An Ultrafinite Set Theory
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From: MoeBlee on 1 Mar 2010 16:48 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? 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. > 5) Axiom of complement: If A is a set there exists a set of urelements > not in A. Okay, you're own axiom. > 6) Axiom of well ordering: The urelements are well ordered. Assuming the ordinary definiton of 'well ordered', I guess. > 7) Axiom of finiteness: There is a largest and smallest urelement. WHAT 'large' and 'small'? According to WHAT relation? What is the purpose of your theory? Do you think it makes ordinary set theory otiose? If you think that, then please show how to derive ordinary mathematics for the sciences from your axioms. > 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. > 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. MoeBlee
From: MoeBlee on 1 Mar 2010 16:50 On Mar 1, 2:49 am, RussellE <reaste...(a)gmail.com> wrote: > The set of all natural numbers is the set of all urelements. Forgive me for skimming at this point, but you proved there is such a set or you took it as an axiom that there is such a set? MoeBlee
From: Transfer Principle on 1 Mar 2010 19:01 On Feb 28, 8:12 pm, Virgil <Vir...(a)home.esc> wrote: > In article > <3c6ba639-63b2-4381-b060-c50a35101...(a)z1g2000prc.googlegroups.com>, > 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. When RE first posted this theory last week, I noticed the thread, but I decided not to post until RE explained his theory more. But now I feel like responding to Virgil's objection here: > 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. This reminds me of the discussion from the Sixth Grade Math thread almost a month ago. In sixth grade math -- indeed, in all K-12 and possibly even undergrad level math -- one thinks about sets as containing elements like natural numbers, real numbers, complex numbers, ordered pairs, functions -- but not other sets. A student _never_ thinks about sets as having other sets as members until learning about standard set theories such as ZFC. Only then does one learn that not only can sets have other sets as elements, but _every_ element of _every_ set is another set. But RE makes it clear that he doesn't want sets to have other sets as elements. To see why, we notice what RE has written: > > Here is a simple ultrafinite set theory (UST). So RE states his goal here -- he wants his theory to be an _ultrafinite_ set theory. So among another things, we expect this theory UST to have a _finite_ model, M. Now we go back to the original post: > > 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. Technically speaking, the Axiom of Pairing doesn't allow the creation of arbitrarily large sets (indeed, one can't even create a set with just _three_ elements, let alone arbitrarily many elements). But RE is on the right track, for any model of the theory with at least these ZFC axioms: Axiom of Extensionality Axiom of Empty Set Axiom of Pairing must be infinite. Even weakening Pairing so that it only gives _singleton_ sets produces (as Patricia Shanahan later alludes to in one of her posts) all of the Zermelo naturals, of which there are infinitely many. So the ultrafinitist requires, in some way, to restrict set formation somehow. In a theory with Extensionality and Empty Set, there _must_ exist some set x such that {x} is not a set, lest infinitely many sets will exist. But Virgil insists that a set theory in which no set may be a member of any other set "would be of damn all use in any part of mathematics," despite the fact that in all of the math one learns from kindergarten to lower-level undergrad, there is no mention of sets containing other sets as elements. (In other words, K-12 and lower-level undergrad math sets contain as elements natural numbers, real numbers, etc., and these numbers are treated in these classes like _urelements_ -- which is sort of like what RE is trying to accomplish!) Virgil, like many standard set theorists, is trying to impose ZFC (at least the ZFC idea of sets being elements of other sets) on everyone, whereas RE is trying to show that this idea is not necessary (or in his case, desirable). And never the twain shall meet? I like RE's idea of having sets have only urelements as elements and being elements only of the proper classes. If we allow sets to have other sets as elements, then there must be some way to limit which sets are allowed to have sets as elements in order to accomplish RE's goal of having a finite universe. But is there any way to do so in a theory that's acceptable to Virgil and other anti-"cranks"? (In other words, instead of saying if x is a set then so is {x}, we must require x to satisfy some suitable property in order for {x} to be a set.) I might look back at some of the old zuhair threads, since this sounds like a trick that zuhair often used in his theories.
From: RussellE on 1 Mar 2010 19:13 On Mar 1, 4:14 am, Patricia Shanahan <p...(a)acm.org> wrote: > RussellE wrote: > >> How do you define the term "natural numbers"? > > > 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. > > My natural numbers serve the same purpose as natural numbers > > in these other systems. Natural numbers have an order. > > I have a well ordering axiom. > > In that case, I suggest you pick a different term, to avoid confusing > yourself and others. Nope. I am calling my numbers the "natural numbers". Mostly to annoy Moeblee. > If you had a combination of zero element and successor operation that > satisfied the Peano axioms, you could use the normal definitions of > natural number arithmetic and any theorem about natural numbers that has > been proved from the Peano axioms. That is how ZFC gets its arithmetic, Natural numbers were around long before Peano and set theories came along. Set theories are an attempt to formalize our intuitive notions of "natural number". The ancient Greeks had intuitive notions of natural numbers. They weren't the same as our intuitive notions. They didn't consider zero to be a number. Many ancient Greeks wouldn't have considered one to be a natural number, either. Many did assume natural numbers could grow without bound. But, there were dissenters even 2500 years ago. I have yet to see a convincing refutation of Zeno's paradoxes. Aristotle couldn't come up believable argument. > using the empty set as zero and the set containing only x as the > successor of x. The empty set is not zero in my set theory. Peano's axioms don't actually define the successor function. PA lists a bunch of properties the successor function must have. Typically, the successor of n is given as "n+1". Unfortunately, Peano's axioms don't define addition. You have to define arithmetic to define addition. And you can't define arithmetic without first defining natural numbers. Any system that defines natural numbers using arithematic is circular. > Obviously, you cannot do that given the fact that your > numbers do not satisfy the Peano axioms. Nor are they meant to. Why would you expect the natural numbers in an ultrafinite set theory to satisfy the axioms of a different set theory, like Peano's axioms? > If you go on using the term "natural numbers" you may fool yourself into > assuming that something is already defined or proved because it has been > defined or proved for systems satisfying the Peano axioms. If you want > arithmetic in your system, you will need to go back to the drawing board > to define it, and prove each theorem you want using only your > definitions, axioms, and any theorems you have already proved. > > I suggest "Easterly numbers" as a placeholder. I will call my numbers Easterly numbers if you call your numbers Peano numbers. > Similarly, you could use > "Easterly arithmetic" for the corresponding system of arithmetic > definitions and theorems. I don't have any problem calling my arithmetic "Easterly" arithmetic. We do talk about "Peano" arithmetic. And my arithematic is very different from Peano arithematic. But, my numbers have just as much right to be called "natural numbers" as Peano's numbers. Russell - 2 many 2 count
From: Patricia Shanahan on 1 Mar 2010 19:27
RussellE wrote: > On Mar 1, 4:14 am, Patricia Shanahan <p...(a)acm.org> wrote: >> RussellE wrote: > >>>> How do you define the term "natural numbers"? >>> 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. >>> My natural numbers serve the same purpose as natural numbers >>> in these other systems. Natural numbers have an order. >>> I have a well ordering axiom. >> In that case, I suggest you pick a different term, to avoid confusing >> yourself and others. > > Nope. I am calling my numbers the "natural numbers". > Mostly to annoy Moeblee. .... I was treating clear communication as desirable, and annoying others as a bad thing. Given your priorities, it probably does not matter what you call your numbers. Patricia |