An Ultrafinite Set Theory
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From: MoeBlee on 2 Mar 2010 16:11 On Mar 2, 2:21 pm, RussellE <reaste...(a)gmail.com> wrote: > On Mar 2, 10:58 am, MoeBlee <jazzm...(a)hotmail.com> wrote: > > > 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. Nope, you don't know what you're talking about. We can define 'is finite' just as we do in ZFC but without using ANY set theory axioms at all. I ALREADY told you: definition of predicates does not depend on axioms. > 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. The axiom of infinity states that there exists a set (we don't even have to say it is a 'set', as we could just say 'object' if we wanted to) having certain properties. Then we prove that there is exactly one set that has those properties and is a subset of any other set having those properties. However, as to definining 'is finite', we don't need the axiom of infinity or any other set theory axioms. > 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. Whatever you mean by a schema for urelements. Also, if your theory is definite, you'd need to say not just that there are finitely many such axioms but give the exact fintite number. > > > 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? Maybe another 2-place predicate (or you could use the 2-place predicate you have, viz. 'e', and stipulate by axiom some kind of ordering with it). But you still need to straighten out other stuff for this to be coherent. > I want to have two types of urelements. One type is a natural number. > The other type is "not a number". Then 'r-tural number' might be a candidate for being a primitive predicate. Then an axiom that says there exist r-tural numbers and an axiom that says there exist objects that are not r-tural numbers. > I think this makes sense for an UST. > The largest "natural number" is not a natural number. Once you get 'largest' straightened out, then the above could be another axiom. > > > For example, NaN+1 = NaN. > > > Freefloating mathematical verbiage. > > Arithmetic is defined for all urelements, I don't know what you mean by 'arithemetic is defined for' in this context. You haven't GIVEN any arithmetical operations. > not just natural numbers. > Assume addition is defined as a two place predicate. You probably mean a 2-place operation. > The sum of NaN and any other urelement is NaN. So 'a' is the operation symbol? What is 'N' again? > > > Arithmetic can be completely defined. > > > Whatever that means. > > It means I can define 2-place predicates that correspond > to addition, multiplication, etc. You probably mean 2-place operations. If you can define them, then go ahead and do it, but you're not allowed to use ANYTHING except your primitives and previously defined predicates and operations. Also, to show that your definitions of operations are "good" you need to prove the appropriate existence and uniqueness theorems. MoeBlee
From: Virgil on 2 Mar 2010 16:25 In article <be3d057d-1a58-4a17-89a7-e312ea28f083(a)b5g2000prd.googlegroups.com>, RussellE <reasterly(a)gmail.com> wrote: > 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. What are 0,1,2,...,n ? You cannot use 0,1,2,...,n unless your world already contains them, i.e., contains numerals for certain natural numbers. > > 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. If NaN - 1 = NaN (i..e., the predecessor of NaN is Nan), your arithmetic is going to be bloody useless. > > Arithmetic can be completely defined. > Every natural number has a unique successor. > The successor of NaN is NaN. > > > Russell > - 2 many 2 count
From: RussellE on 2 Mar 2010 17:10 On Mar 2, 12:54 pm, MoeBlee <jazzm...(a)hotmail.com> wrote: > On Mar 2, 1:28 pm, RussellE <reaste...(a)gmail.com> wrote: > > > > > 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. OK. But, Newton's calculus doesn't correctly predict the orbit of Mercury. > > > > 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. Which properties? I see math more as a science. Natural numbers have properties we can observe. Is geometry "entirely general and abstract"? I get different predictions for the orbit of Mercury depending on which geometry I choose. > > > > 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 think science needs mathematics as good as our ability to measure things. If we can't measure it, science says it doesn't exist. > > > > > > 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? Can I use this theory to better predict the orbit of Mercury? I think such a system could have some interesting properties. Can we prove such a system is consistent? What are the differences between arithmetic on "small" natural numbers and "large" natural numbers? Russell - 2 many 2 count
