An Ultrafinite Set Theory
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From: William Elliot on 2 Mar 2010 00:57 On Mon, 1 Mar 2010, Patricia Shanahan wrote: > RussellE wrote: >> I define a natural number to be an urelement. >> The set of all natural numbers is the set of all urelements. >> This isn't the same definition as Peano's axoims or ZFC. > In that case, I suggest you pick a different term, to avoid confusing > yourself and others. > > I suggest "Easterly numbers" as a placeholder. Similarly, you could use > "Easterly arithmetic" for the corresponding system of arithmetic > definitions and theorems. No. It has nothing to do with any Asian cultures, not even Zen. Send the baby back home with it's father who can christen it as natural computer numbers in honor of his family name.
From: RussellE on 2 Mar 2010 01:48 On Mar 1, 1:48 pm, MoeBlee <jazzm...(a)hotmail.com> wrote: > On Feb 28, 6:34 pm, RussellE <reaste...(a)gmail.com> wrote: > > > 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. > > If they're primitives, then what is the part following the dash > symbol? > Are those definitions or axioms or combination above? Are the > primitives 'collection' and 'element'? Or what? OK. The primitives are element and collection. urelement - Only objects defined to be urelements can be elements of a set set - A collection of elements. proper class - a collection of sets. > PLEASE look up how primitives, defintitions, and axioms work! > > > 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. > > Okay, all Z set theory so far. OK > > 5) Axiom of complement: If A is a set there exists a set of urelements > > not in A. > > Okay, you're own axiom. I am not sure I need this axiom. > > 6) Axiom of well ordering: The urelements are well ordered. > > Assuming the ordinary definiton of 'well ordered', I guess. You got me. I don't define well ordering. I can't define well ordering the way ZFC does. My theory doesn't have sets of ordered pairs. I could define proper classes as ordered pairs. Any suggestions for a well ordering axiom would be welcome. > > 7) Axiom of finiteness: There is a largest and smallest urelement. > > WHAT 'large' and 'small'? According to WHAT relation? Again, this is not a great axiom. A better finitenes axiom would be: 7) Axiom of finitenes: The set U = {u_0, u_1, ..., u_k} exists. u is an element of U implies u is an urelement. > What is the purpose of your theory? I want to show it is possible to have a consistent, finite set theory. > Do you think it makes ordinary set > theory otiose? No. Why would you think that? Are anti-foundational set theories otiose? > If you think that, then please show how to derive > ordinary mathematics for the sciences from your axioms. What do you mean by "ordinary mathematics for the sciences"? Can you derive E=MC^2 from ZFC? If you mean Peano arithematic, my theory can't do that. I can derive part of PA. I can show there is a set of "small" natural numbers for which addition is completely defined. > > 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. > > Why don't you just READ how it's done? > > > Maybe intersection > > can also be derived from the other axioms. > > Yes. You can read about it in virtually any textbook on set theory. ZFC doesn't have an axiom of intersection. I assume intersection can be derived from the axiom schema of specification. My theory doesn't have an axiom schema of specification. > > 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"? Russell - Zeno was right. Motion is impossible.
From: Virgil on 2 Mar 2010 02:29 In article <bbc61265-cb81-487d-95f9-21d2187ada9f(a)k36g2000prb.googlegroups.com>, RussellE <reasterly(a)gmail.com> wrote: > > Assuming the ordinary definiton of 'well ordered', I guess. > > You got me. I don't define well ordering. With your "axioms" you can't define any sort of ordering, so we see that your set theory is disorderly.
From: William Elliot on 2 Mar 2010 02:33 On Mon, 1 Mar 2010, RussellE wrote: > On Mar 1, 12:14�am, William Elliot <ma...(a)rdrop.remove.com> wrote: >>>>> 7) Axiom of finiteness: There is a largest and smallest urelement. >> >>>> That doesn't make U finite. The ordinal number omega_0 + 1 >>>> has a smallest and largest element and isn't finite. >> >>> Yes, I know. I am still having problems coming up with >>> an axiom of finiteness. >> >> You could include in the language, k constant symbols u1,.. uk, >> define U = { u1,.. uk } and state that if u is an urelement, >> then u in U. > > This seems to be the simplest solution. > It would be nice to have something more "elegant". > Simple is elegant. >>>>> The singleton axiom and union axiom are enough to create any set. >>>> No. Even assuming U is finite, you can't construct an empty set. >>> I think I can derive that from intersection. >> >> You can't if there's only one urelement. > > Yes. I am not sure this is a problem. > I could add an empty set axiom. > I want to minimize the number of axioms. > > If I remember correctly, if an axiomatic set theory is > consistent, it is still consistent when we negate an axiom. It is not. You can delete an axiom but to negate an axiom you first have to prove that each axiom is independent of the others. > I am not sure how I can deal with an anti-empty set axiom. > Don't. If you don't need an empty set, then don't create one. > I use to think anti-foundational set theories were strange. > Lately, I have been considering anti-union theories and > anti-comprehension theories. There exists two sets with > the same elements that are not equal. > Read about fuzzy set theory. >> It also excludes the positive integers of Piano's axiom. > > Of course. It's not an UST if it doesn't exclude these. > Then don't call them natural numbers. That expression has been taken. Call them something else. >> Your natural numbers are unnatural. �If it doesn't smell >> like a dog nor bark or look like a dog, then it isn't a dog. > > Are you saying my UST is "counter-intuitive"? No. I'm saying don't call it the natural numbers as they aren't natural and natural numbers already means something that can't be your numbers. Call them something else, like natural computer numbers. > I find it amusing that a finite set theory is "counter-intuitive". > What's finite