An Ultrafinite Set Theory
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From: Virgil on 28 Feb 2010 23:12 In article <3c6ba639-63b2-4381-b060-c50a35101f4b(a)z1g2000prc.googlegroups.com>, RussellE <reasterly(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. These forbid a set being a member of a set, which means that such a set theory would be of damn all use in any part of mathematics.
From: William Elliot on 1 Mar 2010 00:11 On Sun, 28 Feb 2010, RussellE wrote: > Simpler is better. Here is a simple ultrafinite set theory (UST). > > Urelement - an element of a set. A set or proper class can not be an > Urelement. > Set - a collection of urelements. > Proper Class - a collection of sets. > You can model that with U \/ P(U) \/ P(P(U)) where U is the set of urelements. > 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. > That doesn't make U finite. The ordinal number omega_0 + 1 has a smallest and largest element and isn't finite. > 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. > 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. No it doesn't. It shows that the natural numbers cannot be represented in UST. > 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. > Some natural numbers are just too big to be added together. Most natural numbers are too big for computers to comprehend. > This is also true for multiplication and exponentiation (powerset). >
From: Patricia Shanahan on 1 Mar 2010 00:47 RussellE wrote: .... > 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. .... How do you define the term "natural numbers"? Patricia
From: RussellE on 1 Mar 2010 01:30 On Feb 28, 9:11 pm, William Elliot <ma...(a)rdrop.remove.com> wrote: > On Sun, 28 Feb 2010, RussellE 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. I could use my bijection proof. The axiom says if A and B are sets and have a bijection there exists a bijection between A-B and B-A. This would eliminate sets having a bijection with a proper subset. But, I would have to define bijection. > > 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. > > 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. > > No it doesn't. It shows that the natural > numbers cannot be represented in UST. Which natural numbers? This certainly isn't the same set of natural numbers defined by ZFC. ZFC defines natural numbers as the intersection of all inductive sets with the empty set as a member. > > 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? > > Some natural numbers are just too big to be added together. > > Most natural numbers are too big for computers to comprehend. Actually, this is true. At least, it is true for the natural numbers defined by ZFC. Russell - 2 many 2 count
From: William Elliot on 1 Mar 2010 03:14
On Sun, 28 Feb 2010, RussellE wrote: > On Feb 28, 9:11�pm, William Elliot <ma...(a)rdrop.remove.com> wrote: >> On Sun, 28 Feb 2010, RussellE 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. > I could use my bijection proof. The axiom says if A and B > are sets and have a bijection there exists a bijection > between A-B and B-A. > > This would eliminate sets having a bijection with a proper subset. > But, I would have to define bijection. > >>> 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. >>> 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. >> >> No it doesn't. It shows that the natural >> numbers cannot be represented in UST. > > Which natural numbers? Most of them. > This certainly isn't the same set of natural numbers > defined by ZFC. ZFC defines natural numbers > as the intersection of all inductive sets with > the empty set as a member. > It also excludes the positive integers of Piano's axiom. 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. >>> 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. >>> Some natural numbers are just too big to be added together. >> >> Most natural numbers are too big for computers to comprehend. > > Actually, this is true. At least, it is true for the natural > numbers defined by ZFC. > Some numbers are too small for a computer to comprehend and others are too precise for a computer to comprehend. In fact it's worse than computers not being able to comprehend most numbers. All they can ever hope to do is to comprehend almost no numbers. ---- |