An uncountable countable set
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From: stephen on 7 Sep 2006 10:57 Mike Kelly <mk4284(a)bris.ac.uk> wrote: > stephen(a)nomail.com wrote: >> imaginatorium(a)despammed.com wrote: >> >> > Tony Orlow wrote: >> > given that all bit positions are finite, and that >> >> therefore no place in that string can achieve an infinite value, and >> >> that any such number has predecessor and successor. The cute thing is >> >> that the successor to ...1111 is 0, and that ...1111 is essentially -1. :) >> >> > Hmm. So -1 is "essentially" a lot larger than 1, for example, whereas >> > add 5 to both sides and you get the other thing. Well, that's faintly >> > amusing ice pose. >> >> Given that Tony apparently thinks that if you keep adding 1's, you >> eventually get back to zero, I cannot understand why he was >> so upset by the balls and vase problem. >> >> Stephen > I was actually thinking of that problem in the car a week ago. Surely > even if there are "infinite integers" to go on the balls, those balls > get removed infinitesimally before Noon? The nth ball gets removed at > the nth iteration, at time -(1/2) ^ n. Surely Tony would argue that > this is valid in the infinite case. Oh well. Tony's objection was that because you are always adding balls, you can never get 0. But apparently his own math says that if you keep adding 1, you eventually get, ..111111111111, and if you add 1 to that, you get 0. So in Tony's world, if you just add 1 ball at a time, and never remove any balls, you can end up with 0 balls at noon. Stephen
From: imaginatorium on 7 Sep 2006 11:18 stephen(a)nomail.com wrote: > Mike Kelly <mk4284(a)bris.ac.uk> wrote: > > > stephen(a)nomail.com wrote: > >> imaginatorium(a)despammed.com wrote: > >> > >> > Tony Orlow wrote: > >> > given that all bit positions are finite, and that > >> >> therefore no place in that string can achieve an infinite value, and > >> >> that any such number has predecessor and successor. The cute thing is > >> >> that the successor to ...1111 is 0, and that ...1111 is essentially -1. :) > >> > >> > Hmm. So -1 is "essentially" a lot larger than 1, for example, whereas > >> > add 5 to both sides and you get the other thing. Well, that's faintly > >> > amusing ice pose. > >> > >> Given that Tony apparently thinks that if you keep adding 1's, you > >> eventually get back to zero, I cannot understand why he was > >> so upset by the balls and vase problem. > >> > >> Stephen > > > I was actually thinking of that problem in the car a week ago. Surely > > even if there are "infinite integers" to go on the balls, those balls > > get removed infinitesimally before Noon? The nth ball gets removed at > > the nth iteration, at time -(1/2) ^ n. Surely Tony would argue that > > this is valid in the infinite case. Oh well. > > Tony's objection was that because you are always adding balls, > you can never get 0. But apparently his own math says that > if you keep adding 1, you eventually get, ..111111111111, and > if you add 1 to that, you get 0. So in Tony's world, if > you just add 1 ball at a time, and never remove any balls, > you can end up with 0 balls at noon. Oh, but come on! Surely you can see that numbers and billiard balls may behave differently. Particularly when you get close to infinity. Brian Chandler http://imaginatorium.org
From: Tony Orlow on 7 Sep 2006 12:03 MoeBlee wrote: > Tony Orlow wrote: >> My point was that that axiom only applied to sets, but I went further to >> say that even urelements can be considered to be equal to the sets of >> all 1--place predicates which apply to them, > > Then they are not urelements. An urelement has no members. If the set > of 1-place predicates that hold of an object has members, then it can't > be an urelement, since an urelement has no members. Yes, I was saying that there may be no such thing as a urelement, if every object is considered to be the set of all of its attributes. However, what we have here are two different kinds of sets: sets of objects, and sets of attributes. If a urobject, which has no object elements, represents a set of attributes, then this indicates a need to distinguish between sets of objects and sets of attributes. I suppose this is the basis for type theory, which I'll try to read up on. But Virgil says it's an anachronism. We'll see. > > So your passage above is yet another example of what people mean by > your blathering. You just blather whatever mathematical terminology > comes into your head, even if it makes no sense. Blah blah blah. > > One more time: an urelement has no members, so an urelement cannot > equal a set that has members, and the set of 1-place predicates that > hold for an object is a set with members, namely the 1-place predicates > that hold for the object. Focus, Tony, focus. If