An Ultrafinite Set Theory
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From: Jesse F. Hughes on 2 Mar 2010 21:31 Transfer Principle <lwalke3(a)lausd.net> writes: > In computer arithmetic (IEEE 754, which is of course where RE > got the idea of NaN from), NaN-1 is indeed NaN. Here's a link > which explicitly lists NaN-1 as being NaN: > > http://users.tkk.fi/jhi/infnan.html > > 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. > > Ironically, in another thread when I asked about ultrafinitist > theories, Fred Jeffries suggested that I consider the IEEE 754 > standard as an example of ultrafinitism. RE appears to be > heading in that direction with his use of "NaN." Yes, it's very ironic when two wholly unrelated persons have a difference of opinions. -- "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)
From: Transfer Principle on 2 Mar 2010 22:46 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 don't agree with this notion that standard theories have a monopoly on these terms. If we accept ZFC as the standard theory, then what about the theory ZF (or to be explicit, ZF+~AC)? Now ZF+~AC proves the existence of nonempty sets without choice functions. But according to the standard theory ZFC, every nonempty set has a choice function. So what if I were to claim that therefore, these nonempty objects in ZF+~AC that lack choice functions aren't really sets, so we should call them "rets" or "tets" instead? Similarly, ZFA proves the existence of illfounded sets. This is in contrast with the standard theory ZFC, which proves that every set is wellfounded. So what if I were to claim that therefore, these illfounded objects in ZFA aren't really sets, so we should call them "prets" or "vlets" instead? Just as NFU proves the existence of non-Cantorian sets. This is in contrast with the standard theory ZFC, which proves that every set is Cantorian. So what if I were to claim that therefore, these non-Cantorian sets in NFU aren't really sets, so we should call them "nfets" or "wrets" instead? Finally, bringing this back to ultrafinitism, we know that there exist standard naturals n such that Y-V can't answer yes to the question "Is n a natural number?" The sum or product of two standard naturals is also a standard natural, whereas the set of all naturals n such that Y-V can answer "Is n a natural number?" isn't closed under either addition or multiplication. So what if I were to claim that therefore, Y-V isn't really talking about natural numbers, but something called "yvatural numbers" instead? Of course, this is silly. Adherents of ZF+~AC, ZFA, and NFU aren't going to call their objects "rets" just because they aren't sets in ZFC. The "yvatural numbers" example is even worse, since the set of all "yvatural numbers" is a proper subset of the set of all natural numbers, and so every "yvatural number" literally _is_ a natural number, whether the adherents of PA like it or not. Thus, one shouldn't say that the objects that RE describes in his theory aren't sets or natural numbers, unless one is prepared to do the same with NFU's sets or Y-V's naturals. Of course, at this point the standard anti-"cranks" are likely thinking about a "slippery slope" argument -- if one can call RE's objects "natural numbers," what's to stop another so-called "crank" from calling Q the set of naturals, or R the set of naturals, or {e, i, pi, 42} the set of naturals, or some other crazy set? Where do we draw the line? Here's where we draw the line: we can call an object defined in a nonstandard theory by the same name as an object defined in a standard theory, if the nonstandard object is an _analog_ of the standard object in the new theory, satisfying some basic property of the standard object. An example: RE wishes to define "urelement" in his theory. To me, a basic property of "urelements" is that they contain no elements (and aren't the empty set). Since RE's objects don't contain elements, I believe that RE has the right to keep on calling them "urelements." On the other hand, if RE were to define "urelements" so that they have elements, then I'd agree that RE would be disingenuous in calling them "urelements," so that MoeBlee and the others would be justified in making him change their name to "urments" or "burblements." RE's "sets" can contain urelements. Sets in ZFCU and NFU may contain urelements. So I see no reason for RE to change the name "sets," unless we're going to make adherents of ZFCU and NFU stop calling their objects "sets" too. RE's "classes" can contain sets as elements. Classes in NBG may contain sets. So I see no reason for RE to change the name "classes," unless we're going to make adherents of NBG stop calling their objects "classes" too. Finally, ultrafinitists wish to work with a finite subset of the set of standard naturals. So I see no reason for them to stop calling their objects "natural numbers" simply because there are only finitely many of them in their theories. To repeat, ZFC/PA don't have a monopoly on the names "sets," "natural numbers," etc., no matter how much the standard theorists may desire this.
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 |