From: RussellE on 2 Mar 2010 17:21 On Mar 2, 1:11 pm, MoeBlee <jazzm...(a)hotmail.com> wrote: > On Mar 2, 2:21 pm, RussellE <reaste...(a)gmail.com> wrote: > > > On Mar 2, 10:58 am, MoeBlee <jazzm...(a)hotmail.com> wrote: > > > > 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. > > Nope, you don't know what you're talking about. We can define 'is > finite' just as we do in ZFC but without using ANY set theory axioms > at all. I ALREADY told you: definition of predicates does not depend > on axioms. > > > 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. > > The axiom of infinity states that there exists a set (we don't even > have to say it is a 'set', as we could just say 'object' if we wanted > to) having certain properties. Then we prove that there is exactly one > set that has those properties and is a subset of any other set having > those properties. > > However, as to definining 'is finite', we don't need the axiom of > infinity or any other set theory axioms. > > > 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. > > Whatever you mean by a schema for urelements. > > Also, if your theory is definite, you'd need to say not just that > there are finitely many such axioms but give the exact fintite number. > > > > > 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? > > Maybe another 2-place predicate (or you could use the 2-place > predicate you have, viz. 'e', and stipulate by axiom some kind of > ordering with it). > > But you still need to straighten out other stuff for this to be > coherent. > > > I want to have two types of urelements. One type is a natural number. > > The other type is "not a number". > > Then 'r-tural number' might be a candidate for being a primitive > predicate. Then an axiom that says there exist r-tural numbers and an > axiom that says there exist objects that are not r-tural numbers. > > > I think this makes sense for an UST. > > The largest "natural number" is not a natural number. > > Once you get 'largest' straightened out, then the above could be > another axiom. > > > > > For example, NaN+1 = NaN. > > > > Freefloating mathematical verbiage. > > > Arithmetic is defined for all urelements, > > I don't know what you mean by 'arithemetic is defined for' in this > context. You haven't GIVEN any arithmetical operations. > > > not just natural numbers. > > Assume addition is defined as a two place predicate. > > You probably mean a 2-place operation. > > The sum of NaN and any other urelement is NaN. > > So 'a' is the operation symbol? What is 'N' again? NaN is an abbreviation used by many computer languages to mean "not a number". Mathematical functions return a number or "NaN" (which usually means an error). I should have a single symbol for it. I usually call the largest natural number "z". Russell - 2 many 2 count
From: MoeBlee on 2 Mar 2010 17:39
On Mar 2, 4:10 pm, RussellE <reaste...(a)gmail.com> wrote: > On Mar 2, 12:54 pm, MoeBlee <jazzm...(a)hotmail.com> wrote: > > For sake of argument, okay. It doesn't entail that infinite sets don't > > provide a useful and easy to use caclulus. > > OK. But, Newton's calculus doesn't correctly predict the orbit of > Mercury. I never claimed otherwise. Maybe you're confusing 'sciences for the mathematics' with the scientific theories that use mathematics? > > 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. > > Which properties? Those that are defined in a purely abstract way. > I see math more as a science. However you see mathematics, it doesn't vitiate anything I've said here. > Natural numbers have properties we can observe. This subject has been beaten to death by other discussants. In a nutshell: I am aware of experiences that impress me as sensory, and then I am able to form constructs of objects that I categorize as 'physical objects', such as tables and chairs. However, I see no physical object that is, e.g. the natural number 1. > Is geometry "entirely general and abstract"? Sure, pure abstract geometry is. It is not required to reference physical objects or even pictures merely to study the abstract relations of geometry. > I get different predictions for the orbit of Mercury > depending on which geometry I choose. So? > I think science needs mathematics as good as our ability > to measure things. If we can't measure it, science says > it doesn't exist. You're welcome to have your view of things. But nothing you've said vitiates anything I've said. > > 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? > > Can I use this theory to better predict the orbit of Mercury? I quite doubt it. That's my point. > I think such a system could have some interesting properties. > Can we prove such a system is consistent? The one I mentioned. Of course it's consistent. > What are the differences between arithmetic on "small" > natural numbers and "large" natural numbers? Whatever 'small' and 'large' mean to you in this context, I don't know why you're asking ME about this. MoeBlee |