An uncountable countable set
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From: stephen on 6 Sep 2006 10:06 Tony Orlow <tony(a)lightlink.com> wrote: > Your whole point here is ludicrous. You are arguing that the naturals > are not a subset of the rationals, which are not a subset of the reals. > While each superset may require a more complex construction than the > subset, all elements of the subset are covered by the superconstruction. > As points on the real line, they are all real numbers. What you're > saying amounts to what I said above, which is indeed absurd. So, no, I > don't understand a thing you're saying, when you say naturals aren't > reals. Sorry. > Tony int main() { int x=1; float y=1; } Does x equal y? Stephen
From: David R Tribble on 6 Sep 2006 13:23 mueckenh wrote: >> All we can attach to it is the number of elements known >> or existing. Disregarding physical constraints ... > David R Tribble schrieb: >> I was not aware that abstract mathematical concepts (e.g., sets) >> had any physical constraints. > mueckenh wrote: > Then you should learn it. It you are unable to physically (i.e. in > written form or in your mind) distinguish all the elements of a set, > then the set does not exist. Consider the set S formed from the elements 0 and y, where y = x+1 for each x in S For every member x in S, I know there is also a member x+1 in S. So I can distinguish every element of the set. I can also choose x to be as large as I wish - for x=10^1000, I know that 10^1000+1 is also in S. So set S must exist. I really don't see where the physical limitation is for visualizing the elements or the set. More to the point, I don't see how the phyics of the real world have any limiting effect on abstract concepts.
From: MoeBlee on 6 Sep 2006 14:53 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. 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. 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. > 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. > 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. > > 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. > > 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. > 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. > > No, because I never claimed otherwise. I set the set is not the same > > object as the definition of the set. > > Is the set the collection of all the members within it? If so, how does > one describe that collection, if not by properties of those members? There you go again. That we describe and/or define a set by a property or even a set of properties does not entail that the set IS that set of properties. Look, for the most basic example possible, the empty set is defined by the property of having no members; but the empty set is not the set whose only member is the property of having no members, since that set has one member, namely the property of having no members. That is 0 does not equal {x | x is the property of having no members}. Even outside of formal set theory, let's define the set R as R is the set of red things. So R = {x | x is red}, which has a members, for example, a lot of fire trucks and other things. But if R were the set of its defining properties, then R = {x | x is the property of being red}, which has, for example, no fire trucks as members, since a fire truck HAS the property of being red but a fire truck is not ITSELF the property of being red. Again, {x | x is red} has a lot of members, each of which is red. But {x | x is the property of being red} has only one member and it is not even red, since a property is an abstract object that does not itself have a color. So it can't be that {x | x is red} = {x | x is the property of being red}. MoeBlee
From: Virgil on 6 Sep 2006 15:08 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.
From: Virgil on 6 Sep 2006 15:33
In article <44fe29b1(a)news2.lightlink.com>, Tony Orlow <tony(a)lightlink.com> wrote: > Virgil wrote: > > In article <44fdc460(a)news2.lightlink.com>, > > Tony Orlow <tony(a)lightlink.com> wrote: > > > >> MoeBlee wrote: > > > >>> Better would be a=b <-> AP(P(a) <-> P(b)). > >> The difference between = and <-> disappears when logical truth values > >> are quantities from 0 through 1, so I don't see that as any better, but > >> equivalent. > > > > "0 through 1"? > > > > Does TO expect to find any truth values strictly between 0 and 1? > > Yes, they are called probabilities. Then the probability of getting a head in tossing a two headed coin is "true"? > > > > >> I am aware that there are difficulties defining what constitutes a valid > >> property in this sense, as Russell's Paradox demonstrates, but I think > >> the kind of statement that produce such issues can be identified. That > >> would be an interesting discussion.... > > > > In ZF, predicate definition of sets is limited to defining subsets of > > sets which are otherwise known to exist, so that Russell's paradox is > > vanquished. > > > > Absent some mechanism of similar effectiveness, TO's system will crash. > > Well, that doesn't leave the door open to, as Ross would put it, "a > universe in ZFC". However, there is a universe, and sets within it, and > a place, somewhere in the theory, for a universe. So, yes, a > multidimensional universe of properties defining objects needs proper > concoction to work to avoid such problems, but must be possible, given > the universe as prime example. :) The physical universe seems to exist despite our attempts to interpret it. And certainly our interpretations of it do not affect its "reality", so a universe of properties is entirely fictional. |