An Ultrafinite Set Theory
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From: RussellE on 24 Feb 2010 20:52 I searched for "ultrafinite set theory" and all I found was a remark by Zermelo: "The 'ultrafinite antinomies of set theory', which the scientific reactionaries and anti-mathematicians eagerly and delightedly call on in their campaign ..." I get the impression Zermelo didn't like ultrafinitists. There were some articles about Essenin-Volpin's set theory as well as finite abelian groups. I couldn't find an actual ultrafinite set thory. So, I decided to come up with my own set theory. I looked at the axioms of ZFC, but many of these axioms are obviously inconsistent with any fixed finite theory. The axiom of pairing states if A and B are sets, there exists a set with A and B as elements. This allows the creation of arbitrarily large sets. Given the sets: {0} and {1} {{0}, {1}} {{0}, {{0}, {1}} etc. Similarly, the powerset axiom assumes sets can grow without limit. Many set theories use FOL which is based on predicate calculus which is based on propositional calculus. This set theory will use propositional calculus. There are four axioms: 1) The exists N urelements. Each urelement is a Boolean variable. 2) A set is a N-tuple which assigns a truth value to each urelement. 3) A function is N well formed Boolean expressions, one for each urelement. 4) A proper class is a sequence of sets defined by an initial set and a function. Let N = 4. It is simple to show there are exactly 2^4 sets. There are 2^(2^4) possible Boolean expressions with four variables. There can be no more than 2^64 possible functions or 2^68 proper classes. This theory is provably finite. We can define simple mathematical objects with this theory. Assuming N=4, the "natural" numbers can be defined as the 16 possbile sets. Like any set theory. we must define a method of representing natural numbers. Assume we define natural numbers as base 2 binary numbers. Now, we can assign a hexadecimal digit to each set. (d0,c0,b0,a0)=0, (d0,c0,b0,a1)=1, ... (d1,c1,b1,a1)=f. A proper class can be represented by a sequence of unique sets and a "first repeat" set. The first repeat set uniquely determines the function that generated the proper class. Consider this sequence of sets: 0,1,2,3,4,5,6,7,8,9,a,b,c,d,e,f This is not a proper class because there is no first repeat set. We haven't defined the "successor" of set f. There are 16 functions that will generate this sequence of sets. We can arbitrarily define the successor of set f. Assume we define 0 to be the successor of f. This is the proper class of natural numbers: 0,1,2,3,4,5,6,7,8,9,a,b,c,d,e,f,0 We now have the proper class of natural numbers and a unique successor function. We can also define addition and multiplication, but there is a problem. Most theories assume any two abitrarily large numbers can added together. This isn't true in this set theory. Addition is a binary operator. Since there are only four variables, we must split the variables between the two operands. There are functions to add 1-bit with 3-bit numbers and functions to add two 2-bit numbers. For example, there is a function to add any number represented by variables A and B to any number represented by variables C and D. To add 2+2: (d1,c0,b1,a0) +> (d0,c1,b0,a0) A similar function can be found for multiplication. This simple theory shows the natural numbers can be represented as a finite proper class, allows the definition of a unique successor function, and has functions capable of adding and multiplying "small" natural numbers. Does anyone see an obvious inconsistency in this theory? One advantage of an ultrafinite theory is that it should be straightfoward to prove the theory is inconsistent. I find it interesting this theory shows there can be many possible successor functions. It is also interesting to think it may be impossible to define addition and multiplication for all pairs of natural numbers. Russell - 2 many 2 count
From: MoeBlee on 25 Feb 2010 12:03 On Feb 24, 7:52 pm, RussellE <reaste...(a)gmail.com> wrote: > Many set theories use FOL which is based on predicate > calculus which is based on propositional calculus. This > set theory will use propositional calculus. > > There are four axioms: > > 1) The exists N urelements. Each urelement is > a Boolean variable. You just used predicate language (not just propositional). Also, you haven't stated your language and primitives. What is 'N'? What are 'urelements'? What is meant by 'N urelements'? What is a 'Boolean variable'? Without definitions, we need to take those as primitives, in which case your axiom may as well be stated: There exist x burblements. Each burblement is a goolean bairable. > 2) A set is a N-tuple which assigns a truth value > to each urelement. What is a 'set'? What is an 'N-tuple'? What is 'assigns'? What is a 'truth-value'. Might as well be stated: A fret is an x-shoople which fursigns a loosh crabble to each burblement. > 3) A function is N well formed Boolean expressions, > one for each urelement. Might as well be stated: A zumption is x krell dormed megressions flum for each burblement. > 4) A proper class is a sequence of sets defined by an > initial set and a function. Might as well be stated: A slopper trass is a peaquince of frets bemined by an orifal fret and a zumption. MoeBlee
