An Ultrafinite Set Theory
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From: Transfer Principle on 3 Mar 2010 00:11 On Mar 2, 12:54 pm, MoeBlee <jazzm...(a)hotmail.com> wrote: > On Mar 2, 1:28 pm, RussellE <reaste...(a)gmail.com> wrote: > > I have often been told there are no "consistent" ultrafinite set > > theories (UST). > Who told you that? > Here's a consistent "ultrafinite set theory": > Axy x=y. Ah yes, _that_ theory. The theory which spawned a long debate between the standard theorists and Nam Nguyen over whether "Axy (x+y=0)" is provable in the theory. > > > No, I don't. First order PA by itself, is, as far as I know, not > > > adequate for a theory for the sciences. Of course, if MoeBlee doesn't even consider PA to be adequate for the sciences, what chance does RE (or anyone else) have in convincing him that a _weaker_ theory, such as an ultrafinitist theory, is adequate for science? > If all you want are finitely many counting numbers, then maybe > something like this: > First order logic with identity. > Then use the language of identity theory to (theoretically) write out > the formula that says there exist exactly Y number of objects, where Y > is the number 2^500 or whatever you want, but we don't mention "2^500" > in the actual formula as instead we just write the HUGE formula of > identity theory that ensures all and every model of the theory has > exactly 2^500 elements. This is your sole non-logical axiom. Every and > only models that have exactly 2^500 elements are models of this > theory. > Done. Perhaps the following is another way to grasp what RE is thinking, in a way that also sheds light on what many so-called "cranks" are thinking when they try to come up with new theories: Let S be a set of natural numbers (and here we're returning to the standard definition of "natural number"). Then the question is, can we find a theory T such that (ZFC proves that) for every natural number n, n is in S if and only if there exists a set M such that the cardinality of M is n, and M is (a carrier set of) a model of T? Suppose S={1}. Then we need a theory T such that every model of T has cardinality one. Obviously, the theory that MoeBlee mentions, namely the one with lone axiom "Axy (x=y)", qualifies. But suppose S is the set of even natural numbers. So we seek a theory T such that there exists a model of T of each even cardinality, but of no odd cardinality. We may try the following: Assuming FOL with identity: Language: Let Z be a one-place function symbol. Axioms: 1. Ax ~(Zx = x) 2. Ax (ZZx = x) Then every finite model of theory must have an even number of objects since the objects appear in pairs, x and Zx. Now suppose S is the set of odd natural numbers. Then we may replace axiom 1 above with the following axiom: 1'. E!x (Zx = x) But RE is probably thinking about letting S be a more challenging set, such as the set of all powers of two: S = {1, 2, 4, 8, ...} So we need a theory T such that for every power of two there's a model of T with that cardinality, and every finite model of T has a power of two as its cardinality. I have yet to think of such a theory. Such a theory may be the sort of theory that RE has in mind. The objects of this theory may correspond to RE's notion of urelements and sets. (Also, we need to find a way to make all the models _finite_.) (Come to think of it, we may actually want S to be the set of natural numbers of the form n+2^n, not merely 2^n, so that we can have n urelements and 2^n sets.) S = {1, 3, 6, 11, 20, 37, 70, ...} This may be helpful for other "cranks," not just RE. Some "cranks" don't believe in uncountable sets. Of course, with theories with infinite models, we have to be worried about Lowenheim-Skolem. So if we say, "Let T be a theory such that every model of T is countable," we can't say (first-order) PA since by L-S, there exists an uncountable model of PA. (I'm not sure about second-order PA here.) But it's been noted in previous threads that these models of theories that exist via L-S don't necessarily map "e" to anything resembling membership. So we might add a requirement that "e" must be mapped to membership. So one might ask the question: Given a set U, find a theory T such that (ZFC proves that) U is a model of T mapping "e" to membership. The sets U=V_omega and U=V_(omega+omega) have well-known solutions to this problem (viz., ZF-Infinity and ZF-Replacement Schema, respectively.) But what about V_(omega+1)? Notice that the all of the elements of V_(omega+1) are countable, so it might work. (It's possible that NBG-Infinity and Randall Holmes's PST work for V_(omega+1) and V_(omega+2) respectively, but this may be undesirable to the Cantor "cranks" because they want the countably infinite objects to be _sets_, not classes as in these theories.) To me, questions of this type (given a set S, find a theory T such that S is a model of T, with certain requirements in order to avoid trivial answers) are interesting, though they might not be interesting to the standard theorists.
