An Ultrafinite Set Theory
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From: RussellE on 26 Feb 2010 21:22 On Feb 25, 11:42 pm, William Elliot <ma...(a)rdrop.remove.com> wrote: > On Thu, 25 Feb 2010, RussellE wrote: > > On Feb 25, 9:03�am, MoeBlee <jazzm...(a)hotmail.com> wrote: > >> On Feb 24, 7:52�pm, RussellE <reaste...(a)gmail.com> wrote: > > You could have a FOL with n constants, u_1,.. u_N > which can be considered as urelements (in the metalanguage). I was trying to come up with a type of "circuit board" system with a finite number of inputs and a finite number of outputs. > For the object language, first pick your primitives. Then make > definitions in terms of the primitives. After you've done that, > then you may indicate in the meta-language some intended sematics > for the primitives. OK. I can simplify the theory and better define my primitives. Some definitions. Urelement - a mathematical object that can be an element of a set. An urelement can not be a set in this theory. Set - a collection of urelements. In this theory, sets can only have distinct urelements as members. A set can be empty. Urelement variable - Define X to be the urelement variable for urelement x. If a set has x as a member then X is true, else X is false. Function - a process to convert an input set into an output set. A function has a well formed Boolean expression for each urelement. This expression determines if the urelement is present in the output set. If the expression for urelement x is true, then x is a member of the output set. A Boolean expression can consist of parenthesis, OR, AND, NOT, the constants True or False, and/or urelement variables. Proper class - a sequence of sets defined by repeatedly applying a function to an initial set. Now, I will try to steal axioms from ZFC. http://en.wikipedia.org/wiki/Zermelo%E2%80%93Fraenkel_set_theory 1) Axiom of extensionality: Two sets are equal if they have the same elements. This theory doesn't need the Axiom of foundation. Sets can't be an element of a set by definition. The axiom schema of specification defines the existence of certain sets. I will define the empty set and use the empty set to derive all other sets. 2) Axiom of the empty set: The empty set exists. 3) Axiom of replacement: S is a set if a function exists to convert the empty set into S. As an example, assume x is the only urelement. X is the urelement variable. Define the function: X_out = ~X_in Apply this function to the empty set to get {x}. The empty set and {x} are the only sets if x is the only urelement. The axioms of pairing, union, and collection are a problem. These axioms assume sets can be elements. They also assume we can perform operations on more than one set at a time. I deliberately defined functions to be unary operators. A function is defined over all urelements. Maybe this is too restrictive. I can't define binary operators like pairing or union because I can't have two input sets. The only way I can specify more than one set is as a proper class. The axiom of infinity is also a problem. I want to replace AoI with something like "there are a finite number of urelements". Unfortunately, I haven't defined finite yet. The only solution I can come up with is to specify the existence of certain urelements. 4) Axiom of Finiteness: The urelements a,b,c, and d exist. I think I can derive the powerset axiom from these axioms. The powerset of the urelements is a proper class. And I don't need a well ordering axiom. A proper class is ordered by definition. It would be nice if I could come up with a general method for binary operators. Maybe I could define proper classes in such a way as to allow n-ary functions. Any ideas are welcome. Russell - 2 many 2 count