you reread what I wrote before, and focus, you'll see I was drawing a distinction between two perspectives on the matter. It might be best to call the set of attributes of an object its "nature" or "character", rather than a set, and reserve "set" for objects. > >> No, it doesn't. The axiom which DOES mention properties states that, >> given a set S and a property P, we can form a subset of S which includes >> all elements x of S for which P(x) is true. Given the universe as a set >> (possible, depending on how you define the universe), this would imply >> exactly what I'm saying, that every property defines a set. > > You are conflating the fact that properties define sets with your own > notion that the sets ARE the set of properties that define the set. > That is exactly the point of the video image analogy I made, which you > dismissed. Just because a property (actually, in Z set theories, since > Skolem, for precision, we use formulas rather than properties, but that > is a technical point here) defines a set does not entail that the set > IS the set of properties that define it. When you say you use formulas, that is for sets of real numbers (including rational and naturals). Now, please distinguish the curve defined by a formula and the set of points in that curve. I am cnflating the curve and the formula. They are one. > >> Yes, at times you've been very generous, and I appreciate your >> contributions. You just seem rather cranky lately. > > Because it is frustrating talking with you, as I prefer not to consider > you stupid, but your willful ignorance (not just in refusing to a read > a single book on the subject but also in ignoring so many crucial > points made in posts made to you) and arrogance cause you to say so > many stupid things. My arrogance is driven by conviction and success in formulating an alternative. Your frustration comes from trying to prove me wrong in that. > >>> I never said any such thing. I said that it remains to be seen how you >>> could have an object BE the set of its defining properties. >> See above. I asked you if the idea was reasonable, you said no, and went >> on a diatribe about my ignorance. But, okay, whatever you say. > > The thust of what I said is that the idea is inconsistent with Z set > theories, so you need to devise some other system if you want to > implement the idea; also, I mentioned that even in some system of your > own, I suspect (suspect, not proven) that your idea entails a vicious > circle that will cause a contradiction within its own system. I haven't disagreed with that, but I do believe there is a way out of the vicious cycles you mention, even when trying to incorporate the fundamental notion of discernibility by property into set theory. So far, there seems to be some perceptions of contradiction, but as far as I can tell, no actual contradictions between any ideas I've put forth. > >>> The set is an object in some domain of discourse. The definition is a >>> syntactical object, which is a member of the theory but almost never >>> (if ever) a member itself of the domain of discourse. >> So, there are members of the theory that transcend the domain of >> discourse? > > No, that's nonsense what you just said, and nothing I said deserves > such a nonsensical reply. You said a syntactical object may be a member of the theory, but not of the domain of discourse. It's right above. > >> Somehow, I thought the theory and its objects WAS the domain >> of discourse, > > You thought that because you have no idea about any of this. Why do you > make up such thoughts out of the blue in your mind when you could just > open a book and get the correct formulations? > > A theory is a set of sentences closed under entailment. > > A domain of discourse (a universe) for a language is the value of the > mapping from certain symbols of the language with the universal > quantifier as the argument. If the mapping is a model of the theory (in > that language), then said value is a domain of discourse (universe) for > the theory. More roughly put, a domain of discourse (universe) for a > theory is a set, that has along with it relations and possibly > functions on that set, such that the sentences of the theory are true > for that set and its relations and functions. Please read a book on > mathematical logic so that you'll know what these things are instead of > being stuck with your own completely incorrect guess as to what they > are. If the set is a member of the domain of discourse, rather than a member of the theory itself, why is the definition of the set not a member of the domain of discourse? Isn't the definition of a set generally some kind of mapping from a standard set such as the naturals, that is, some kind of relation? It's not just a string, but usually a function used as this relation or mapping. > >>> No, because I neve
