An Ultrafinite Set Theory
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From: RussellE on 2 Mar 2010 13:12 Simpler is better. 1) Axiom of extensionality - two sets are equal if they have the same elements. 2) Axiom of finiteness - the set U = {u0,u1,u2,...,uk) exists. u is an element of U iff u is an urelement. 3) Axiom of well ordering - the urelements have the following order: u0 < u1 < u2 < ... < uk 4) Axiom of singleton - if u is an urelement there exists a set with u as the only element. 5) Axiom of union - if A and B are sets there exists a set with all the elements of both A and B. 6) Axiom of intersection - if A and B are sets there exists a set with all elements common to A and B. 7) Axiom of complement - if A is a set there exists a set of urelements not in A. Define every urelement except the largest to be a natural number. The largest urelement is defined as NaN - not a number. Any arithmetic operation with NaN as an operand equals NaN. For example, NaN+1 = NaN. Arithmetic can be completely defined. Every natural number has a unique successor. The successor of NaN is NaN. Russell - 2 many 2 count
From: MoeBlee on 2 Mar 2010 13:58 On Mar 2, 12:12 pm, RussellE <reaste...(a)gmail.com> wrote: > Simpler is better COHERENT would be nice too. > 1) Axiom of extensionality - two sets are equal if they have the same > elements. I guess you're adopting identity theory also with this. > 2) Axiom of finiteness - the set U = {u0,u1,u2,...,uk) exists. > u is an element of U iff u is an urelement. You skipped what I said about that. > 3) Axiom of well ordering - the urelements have the following order: > u0 < u1 < u2 < ... < uk This is not an axiom in the language of your system. PLEASE, you tell us you're giving us primitives and stuff, but then you just skip right past the matter. > 4) Axiom of singleton - if u is an urelement there exists a set with u > as the only element. Okay, but only assuming that you have identity theory (or first order logic with identity) to work with to explicate the expression "the only". > 5) Axiom of union - if A and B are sets there exists a set with all > the elements of both A and B. Okay. > 6) Axiom of intersection - if A and B are sets there exists a set with > all elements common to A and B. Okay. > 7) Axiom of complement - if A is a set there exists a set of > urelements not in A. Okay. > Define every urelement except the largest to be a natural number. This depends on your Axiom 3, which is still just floating mathematical verbiage. > The largest urelement is defined as NaN - not a number. > Any arithmetic operation with NaN as an operand equals NaN. Freefloating mathematical verbiage. > For example, NaN+1 = NaN. Freefloating mathematical verbiage. > Arithmetic can be completely defined. Whatever that means. > Every natural number has a unique successor. > The successor of NaN is NaN. To quote the scribe, "Huh?" MoeBlee
From: RussellE on 2 Mar 2010 14:28 On Mar 2, 10:03 am, MoeBlee <jazzm...(a)hotmail.com> wrote: > On Mar 2, 12:48 am, RussellE <reaste...(a)gmail.com> wrote: > > > What is the purpose of your theory? > > > I want to show it is possible to have a consistent, finite set theory. > > What do you MEAN by a 'finite theory'? You seem to have your own > definition of 'finite'. The title of the thread is "An Ultrafinite Set Theory". I want a theory with a largest natural number. This is more restrictive than just a finite theory. > (1) Do you mean (given the ordinary definition of 'finite'), a theory > that has a theorem that there exist only finite sets? Then that is > easy: The theory, in the language of set theory, whose sole axiom is > "There exist only finite sets" is a consistent theory. > > (2) Do you mean (given the ordinary definition of 'finite'), a theory > that has only models with finite domains? Again, that is easy. The > theory, in the langauge of set theory, whose sole axiom is "Axy(x=y)" > is consistent and has models only with finite domains. > > So what? I have often been told there are no "consistent" ultrafinite set theories (UST). I suspect people don't mean we can always derive a contradiction from the axioms of a UST. I think they mean UST's aren't consistent with their idea of arithematic. > > > Do you think it makes ordinary set > > > theory otiose? > > > No. Why would you think that? > > I'm just asking? > > > What do you mean by "ordinary mathematics for the sciences"? > > I have no PRECISE definition. It is left open to reasonable > interpretation. But a typical minimum would be some calculus for > predicting the motions of objects, for caclulating probabilites and > for statistics. The philosophy of science says truth can only be determined by measurement and repeatable experiments. In some ways, science is the antithesis of mathematics which says truth can be derived by pure reasoning. Could a UST predict the orbit of Mercury? I think one could. Assume a set of 2^500 natural numbers. Predicting the orbit of Mercury to within experimental error should not be much harder than writnig a video game for a