From: William Elliot on 26 Feb 2010 22:50 On Fri, 26 Feb 2010, RussellE wrote: > On Feb 25, 11:42�pm, William Elliot <ma...(a)rdrop.remove.com> wrote: >> On Thu, 25 Feb 2010, RussellE wrote: > >> You could have a FOL with n constants, u_1,.. u_N >> which can be considered as urelements (in the metalanguage). > > I was trying to come up with a type of "circuit board" system > with a finite number of inputs and a finite number of outputs. > Consider automata. >> For the object language, first pick your primitives. �Then make >> definitions in terms of the primitives. �After you've done that, >> then you may indicate in the meta-language some intended sematics >> for the primitives. > > OK. I can simplify the theory and better define my primitives. > > Some definitions. > > Urelement - a mathematical object that can be an element > of a set. An urelement can not be a set in this theory. > > Set - a collection of urelements. In this theory, sets can > only have distinct urelements as members. A set can be empty. > > Urelement variable - Define X to be the urelement variable > for urelement x. If a set has x as a member then X is true, > else X is false. > The urelement variable X is true because x is in the set {x}. > Function - a process to convert an input set into an output set. > A function has a well formed Boolean expression for each urelement. > This expression determines if the urelement is present in the output > set. All those definitions are in the metalanguage. > If the expression for urelement x is true, then x is a member of the > output set. > Vague. > A Boolean expression can consist of parenthesis, OR, AND, NOT, the > constants True or False, and/or urelement variables. > > Proper class - a sequence of sets defined by repeatedly applying a > function to an initial set. > Consider recursive sets. > Now, I will try to steal axioms from ZFC. > http://en.wikipedia.org/wiki/Zermelo%E2%80%93Fraenkel_set_theory > You'll need to modify ZFC for if u is an object (urelements), then by ZFC, u = empty set because for all x, x not in u and for all x, x not in empty set. Thus for all x, (x in u iff x in empty) and u = empty set. > As an example, assume x is the only urelement. > X is the urelement variable. Define the function: > > X_out = ~X_in > X_out and X_in are undefined. > Apply this function to the empty set to get {x}. > The empty set and {x} are the only sets if x > is the only urelement. > Doesn't make any sense. > I deliberately defined functions to be unary operators. > A function is defined over all urelements. Maybe this > is too restrictive. I can't define binary operators > like pairing or union because I can't have two input sets. > The only way I can specify more than one set is > as a proper class. > > The axiom of infinity is also a problem. > I want to replace AoI with something like > "there are a finite number of urelements". Both statements can be true. > Unfortunately, I haven't defined finite yet. > That and a lot of other stuff and a bunch of vaguery. > The only solution I can come up with is to > specify the existence of certain urelements. > You're solving a problem? > 4) Axiom of Finiteness: The urelements a,b,c, and d exist. > That doesn't say there are finite many objects (urelements), only that there are four objects, a, b, c, and d. > I think I can derive the powerset axiom from these axioms. Of course, it's part of the ZFC package. > The powerset of the urelements is a proper class. > And I don't need a well ordering axiom. You get that with ZFC. > A proper class is ordered by definition. It is? Oh yes, sequences are ordered. Calling a sequence a class is a misnomer. > It would be nice if I could come up with a general method > for binary operators. Maybe I could define proper classes > in such a way as to allow n-ary functions. > > Any ideas are welcome. > Learn up on propositional and predicated calculus. Take special note how a formal object language is constructedd and the distinction between the object language and the metalanguage. Clarify your ideas before trying to fomalize them. You've got a set O of objects. You've got functions from subsets of O to subsets of O. You've got inputs and outputs of some mysterious things.
From: Virgil on 26 Feb 2010 23:48 In article <b4492d25-d033-4b65-bd95-f9997502009b(a)s25g2000prd.googlegroups.com>, RussellE <reasterly(a)gmail.com> wrote: > Some definitions. > > Urelement - a mathematical object that can be an element > of a set. An urelement can not be a set in this theory. > > Set - a collection of urelements. In this theory, sets can > only have distinct urelements as members. A set can be empty. According to the two definitions above, it is impossible for any set to be a member of any set, which is going to lead to an excessively constrictive set theory of little if any use to mathematics. Typical of RussellE.