From: Tony Orlow on 7 Sep 2006 12:05 Virgil wrote: > In article <44fe28c8(a)news2.lightlink.com>, > Tony Orlow <tony(a)lightlink.com> wrote: > >>> In ZF, et all, sets are not defined, they are among the undefined terms >>> which every mathematical theory must have. >>> >>> Whenever one attempts to define everything, one ends, at some point, >>> with circularity. >> Can I quote you on that? > > It is a truth that appears in almost every text on logic or on the > foundations of mathematicsthat there must be some terms left undefined. > > So use it as you will, but do not attribute it to me, as it is not an > idea originating with me. I certainly didn't mean to imply that any idea had originated with you, Virgil, so don't get bent out of shape. :)
From: Tony Orlow on 7 Sep 2006 12:20
David R Tribble wrote: > Tony Orlow wrote: >> So, what is it you think I DON'T get about inductive proof, sets, and >> recursion? > > For one thing, you seem to think that successively adding 1 to > the naturals, starting with 0, leads to an infinite value. That shows > that you don't understand induction over the naturals. > > Another is the total absence of stated axioms for your > alleged system of "infinite arithmetic". > > Yet another is your confusion between "countable", "uncountable", > and "infinite". > > >> As far as I can tell, very few dare to question the >> predefined rules as set forth, > > Actually quite a few people question them. It would be fair > to say that most of us questioned the theorems of set theory > when we first encountered them. > > However, most folks eventually understand the logical consistency > of set theory. It's only the minority who continue to misunderstand > it. > > >> but I have yet to see any valid >> counterexample to my rules regarding inductive proof in the infinite >> case. > > Since you still have not provided a meaningful definition of your > "infinite case", that should not be surprising. > > I suggested that you show a proof in your system (using your > new axioms) that the number of reals in [0,2] is twice the number > of reals in [0,1], and that both intervals are dense in the reals. > That should be easy for you, right? It's trivial given the axiom that there are Big'un reals in every unit interval. Denseness is defined for any real interval givent he axiom of internal infinity: x<z -> Ey x<y<z > > Assuming that you could prove such a thing, you then have > to realize that such a system is incompatible with standard > arithmetic and set theory. You see why, don't you? > Yes, of course it's incompatible. That's because set theory ignores measure. > >> Where an equality between expressions is proven for all n, it is >> valid for infinite n. > > Again, this is something you have to prove instead of just claiming > as true. A good start is to define exactly what you mean by > "infinite", since you reject the accepted definitions everyone else > uses. > If all infinite values are greater than finite values, and I prove P(n) for all n>m (finite m), then I have proven it fall all infinite n, since they are greater than m. > >> Where an inequality is proven for all n greater >> than some finite m, and the difference upon which the inequality is >> based does not have a limit of 0 as n->oo, it holds also in the infinite >> case. > > What does "in the infinite case" mean? > When n is an actually infinite value, such as the number of points in a unit interval (Big'un). > >> The staircase in the limit vs. the diagonal line was a valiant >> effort at a counterexample, but obviously flawed in its reliance on the >> limitations of point set topology, and the only other attempt was based >> on a half hidden limit of 0 for the nested inequality as n->oo, based on >> a function discontinuity, so was dismissible without any fanfare. > > You're starting to sound like Ross. Slow down, you're using too > many loaded words in too small a space. > Read more slowly then. My grammar is correct. If you remembered those discussions, you'd know what I was saying. > >> So, if >> you have what you think is a valid counterexample which shows how little >> I know about inductively defined structures and how they suddenly change >> character when n=oo, well, bring it on. :) > > It would help if you could define what it means for n=oo in the > first place. But be warned that since that has no meaning in > standard arithmetic, whatever system you come up with in which > it does have some kind of meaning is a system that cannot be > compatible with standard arithmetic. > Yeah, I know. I've known that for 25 years. That doesn't stop me from considering alternatives to transfinitology any more than being surrounded by McDonald's' and Pizza Huts prevents me from cooking real food. Tony |