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: 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
From: MoeBlee on 3 Mar 2010 11:21 On Mar 2, 9:46 pm, Transfer Principle <lwal...(a)lausd.net> wrote: First, Transfer Prinicple, you blowhard, you lied again about me in your previous posts. And, so far, you've not responded to my latest requests that you stop doing that. Over a few years now, you've made an actual pattern out of lying about my posts. And you continue without blinking. You have my contempt for that. > On Mar 2, 10:03 am, MoeBlee <jazzm...(a)hotmail.com> wrote: > This is the second time that MoeBlee has played around with > rhyming words like "ret" and "urment" in trying to describe > RE's theory. > I don't agree with this notion that standard theories have > a monopoly on these terms. Who said standard theories have such a monopoly? I suggest different terms merely for the sake of avoiding confusion with their more ordinary senses in mathematics. (Of course, though, what I'm really trying to do is enforce a sinister program of mind control.) > If we accept ZFC as the standard > theory, then what about the theory ZF (or to be explicit, > ZF+~AC)? Just to be clear, it is not the case that ZF = ZF+~AC. (Whether you meant that or not.) > 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? Bad argument. Different theories do prove sets to have different characteristics. But still there are some basic agreements so that entirely new terminology is not needed. Anyway, in ordinary mathematics we can find many differences in uses of terms - incompatible from one author to the next. Sometimes fairly serious confusions may result due to such differences. And so, ideally, it would be nice if the community of mathematicians adopted a more or less universal canonical system of terminology. But it's not likely that will happen, especially as each author will adopt his own terminology as best suits his own context, which makes sense in its way too. But that said, still, when someone uses ordinary terminology in a way that is RADICALLY different, then it may make good sense to suggest using different words so that we can discuss the new system without confusing its terminology with that of more ordinary mathematics, especially as we may wish to discuss comparisons between the new system and more ordinary mathematics. Also, for someone such as RussellE who does not understand the axiomatic method, using words like 'ret' and 'rment' emphasizes that his actual mathematical arguments may not make any use of the connotations, suggested associations, and other non-formal baggage associated with the terminology, but rather that the formal reasoning must be purely from the axioms and definitions (i.e., Hilbert's famous 'tables and beer mugs' explanation). (But of course, by suggesting distinct terminology, actually what I'm trying to do is to impose a quite politically incorrect brainwashing so that all alternatives to standard mathematics will, under my boot of fascism, be crushed before they can barely be born.) > 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." RussellE has a "right" to use terminology any way he wants to. In your paranoid fantasy, you think this is some kind of battle between the behemoth establishment of standard mathematics and poor plucky underdogs like RussellE. Sorry, but, for my part, this is not a territorial battle. Rather, my suggestion is just to most easily avoid confusions (in fact, if even made such suggestions in conversations regarding clashes of terminology WITHIN "standard mathematics"). As to your suggestion about analogs, there's no need even for that amount of complication. Rather, since RussellE is proposing quite different senses and since his system is in flux as he revises it (thus we don't know where is terminology will finally end up), it's just simpler to let him use a different set of terminology. It's not as if the standard terminology is privileged in some unfair way. If it would make you happy, I'd just as soon oblige by using 'zet' instead of 'set' and 'zember of' instead of 'member of', etc., in standard mathematics. In a strictly MATHEMATICAL sense it matters not to me. The only problem though is that requires then a lot of REwriting and EXPLAINING (that the terminology is now changed). So, since RussellE is just STARTING to write his system, it makes more sense to ask him to use distinct terminology. I'd think reasonable people would understand such things. But you transcend reason, as in your view from high above you can see that really what I'm trying to do is impose a kind of thought control to exclude alternative mathematics from the git-go. MoeBlee
From: MoeBlee on 3 Mar 2010 11:31
On Mar 2, 11:03 pm, RussellE <reaste...(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". No, FIRST ORDER PA doesn't define 'natural number' AT ALL. (Well, trivially, we could define in PA: x is a natural number <-> x=x but that's not what's at stake here.) > PA says 0, S(0), S(S(0)), ... are natural numbers. No it doesn't. (We're talking about first order PA). > 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. Fine. Now, we're waiting for your definition of 'natural number'. > Let U = {a,b,c,d} What are a, b, c, d? (Below, I say what I think you're driving at.) > Let a < b < c < d > > 0 = {a} > 1 = {a,b} > 2 = {a,b,c} > 3 = {a,b,c,d} I think what you mean to do is to say there is a unique least urelement, then a unique least urelement greater than the unique least urelement, etc. And then you form singletons and unions thereof. Sure, no problem, and then you can one-by-one define 0, 1, 2, 3, etc. (You could even simplify by taking a, b, c, etc., as defined above, to be each 0, 1, 2, etc.) But that is not a definition of the PREDICATE 'is a natural number'. Damn, you don't understand any of this stuff, because you continue to insist on remaining ignorant of what other human beings have done previous to you to work on such problems. MoeBlee |