From: RussellE on 27 Feb 2010 00:04 On Feb 26, 7:50 pm, William Elliot <ma...(a)rdrop.remove.com> wrote: > On Fri, 26 Feb 2010, RussellE wrote: > > On Feb 25, 11:42�pm, William Elliot <ma...(a)rdrop.remove.com> wrote: > >> On Thu, 25 Feb 2010, RussellE wrote: > > >> You could have a FOL with n constants, u_1,.. u_N > >> which can be considered as urelements (in the metalanguage). > > > I was trying to come up with a type of "circuit board" system > > with a finite number of inputs and a finite number of outputs. > > Consider automata. The system I am trying to describe is very similar to cellular automata. > >> For the object language, first pick your primitives. �Then make > >> definitions in terms of the primitives. �After you've done that, > >> then you may indicate in the meta-language some intended sematics > >> for the primitives. > > > OK. I can simplify the theory and better define my primitives. > > > Some definitions. > > > Urelement - a mathematical object that can be an element > > of a set. An urelement can not be a set in this theory. > > > Set - a collection of urelements. In this theory, sets can > > only have distinct urelements as members. A set can be empty. > > > Urelement variable - Define X to be the urelement variable > > for urelement x. If a set has x as a member then X is true, > > else X is false. > > The urelement variable X is true because x is in the set {x}. Yes. I keep forgetting everyone isn't a computer programmer. > > Function - a process to convert an input set into an output set. > > A function has a well formed Boolean expression for each urelement. > > This expression determines if the urelement is present in the output > > set. > > All those definitions are in the metalanguage. OK. I see now functions must be defined in the meta-language. > > If the expression for urelement x is true, then x is a member of the > > output set. > > Vague. > > > A Boolean expression can consist of parenthesis, OR, AND, NOT, the > > constants True or False, and/or urelement variables. > > > Proper class - a sequence of sets defined by repeatedly applying a > > function to an initial set. > > Consider recursive sets. > > > Now, I will try to steal axioms from ZFC. > >http://en.wikipedia.org/wiki/Zermelo%E2%80%93Fraenkel_set_theory > > You'll need to modify ZFC for if u is an object (urelements), > then by ZFC, u = empty set because for all x, x not in u > and for all x, x not in empty set. > Thus for all x, (x in u iff x in empty) and u = empty set. Yes. I think i can make some simple modifications to ZFC. > > As an example, assume x is the only urelement. > > X is the urelement variable. Define the function: > > > X_out = ~X_in > > X_out and X_in are undefined. X_out determines if urelement x is in the output set. X_in is true if x is a member of the input set. > > I think I can derive the powerset axiom from these axioms. > > Of course, it's part of the ZFC package. ZFC has a powerset axiom. We can't derive the powerset of all sets exist from the other axioms. > > The powerset of the urelements is a proper class. > > And I don't need a well ordering axiom. > > You get that with ZFC. Again, as an axiom. > > A proper class is ordered by definition. > > It is? Oh yes, sequences are ordered. > Calling a sequence a class is a misnomer. Yes. I should call them proper sequences. > > It would be nice if I could come up with a general method > > for binary operators. Maybe I could define proper classes > > in such a way as to allow n-ary functions. > > > Any ideas are welcome. > > Learn up on propositional and predicated calculus. > Take special note how a formal object language is constructedd > and the distinction between the object language and the metalanguage. > > Clarify your ideas before trying to fomalize them. > > You've got a set O of objects. > You've got functions from subsets of O to subsets of O. > You've got inputs and outputs of some mysterious things Is there a version of ZFC where sets can only have urelements as members? It would still have the axiom of extensionality. It doesn't need foundation. It could have a union axiom. It would need a different schema of specification. The schema would have to differentiate between sets and elements. Russell - 2 many 2 count
From: RussellE on 28 Feb 2010 19:34
Simpler is better. Here is a simple ultrafinite set theory (UST). Primitives: Urelement - an element of a set. A set or proper class can not be an urlelement. Set - a collection of urelements. Proper Class - a collection of sets. 1) Axiom of extensionality: Two sets are equal (are the same set) if they have the same elements. 2) Axiom of singletons: If x is an urelement there exists a set, {x}, with x as its only element. 3) Axiom of union: If A and B are sets there exists a set with the elements of both A and B. 4) Axiom of intersection: If A and B are sets there exists a set with the elements common to both A and B. 5) Axiom of complement: If A is a set there exists a set of urelements not in A. 6) Axiom of well ordering: The urelements are well ordered. 7) Axiom of finiteness: There is a largest and smallest urelement. I probably don't need the axiom of complement. It can be derived from the other axioms. I included the axiom of intersection because I don't really understand how set theories like ZFC define intersection. Maybe intersection can also be derived from the other axioms. I don't need an axiom schema of specification. The singleton axiom and union axiom are enough to create any set. Many people have pointed out the difference between "all sets are finite" which is consistent with ZFC-Inf, and "there exists a largest element" which is not consistent with the other axioms of ZFC. The axiom of finiteness make this an UST. The simplest way to represent natural numbers in this system is to assume each natural number is an urelement. This gives us the finite set of all natural numbers. Many people have told me all known UST's are inconsistent. Obviously, no UST will be consistent with axioms from other set theories. No UST will be consistent with the axiom "if n is a natural number then n+1 is a natural number". My UST doesn't have this axiom. This UST shows some of the properties I think all UST's must have. For example, we can not define addition for every pair of natural numbers. Some natural numbers are just too big to be added together. This is also true for multiplication and exponentiation (powerset). Russell - 2 many 2 count |