really large computer monitor with a finite number of pixels. > > Can you derive E=MC^2 from ZFC? > > As I understand, that is a statement of physics, not just of > mathematics. I'm referring to the mathematics uses for physics, not > the physics itself. The first experimental evidence for the theory of relativity was a deviation in the orbit of Mercury as calulated using Newton's laws. How do you draw a line between math and science? > > If you mean Peano arithematic, > > No, I don't. First order PA by itself, is, as far as I know, not > adequate for a theory for the sciences. What does "adequate" mean? For the most part, science doesn't care about theory. > > > > I don't need an axiom schema of specification. > > > > The singleton axiom and union axiom are enough to create any set. > > > > Depends on what you think are "enough" sets. > > > Do we really need a continuum number of sets to do "science"? > > I don't know. But that doesn't obviate the sense of my question as to > what you consider enough sets. Why do you need sets at all? I don't need sets. I really want natural numbers. > Why do you > need a set theory? To formalize intuitive notions of "natural numbers". > Why do you need any theory at all? So I can experiment on them. Before I started playing with this UST, I might not have considered the idea that arithmetic may not be defined for every pair of natural numbers. What if some numbers are so big they can't be added to another number. How would this change our intutions about natural numbers. Can I come up with a test for this in the real world? Russell - The universe is one dimensional
From: RussellE on 2 Mar 2010 15:21 On Mar 2, 10:58 am, MoeBlee <jazzm...(a)hotmail.com> wrote: > On Mar 2, 12:12 pm, RussellE <reaste...(a)gmail.com> wrote: > > > Simpler is better > > COHERENT would be nice too. > > > 1) Axiom of extensionality - two sets are equal if they have the same > > elements. > > I guess you're adopting identity theory also with this. > > > 2) Axiom of finiteness - the set U = {u0,u1,u2,...,uk) exists. > > u is an element of U iff u is an urelement. > You skipped what I said about that. You said I needed to define finite ("r-nite"). This may be difficult to do with one axiom considering the contortions we have to go through to define finite in ZFC. The simplest method does seem to be to define a set of urelements (you called it a ret). The axiom of infinity defines a set. I could have an axiom schema for urelements. Each urelement is a new axiom. Then, I could require the theory to have a "finite" number of axioms. > > 3) Axiom of well ordering - the urelements have the following order: > > u0 < u1 < u2 < ... < uk > > This is not an axiom in the language of your system. Do I need something like a 1-place predicate? > PLEASE, you tell us you're giving us primitives and stuff, but then > you just skip right past the matter. > > > 4) Axiom of singleton - if u is an urelement there exists a set with u > > as the only element. > > Okay, but only assuming that you have identity theory (or first order > logic with identity) to work with to explicate the expression "the > only". > > > 5) Axiom of union - if A and B are sets there exists a set with all > > the elements of both A and B. > > Okay. > > > 6) Axiom of intersection - if A and B are sets there exists a set with > > all elements common to A and B. > > Okay. > > > 7) Axiom of complement - if A is a set there exists a set of > > urelements not in A. > > Okay. > > > Define every urelement except the largest to be a natural number. > > This depends on your Axiom 3, which is still just floating > mathematical verbiage. > > > The largest urelement is defined as NaN - not a number. > > Any arithmetic operation with NaN as an operand equals NaN. > > Freefloating mathematical verbiage. I want to have two types of urelements. One type is a natural number. The other type is "not a number". I think this makes sense for an UST. The largest "natural number" is not a natural number. > > For example, NaN+1 = NaN. > > Freefloating mathematical verbiage. Arithmetic is defined for all urelements, not just natural numbers. Assume addition is defined as a two place predicate. The sum of NaN and any other urelement is NaN. > > Arithmetic can be completely defined. > > Whatever that means. It means I can define 2-place predicates that correspond to addition, multiplication, etc. Russell - 2 many 2 count
From: MoeBlee on 2 Mar 2010 15:54
