Cantor Confusion
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From: stephen on 23 Oct 2006 12:08 Randy Poe <poespam-trap(a)yahoo.com> wrote: > Lester Zick wrote: >> On Mon, 23 Oct 2006 02:00:06 GMT, "Dik T. Winter" <Dik.Winter(a)cwi.nl> >> wrote: >> >> >In article <1161518242.756958.103660(a)h48g2000cwc.googlegroups.com> mueckenh(a)rz.fh-augsburg.de writes: >> >> [. . .] >> >> > Virgils definition, although not wrong, definitions >> >can not be wrong, does not lead to desirable results. >> >> Since when can definitions not be wrong? > A definition is a statement by a person that a certain > symbol will be used by them to stand for some concept. > What is there in such a statement that can be wrong? > Example: I will use the term "gleeb" to refer to an integer > which is divisible by 2. > How can that statement be wrong? How would you > define "wrong" for such a statement? > - Randy But it is horribly confusing when somebody insists on using private definitions for common words, especially when they do not explicitly state their definition. If someone insists on using "even" to mean "numbers divisible by 3" they are not likely to be well received. Stephen Stephen
From: Sebastian Holzmann on 23 Oct 2006 12:12 David Marcus <DavidMarcus(a)alumdotmit.edu> wrote: > Sebastian Holzmann wrote: >> mueckenh(a)rz.fh-augsburg.de <mueckenh(a)rz.fh-augsburg.de> wrote: >> > How would you construct an actually infinite set? Pair, power, union? >> > They all stay in the finite domain if you start with existence of the >> > empty (or any other finite) set. Comprehension or replacement cannot go >> > further. So, how would you like to achieve it? >> >> Let's assume that ZFC - AoI is consistent, otherwise the statement is >> void. Let \phi_n be a sentence in the language of set theory that >> encodes the statement "There exists a set that has at least n elements" >> (you can formulate this one as a homework problem). >> >> Since for every finite subset of the set of \phi_n, there exists a model >> of ZFC - AoI where this subset is true (any model of ZFC - AoI should do >> the trick), by compactness, there exists a model of ZFC - AoI where >> _all_ \phi_n are simultaneously true. ("True" might not be the correct >> English word for it, please do someone correct me in this.) A set that >> has, for any n, at least n elements cannot be finite. Therefore it must >> be infinite. > > Are you sure this argument works in ZFC - AoI? I really don't know, but > I suspect you might have trouble proving the compactness theorem. Also, > the fact that there is a model where the set is infinite doesn't mean > ZFC - AoI knows the set is infinite. ZFC is a FO theory. Compactness for FO has been established a long time ago. I might have enormous problems trying to prove that there is a model of ZFC (with or without AoI) and hopefully I'll never succeed. But apart from that I don't see a problem. If "ZFC - AoI knows the set is infinite" should mean "in this model there is a bijection of this set to a proper subset", that, indeed, might not be true (I haven't thought about this). But certainly this set is not finite, as that would imply that it has at most n elements, for some n in (a naively given) IN.
From: MoeBlee on 23 Oct 2006 13:06 mueckenh(a)rz.fh-augsburg.de wrote: > MoeBlee schrieb: > > > mueckenh(a)rz.fh-augsburg.de wrote: > > > "Continue in this manner" is just what is used to describe how the > > > infinite is realized. > > > > Not in formal axiomatic set theory. If you can't define your "continue > > in this manner" in the language of set theory, then your argument is > > not, contrary to your claim, in set theory, and so, if your argument > > does depend on "continue in this manner" and you won't define it in > > some set theory such as Z set theory, then, of course, we're done as > > far as me taking you seriously in regards to your argument and claim > > that it shows anything about set theory. > > "Continue in this manner" is Cantor's definition, We were talking about Z set theory or one of its variants, were we not? I thought your argument about trees was supposed to occur in some axiomatic set theory such as Z set theory. In such set theories, expressions of the kind "continue in this manner'" must be made rigorous by proving the existence of an infinite function that "plays the role" of "continuning in this manner without end". > Cantor laid the foundations of sets theory. If modern set theory would > not accept his definition, then it should not call itself set theory. It's called Z set theory. If you don't want it to be called 'set theory' just because it has refinements not in Cantor's work, then I guess we could call it 'zet zery' or whatever. But that's missing the point. Upon Skolem's refinement of Zermelo's set theory, we have a formal set theory. This and other formal set theories are what we mean by set theory for about the last eighty years. We are not obliged to remain with Cantor's seminal work. > But this same definition is the *only* one defined in ZFC: There exists > a set such that x u {x} is in it if x is in it, abbreviated according > to Peano: if n is in it, then n+1 is in it. No other than this > "continue in this manner"-definition can be claimed correct in modern > set theory. You're the one who wants to use "continue in this manner" as part of an argument, not