From: Marshall on 3 Mar 2010 00:17 On Mar 2, 7:31 pm, "Jesse F. Hughes" <je...(a)phiwumbda.org> wrote: > "It is my opinion that since neither Spight nor Hughes can see or > understand their moral trespass [namely, quoting AP in .sigs], that > their degrees from whatever university they earned their degree should > be annulled." -- Archimedes Plutonium (12/1/09) <zoidberg>What an honor!</zoidberg> I've made it into Jesse Hughes' quotes file! I can now take my rightful place, albeit as a junior member, alongside such giants as Archimedes Plutonium and James Harris, as one of the Greats of Usenet. Let me now quote our lovable mascot John Jones, extraneous comma and all: "I am truly, a giant among mortals." Marshall
From: Virgil on 3 Mar 2010 03:47 In article <9ebc97a3-3dc7-4583-96cc-af6408139ba0(a)u15g2000prd.googlegroups.com>, RussellE <reasterly(a)gmail.com> wrote: > I looked at how Peano arithmetic is formalized: > http://en.wikipedia.org/wiki/Peano_axioms > > I can define arithmetic the same way by > changing my definition of natural number. > PA defines natural numbers in "unary". > PA says 0, S(0), S(S(0)), ... are natural numbers. > We just count the calls to successor. > > I can define natural numbers as sets just like PA. > With this definition, I don't assume the urlements > are natural numbers. I only assume they are ordered. You cannot even define order until you have a theory sufficient to allow definition of relations . > > Define 0 as the singleton set containing the smallest urelement. How do you know there is a "smallest" urelement? > > Define successor of set X to be the union of X and > the singleton set of the smallest urelement not in X. > > Let U = {a,b,c,d} > Let a < b < c < d > > 0 = {a} > 1 = {a,b} > 2 = {a,b,c} > 3 = {a,b,c,d} > > The set U is closed under my successor function. > The successor of {a,b,c,d} is {a,b,c,d} U {}. By what definition of function? What is the domain of that "successor" function, and what is its range, and is it a bijective function or not?
From: Jesse F. Hughes on 3 Mar 2010 06:44 Transfer Principle <lwalke3(a)lausd.net> writes: > On Mar 2, 12:54 pm, MoeBlee <jazzm...(a)hotmail.com> wrote: >> On Mar 2, 1:28 pm, RussellE <reaste...(a)gmail.com> wrote: >> > I have often been told there are no "consistent" ultrafinite set >> > theories (UST). >> Who told you that? >> Here's a consistent "ultrafinite set theory": >> Axy x=y. > > Ah yes, _that_ theory. The theory which spawned a long > debate between the standard theorists and Nam Nguyen over > whether "Axy (x+y=0)" is provable in the theory. You focus on the most inconsequential coincidences as if they were central. -- Jesse F. Hughes Quincy (age 3 1/2, looking at a picture): Are these people Canadians? Me: Uh, no, they're Australian Aborigines. Quincy: Do they fight Canadians?
From: Jesse F. Hughes on 3 Mar 2010 06:47
Transfer Principle <lwalke3(a)lausd.net> writes: > Let S be a set of natural numbers (and here we're returning to the > standard definition of "natural number"). Then the question is, > can we find a theory T such that (ZFC proves that) for every > natural number n, n is in S if and only if there exists a set M > such that the cardinality of M is n, and M is (a carrier set of) a > model of T? > > Suppose S={1}. Then we need a theory T such that every model of T > has cardinality one. Obviously, the theory that MoeBlee mentions, > namely the one with lone axiom "Axy (x=y)", qualifies. > > But suppose S is the set of even natural numbers. [...] As far as I can tell, Russell is only interested in taking finite initial segments of N as his urelements and has not yet mentioned a connection between cardinality and those urelements. Maybe I've missed something, but what you're focusing on here doesn't look at all like Russell's work. -- "There's lots of things in this old world to take a poor boy down. If you leave them be, you can save yourself some pain. You don't have to live in fear, but you best have some respect, For rattlesnakes, painted ladies and cocaine." -- Bob Childers |