On Mar 2, 1:28 pm, RussellE <reaste...(a)gmail.com> wrote: > On Mar 2, 10:03 am, MoeBlee <jazzm...(a)hotmail.com> wrote: > > > On Mar 2, 12:48 am, RussellE <reaste...(a)gmail.com> wrote: > > > > What is the purpose of your theory? > > > > I want to show it is possible to have a consistent, finite set theory.. > > > What do you MEAN by a 'finite theory'? You seem to have your own > > definition of 'finite'. > > The title of the thread is "An Ultrafinite Set Theory". > I want a theory with a largest natural number. > This is more restrictive than just a finite theory. It's no problem to have a theory that has a theorem "there exists a largest natural number". So what? What ELSE does the theory prove? What interesting and/or useful mathematics can you prove with your theory? > > (1) Do you mean (given the ordinary definition of 'finite'), a theory > > that has a theorem that there exist only finite sets? Then that is > > easy: The theory, in the language of set theory, whose sole axiom is > > "There exist only finite sets" is a consistent theory. > > > (2) Do you mean (given the ordinary definition of 'finite'), a theory > > that has only models with finite domains? Again, that is easy. The > > theory, in the langauge of set theory, whose sole axiom is "Axy(x=y)" > > is consistent and has models only with finite domains. > > > So what? > > I have often been told there are no "consistent" ultrafinite set > theories (UST). Who told you that? Here's a consistent "ultrafinite set theory": Axy x=y. So what? > I suspect people don't mean we can always derive a contradiction from > the axioms of a UST. I think they mean UST's aren't consistent with > their idea of arithematic. Yeah, okay. So it's not consistent with their idea of arithmetic. What of it? > > > What do you mean by "ordinary mathematics for the sciences"? > > > I have no PRECISE definition. It is left open to reasonable > > interpretation. But a typical minimum would be some calculus for > > predicting the motions of objects, for caclulating probabilites and > > for statistics. > > The philosophy of science says truth can only be determined > by measurement and repeatable experiments. > In some ways, science is the antithesis of mathematics which > says truth can be derived by pure reasoning. For sake of argument, let's say you're right. So what? > Could a UST predict the orbit of Mercury? > I think one could. Assume a set of 2^500 natural numbers. > Predicting the orbit of Mercury to within experimental error > should not be much harder than writnig a video game for a > really large computer monitor with a finite number of pixels. For sake of argument, okay. It doesn't entail that infinite sets don't provide a useful and easy to use caclulus. > > > Can you derive E=MC^2 from ZFC? > > > As I understand, that is a statement of physics, not just of > > mathematics. I'm referring to the mathematics uses for physics, not > > the physics itself. > > The first experimental evidence for the theory of relativity > was a deviation in the orbit of Mercury as calulated using > Newton's laws. How do you draw a line between math and science? I don't have a comprehensive answer. Personally, I would say that (pure, or theoretical) mathematics concerns deductions about relations among purely abstract objects. That is, objects whose properties are entirely general and abstract. > > > If you mean Peano arithematic, > > > No, I don't. First order PA by itself, is, as far as I know, not > > adequate for a theory for the sciences. > > What does "adequate" mean? For the most part, > science doesn't care about theory. 'adequate' in its ordinary English meaning. Science uses a certain amount of mathematics. By adequate I mean such mathematics as needed for ordinary calculus, finite combinatorics, probablity, statistics. Then also for whatever other mathematics is needed for such constructs as relativity and quantum mechanics. > > > > > I don't need an axiom schema of specification. > > > > > The singleton axiom and union axiom are enough to create any set. > > > > > Depends on what you think are "enough" sets. > > > > Do we really need a continuum number of sets to do "science"? > > > I don't know. But that doesn't obviate the sense of my question as to > > what you consider enough sets. Why do you need sets at all? > > I don't need sets. I really want natural numbers. You contradict below: > > Why do you > > need a set theory? > > To formalize intuitive notions of "natural numbers". So do you want 'set' to be a concept in your theory or not? If all you want are finitely many counting numbers, then maybe something like this: First order logic with identity. Then use the language of identity theory to (theoretically) write out the formula that says there exist exactly Y number of objects, where Y is the number 2^500 or whatever you want, but we don't mention "2^500" in the actual formula as instead we just write the HUGE formula of identity theory that ensures all and every model of the theory has exactly 2^500 elements. This is your sole non-logical axiom. Every and only models that have exactly 2^500 elements are models of this theory. Done. Now so what? > > Why do you need any theory at all? > > So I can experiment on them. Before I started playing > with this UST, I might not have considered the idea > that arithmetic may not be defined for every pair of > natural numbers. What if some numbers are so big > they can't be added to another number. How would > this change our intutions about natural numbers. > Can I come up with a test for this in the real world? Fine. But then first please find out how formal theories actually work. MoeBlee |