An Ultrafinite Set Theory
in [Theory]
|
Prev: Now Win a laptop computer from (EZ laptop) free of charge
Next: Prizes worth 104,000 INR at stake for RAD. Certificates to all participants.
From: William Elliot on 2 Mar 2010 06:37 On Tue, 2 Mar 2010, RussellE wrote: >>> On Mar 1, 12:14�am, William Elliot <ma...(a)rdrop.remove.com> wrote: >>>> 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. > > I can define successor as a "circuit". What's 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. Crazy. What you doing? Your syntax doesn't make any sense. For each urelement you're labeling a variable? No. Doesn't even make basic programming sense. You are defining a proposition over urelements. I(k) when k in U and some set A with k in A. > Define K_out to be true if the successor has k as an element. > That defies any sense whatever. Are you sure you wrote what you intended to write? > A_out = B_in > B_out = C_in > C_out = D_in > D_out = A_in > Meaningless other than it seems to be a circuit, a cycle. Sure you could give U a circular order like the integers modulus |U| > 1. > This is a successor function for the elements of U. > It doesn't actually define the "first" element. Since U is well ordered, isn't the successor of k the successor of k in the well ordering? >>> 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. > Make up your mind. >>> Position and momentum can't be computed beyond a certain precision. >> >> Commonplace physics. The national debt can't be computer beyond a >> certain precision 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. > How would the others using your hippy numbers know that it was done and what it was. How would you convert all previous work to the new augmented yippy numbers. Most of all, who decides when and if more urelements need to be included, how many are needed and what new urelement will be? I can see it now: Natural Computer Numbers 2011.3.10. >>> 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. If you knew that the US dollar was about to collapsed with an 80% assurance you'd keep 20% of your money in US dollars and 80% in gold or in food, land and guns, depending upon how much of an optimist you are. > 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. > You'd also like the tribe that counts one, two, many. Most language count one, many; singular and plural. Sanskrit counts, one, two, many; singular, dual and plural. s + s = d; s+d = d+d = s+p = d+p = p+p = p S(s) = d; S(d) = p; S(p) = p
From: Jesse F. Hughes on 2 Mar 2010 09:38 RussellE <reasterly(a)gmail.com> writes: > 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. It seems to me that you also have predicates for Urelement, Set and Class. > > urelement - Only objects defined to be urelements can be elements of a > set > set - A collection of elements. > proper class - a collection of sets. So, it seems to me that you want the following axioms (Ax)(Urelement(x) -> (Ay)~y e x) (Ax)(Set(x) -> (Ay)(y e x -> Urelement(y)) (Ax)(Class(x) -> (Ay)(y e x -> Set(y)) (Ax)(Set(x) -> (~Urelement(x) & ~Class(x)) (Ax)(Class(x) -> ~Urelement(x)) >> > 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. Just add another relation < and axioms (Ax)(Ay)(x < y -> (Urelement(x) & Urelement(y))) and the usual axioms specifying that < is a well-ordering. >> > 7) Axiom of finiteness: There is a largest and smallest urelement. Simply add another axiom that <^op (i.e., >) is also a well-ordering. Every well-ordering has a least element. You'll have to check, of course, that you *can* write down the axioms for well-ordering. I don't see any issues, but I haven't thought it through. -- "At some point in the future history of humanity, AP will eclipse even Jesus." -- Archimedes Plutonium, 10/21/07 "I wrote those lines because I am not a megalomania [sic] but rather very humble and down to earth." -- Archimedes Plutonium, 10/22/07
From: MoeBlee on 2 Mar 2010 13:03 On Mar 2, 12:48 am, RussellE <reaste...(a)gmail.com> wrote: > On Mar 1, 1:48 pm, MoeBlee <jazzm...(a)hotmail.com> wrote: > 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. That is quite confused. The following is the best I can make sense of what you might be driving at. (By the way, the reason yours is confused is not because of the words 'set', etc., that you use but rather your table of the words is mixed up. However, I've proposed new words since there is no sense in confusing with the ordinary use of words in set theory): Primitives: 1-place predicate - 'x is a ret' 1-place predicate - 'x is an urment' 2-place predicate - 'xey' ('x is an element of y') Axiom: x is a ret -> Ay(yex -> y is an urment) Definition: x is a rass <-> Ay(yex -> y is a ret) (So you don't need 'collection'.) Maybe you still want to revise what I did, but at least mine is clear. > > > 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. Sure you can. Definitions of these kinds of predicates (in a language with 'e') don't depend on axioms. > 7) Axiom of finitenes: The set U = {u_0, u_1, ..., u_k} exists. Nope. What are '0', '1'? What does "..." mean? You're basically question begging by using "..." in this way to mean something like "finite" when what you need to do is DEFINE 'finite' (I'd call it 'r-nite') in your language. > > What is the purpose of your theory? > > I want to show it is possible to have a consistent, finite set theory. What do you MEAN by a 'finite theory'? You seem to have your own definition of 'finite'. (1) Do you mean (given the ordinary definition of 'finite'), a theory that has a theorem that there exist only finite sets? Then that is easy: The theory, in the language of set theory, whose sole axiom is "There exist only finite sets" is a consistent theory. (2) Do you mean (given the ordinary definition of 'finite'), a theory that has only models with finite domains? Again, that is easy. The theory, in the langauge of set theory, whose sole axiom is "Axy(x=y)" is consistent and has models only with finite domains. So what? > > Do you think it makes ordinary set > > theory otiose? > > No. Why would you think that? I'm just asking? > 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. > 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. > 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. > > > 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? Why do you need a set theory? Why do you need any theory at all? What is your purpose in having a theory (other than merely to have a consistent one such that (1) or (2) from above) or in having a foundational theory? MoeBlee
From: RussellE on 2 Mar 2010 13:12 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. 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. Arithmetic can be completely defined. Every natural number has a unique successor. The successor of NaN is NaN. Russell - 2 many 2 count
From: MoeBlee on 2 Mar 2010 13:58
On Mar 2, 12:12 pm, RussellE <reaste...(a)gmail.com> wrote: > Simpler is better COHERENT would be nice too. > 1) Axiom of extensionality - two sets are equal if they have the same > elements. I guess you're adopting identity theory also with this. > 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. > 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. PLEASE, you tell us you're giving us primitives and stuff, but then you just skip right past the matter. > 4) Axiom of singleton - if u is an urelement there exists a set with u > as the only element. Okay, but only assuming that you have identity theory (or first order logic with identity) to work with to explicate the expression "the only". > 5) Axiom of union - if A and B are sets there exists a set with all > the elements of both A and B. Okay. > 6) Axiom of intersection - if A and B are sets there exists a set with > all elements common to A and B. Okay. > 7) Axiom of complement - if A is a set there exists a set of > urelements not in A. Okay. > Define every urelement except the largest to be a natural number. This depends on your Axiom 3, which is still just floating mathematical verbiage. > The largest urelement is defined as NaN - not a number. > Any arithmetic operation with NaN as an operand equals NaN. Freefloating mathematical verbiage. > For example, NaN+1 = NaN. Freefloating mathematical verbiage. > Arithmetic can be completely defined. Whatever that means. > Every natural number has a unique successor. > The successor of NaN is NaN. To quote the scribe, "Huh?" MoeBlee |