me. In set theory, we prove the existence of certain functions if we want to use a 'continue in this manner' type of reasoning. > If you claim that what you call modern set theory has a deviating > definition of infinity, then I am not interested in your theory. This is not a matter of defining 'is infinite'. And if you're not interested in Z set theory, then fine. But then you don't claim that your argument about trees in Z set theory? I take it that your argument about trees is in your informal understanding of pre-formal Cantorian set theory. And I am not interested in your informal understanding of pre-formal Cantorian set theory as if your informal understanding of pre-formal Cantorian set theory has anything to do with formal mathematics. > You > (not I) may translate my arguing into your theory, if you cannot > understand other languages. if your translation leads to my results, > you may be satisfied, and if it does not, then your translation or your > theory is a mess. You were the one who said your argument is in set theory. If by 'set theory', you mean your own informal understanding of pre-formal Cantorian set theory, then I'm not interested. > > You don't know what your're talking about. The convergence is proven. > > The identity of 1 and 0.999... in analysis is proven. And it's proven in set theory (by 'set theory', I mean such formal theories as Z set theory), as are the usual theorems of analysis. > The difference > 10^(-n) disappears for n --> oo (and not earlier!). The identity is not > proven in case of the diagonal number because it cannot be proven, > because of the lacking factor 10^(-n). Even for the digit with index n > --> oo the difference is as important as for the fist digit. That's all irrelevent to the diagonal argument. If you read a textbook such as Suppes's 'Axiomatic Set Theory', then you'd so the steps in the argument. Then you could say what step you object to, and we can explain it to you. But instead you just want to talk as if those steps don't exist. No rational mathematical conversation is possible with you on this subject if you simply will not look at the steps in the argument. > Here is the schisma, the point where mathematics is inconsistent: > Either the digit becomes negligible in both cases or in none of the > cases, i.e., either 1 =/= 0.999... in analysis or the later digits of > the diaonal number become more and more unimportant such that for an > infinite number there are negligible digits and Cantor's argument > breaks down. No, the set theoretic arguments are rigorous. They are not vitiated by your informal posits about digits becoming negligible. If the diagonal argument is flawed, then there is a step that is not authorized by the axioms and first order logic. You have not shown such a flawed step. I really should not continue arguing with someone who simply will not direct himself to the actual mathematics presented but instead wants to talk about his own personal, informal, non-axiomatic, notions about what he thinks the arguments must address. MoeBlee
From: MoeBlee on 23 Oct 2006 13:16 mueckenh(a)rz.fh-augsburg.de wrote: > MoeBlee schrieb: > > Yes, just as I said, the discussion is about the philosophy of > > mathematics and set theory (and, I should add, about informal concerns > > and motivations), but there is not, WITHIN the set theory discussed > > there, a definition of 'actually infinite' and 'potentially infinite'. > > Does the study of formal languages really make incapable of > understanding plain text? No, but, things being equal, a formalized theory is preferable to an unformalized one. > What is written above means: "INFINITY" IN > SET THEORY IS ALWAYS "ACTUAL INFINITY". This could be translated as > completed or finished infinity but usually is not, because that would > deter new students from studying this matter. And most of them never > get a grasp of that fact. Yes, as I said, in an informal description, set theory takes infinite sets as objects that are infinite as opposed to "unended". And, again, as I said, in the formal theory, there are no predicate symbols for 'is actually infinite' or 'is potentially infinite'. MoeBlee
From: MoeBlee on 23 Oct 2006 13:24
Han.deBruijn(a)DTO.TUDelft.NL wrote: > MoeBlee schreef: > > > "built in" provision that equality be interepreted as the actual > > identity relation. > > Okay. Define your "identity relation". If it's primitive, then it's not defined. Usually, in mathematical logic, the symbol for identity is the only primitive predicate symbol. And it cannot be defined, even if we wanted to, in first order logic, which goes along with what I mentioned in my earlier post. Again, I have to say, it is not efficient for me to answer questions going in reverse. That is, if you say, "Why?" or "Define", then you can ask "Why?" of "Define" after every of my answers to a previous question of "Why?" or "Define". And I can keep answering until I finally arrive at the primitives of the theory or even the primitive notions of the informal meta-meta-theory. But it is much more efficient for us to START with those primitives and then demonstrate theorem by theorem. Thus, I suggest you just get a book on the subject and learn it like anyone else has ever had to learn a technical subject. Then, I hope that I could help you with any questions you have along the way. MoBlee |