhair Al-Johar