An Ultrafinite Set Theory
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From: Patricia Shanahan on 2 Mar 2010 23:34 Transfer Principle wrote: > On Mar 2, 10:03 am, MoeBlee <jazzm...(a)hotmail.com> wrote: >> On Mar 2, 12:48 am, RussellE <reaste...(a)gmail.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. >> 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') > > This is the second time that MoeBlee has played around with > rhyming words like "ret" and "urment" in trying to describe > RE's theory. (The first time was back in the second post of > this thread.) Also the standard theorists Patricia Shanahan > William Eliot have also criticized RE for trying to steal > terminology from the standard theories (such as ZFC and PA) > and use them in his own theory. I'm not a theorist at all, standard or otherwise. I'm a practical programmer and computer architect. I do think it would reduce confusion if the term "natural numbers" were used for a structure that does conform to the Peano Postulates, and other terms were used for structures that don't. There are examples of errors in algorithms that may have been due to thinking "integer", and applying a formula that works for integers, when the reality is a bounded range number type. Patricia
From: RussellE on 3 Mar 2010 00:03 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. Define 0 as the singleton set containing the 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 {}. Russell - 2 many 2 count
From: Marshall on 3 Mar 2010 00:09 On Mar 2, 7:22 pm, Transfer Principle <lwal...(a)lausd.net> wrote: > > Therefore, by Virgil's standards, IEEE 754 arithmetic must be > "bloody useless," even though Virgil probably uses software > that adheres to IEEE 754 every time he turns on his computer. IEEE 754 is of course quite useful for doing calculations. It's not something that qualifies as a model of anything the least bit applicable to mathematical proof, which means that *in context* Virgil's claim is entirely correct, even if perhaps a bit dramatically phrased. Algebraic properties so basic and fundamental as associativity of addition and multiplication do not hold in IEEE 754. Marshall
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 |