From: RussellE on 25 Feb 2010 21:57 On Feb 25, 9:03 am, MoeBlee <jazzm...(a)hotmail.com> wrote: > On Feb 24, 7:52 pm, RussellE <reaste...(a)gmail.com> wrote: > > > Many set theories use FOL which is based on predicate > > calculus which is based on propositional calculus. This > > set theory will use propositional calculus. > > > There are four axioms: > > > 1) The exists N urelements. Each urelement is > > a Boolean variable. > > You just used predicate language (not just propositional). I used a quantifier? I guess I did use the word "each". http://en.wikipedia.org/wiki/Predicate_logic > Also, you haven't stated your language and primitives. > > What is 'N'? N is some number. http://en.wikipedia.org/wiki/Number > What are 'urelements'? An urelement is an object that can be a member of a set, but is not itself a set. http://en.wikipedia.org/wiki/Urelement > What is meant by 'N urelements'? This theory has a fixed number, N, of urelements. > What is a 'Boolean variable'? Its this thing a guy named George Boole invented. http://en.wikipedia.org/wiki/Boolean_algebra_(logic) > Without definitions, we need to take those as primitives, in which > case your axiom may as well be stated: > > There exist x burblements. Each burblement is a goolean bairable. These axioms define four "primitives": urelement, set, function, and proper class. > > 2) A set is a N-tuple which assigns a truth value > > to each urelement. > > What is a 'set'? A set is an ordered list of assignments for the Boolean variables defined by axiom 1. Many set theories don't define set. Set is a primitive. Of course, we can call them "fret" if you want. > A fret is an x-shoople which fursigns a loosh crabble to each > burblement. Russell - 2 many 2 count
From: William Elliot on 26 Feb 2010 02:42 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: >> >>> Many set theories use FOL which is based on predicate >>> calculus which is based on propositional calculus. This >>> set theory will use propositional calculus. >> >>> There are four axioms: >> >>> 1) The exists N urelements. > Huh? Do you mean "There exists"? >>> Each urelement is a Boolean variable. That's predicate calculus. forall x, (x urelement -> x Boolean_variable) > N is some number. You could have a FOL with n constants, u_1,.. u_N which can be considered as urelements (in the metalanguage). > An urelement is an object that can be a member > of a set, but is not itself a set. What's an object? What is "a member of". > This theory has a fixed number, N, of urelements. > >> What is a 'Boolean variable'? > > Its this thing a guy named George Boole invented. > http://en.wikipedia.org/wiki/Boolean_algebra_(logic) > > These axioms define four "primitives": urelement, set, function, and > proper class. > >>> 2) A set is a N-tuple which assigns a truth value >>> to each urelement. >> >> What is a 'set'? > > A set is an ordered list of assignments for > the Boolean variables defined by axiom 1. > What's an ordered list? What are assignments? > Many set theories don't define set. Set is a primitive. You however prefered to defined set by undefined terms. 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. ----
From: MoeBlee on 26 Feb 2010 15:21
On Feb 25, 6:57 pm, RussellE <reaste...(a)gmail.com> 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: > > > > Many set theories use FOL which is based on predicate > > > calculus which is based on propositional calculus. This > > > set theory will use propositional calculus. > > > > There are four axioms: > > > > 1) The exists N urelements. Each urelement is > > > a Boolean variable. > > > You just used predicate language (not just propositional). > > I used a quantifier? I guess I did use the word "each".http://en.wikipedia.org/wiki/Predicate_logic You used 'there exists'. > > Also, you haven't stated your language and primitives. > > > What is 'N'? > > N is some number.http://en.wikipedia.org/wiki/Number > > What are 'urelements'? > > An urelement is an object that can be a member > of a set, but is not itself a set.http://en.wikipedia.org/wiki/Urelement > > > What is meant by 'N urelements'? > > This theory has a fixed number, N, of urelements. > > > What is a 'Boolean variable'? > > Its this thing a guy named George Boole invented.http://en.wikipedia.org/wiki/Boolean_algebra_(logic) Sorry, I thought I could make my point with you without bludgeoning you over the head with the obvious. I'm not asking what are the ordinary mathematical definitions of your terminology, but rather whether you are taking this terminology as primitive or defined per YOUR SYSTEM. > > Without definitions, we need to take those as primitives, in which > > case your axiom may as well be stated: Oh, sorry, I did bludgeon you with it after all, and you still didn't get it. > > There exist x burblements. Each burblement is a goolean bairable. > > These axioms define four "primitives": urelement, set, function, and > proper class. Axioms don't define primitives (except possibly in an informal sense of 'define'). So your four primitives are 'urelement, 'set', 'function', 'proper class'? Any others? > > > 2) A set is a N-tuple which assigns a truth value > > > to each urelement. > > > What is a 'set'? > > A set is an ordered list of assignments for > the Boolean variables defined by axiom 1. Please, what is 'ordered', 'list', 'assignments', 'and defined by axiom' in YOUR SYSTEM? Don't answer that, please, since it's a rhetorical question. It's apparent that you're clueless as to how axiomatic systems work. > Many set theories don't define set. Set is a primitive. > Of course, we can call them "fret" if you want. You are virtually completely uninformed about how 'set' may be defined in certain ordinary set theories (which does not contradict that also we may take the basic intuitive notion of set to be undefined). You have no system or theory you've presented at all. What you have is merely a bunch of mathematical terminology thrown together. If you wish to have an intelligible system, you'd do well to specify: the logical system (are you using classical predicate logic with identity?) the entire list of non-logical primitives the axioms written only with the primitives (or, written with symbols properly defined from primitives) MoeBlee |