set theory? The theory of finite sets? > We all have pre-conceived intuitions about "natual numbers". > I don't think natural numbers can grow without limit. That's because you're limited by the lack of visualization of computers. > I want my set theory to formalized my pre-conceived notion > of natural numbers. I myself, have consider bounded integers and found the complexity too much to be of worth. >>>>> 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. >> >>>> Of course it doesn't. You haven't even defined incrementation. >> >>> I have a well ordering axiom. What else do I need? >> >> A definition of n + 1 as the successor urelement. > > How does ZFC define the successor function? S(x) = x \/ {x} > Is there a "successor" axiom? It's one of Peano's axioms. > "n+1" is meaningless for certain n in my UST. > Which ones? You could call some large urelement oo (ie infinity) or overflow and instead of a + b and S(u) being undefined for certain urelements, you could say a + b = oo and S(oo) = oo. > Some people think the universe is a computer. > If so, there are numbers too big and too small > for the universe to comprehend. > Their universe excludes the mind. > Position and momentum can't be computed beyond a certain precision. > Commonplace physics. The national debt can't be computer beyond a certain percision also. What happens when you've a finite set of integers and some physicists has need for a larger or more precise number that what you provide? > It could be worse. If physicists come up with a set theory it will be > something like "there is a probability 1=1, a probability 1=2, ..." > Look into fuzzy set theory. ----
From: RussellE on 2 Mar 2010 03:16
On Mar 1, 11:33 pm, William Elliot <ma...(a)rdrop.remove.com> wrote: > On Mon, 1 Mar 2010, RussellE wrote: > > On Mar 1, 12:14 am, William Elliot <ma...(a)rdrop.remove.com> wrote: > >>>>> 7) Axiom of finiteness: There is a largest and smallest urelement. > > >>>> That doesn't make U finite. The ordinal number omega_0 + 1 > >>>> has a smallest and largest element and isn't finite. > > >>> Yes, I know. I am still having problems coming up with > >>> an axiom of finiteness. > > >> You could include in the language, k constant symbols u1,.. uk, > >> define U = { u1,.. uk } and state that if u is an urelement, > >> then u in U. > > > This seems to be the simplest solution. > > It would be nice to have something more "elegant". > > Simple is elegant. > > >>>>> The singleton axiom and union axiom are enough to create any set. > >>>> No. Even assuming U is finite, you can't construct an empty set. > >>> I think I can derive that from intersection. > > >> You can't if there's only one urelement. > > > Yes. I am not sure this is a problem. > > I could add an empty set axiom. > > I want to minimize the number of axioms. > > > If I remember correctly, if an axiomatic set theory is > > consistent, it is still consistent when we negate an axiom. > > It is not. You can delete an axiom but to negate an axiom > you first have to prove that each axiom is independent of > the others. Thanks. > > I am not sure how I can deal with an anti-empty set axiom. > > Don't. If you don't need an empty set, then don't create one. > > > I use to think anti-foundational set theories were strange. > > Lately, I have been considering anti-union theories and > > anti-comprehension theories. There exists two sets with > > the same elements that are not equal. > > Read about fuzzy set theory. > > >> It also excludes the positive integers of Piano's axiom. > > > Of course. It's not an UST if it doesn't exclude these. > > Then don't call them natural numbers. > That expression has been taken. Call them something else. > Call > them something else, like natural computer numbers. OK. We can call them natural computer numbers. > > I want my set theory to formalized my pre-conceived notion > > of natural numbers. > > I myself, have consider bounded integers and > found the complexity too much to be of worth. Most computer engineers agree with you. > >>>>> 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. > > >>>> Of course it doesn't. You haven't even defined incrementation. > > >>> I have a well ordering axiom. What else do I need? > > >> A definition of n + 1 as the successor urelement. > > > How does ZFC define the successor function? > > S(x) = x \/ {x} My theory won't allow the union of an element and a set. I can define successor as a "circuit". Assume we have the set U = {a,b,c,d}. We arbitrarily choose a singleton set like {a}. Define the variable K_in to be true if a set has k as an element. Define K_out to be true if the successor has k as an element. A_out = B_in B_out = C_in C_out = D_in D_out = A_in This is a successor function for the elements of U. It doesn't actually define the "first" element. > > Is there a "successor" axiom? > > It's one of Peano's axioms. > > > "n+1" is meaningless for certain n in my UST. > > Which ones? You could call some large urelement oo (ie infinity) > or overflow Programmers use NaN. Not a number. > and instead of a + b and S(u) being undefined for certain > urelements, you could say a + b = oo and S(oo) = oo. The simplest is to define "0" as a successor. I can also define modulo arithmetic. > > Some people think the universe is a computer. > > If so, there are numbers too big and too small > > for the universe to comprehend. > > Their universe excludes the mind. > > > Position and momentum can't be computed beyond a certain precision. > > Commonplace physics. The national debt can't be computer beyond a > certain percision also. What happens when you've a finite set > of integers and some physicists has need for a larger or more > precise number that what you provide? Add another urelement. > > It could be worse. If physicists come up with a set theory it will be > > something like "there is a probability 1=1, a probability 1=2, ..." > > Look into fuzzy set theory. I never found much use for fuzzy logic. It works well for some things, but, knowing something is 80% true doesn't help in a lot of situations. There are easier ways to calculate odds than fuzzy logic. I like multi-valued logics. A tri-value logic with "true", "false", and "don't know" is interesting. Russell - 2 many 2 count |