Cantor Confusion
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From: mueckenh on 24 Oct 2006 06:33 Sebastian Holzmann schrieb: > Oh, I think I begin to see your problem here. But before we can speak of > ZFC as a theory, we must first have some sort of "background set theory" > available. And if we do not allow that background theory to "have" > infinite sets (in some naive way), we cannot even formulate Z, because > it consist of infinitely many sentences... So in principle we need the set of all sets in order to talk about every thing including the fact that it does not exist and that we cannot have any infinity unless we have the axiom of infinity. Yes, some very naive (formalized?) opinion. Regards, WM
From: Sebastian Holzmann on 24 Oct 2006 06:34 mueckenh(a)rz.fh-augsburg.de <mueckenh(a)rz.fh-augsburg.de> wrote: > > Sebastian Holzmann schrieb: >> 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). > > Perhaps you do another homework first: Why should such such a sentence > be correct? It is not difficult to encode the sentence: Card(Q) > > Card(R). Is that a proof? Such a sentence should not be "correct". Perhaps you should think of it as a request for a certain property. Can you formulate it in the language of set theory? It is a good exercise when you want to take part in discussions about mathematical logic. Of course a sentence "Card(Q) > Card(R)" is not a proof. Please consult a book on mathematical logic, set theory and model theory. >> 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. > > Further in this case it would be nonsense to introduce the axiom of > infinity. The axiom of infinity is added in ZFC to ensure that an infinite set exists in _any_ model of the theory.
From: Sebastian Holzmann on 24 Oct 2006 06:35 mueckenh(a)rz.fh-augsburg.de <mueckenh(a)rz.fh-augsburg.de> wrote: > > Example: I will use the term "gleeb" to refer to an integer > which is divisible by pi. > > This statement is wrong. And it is nonsense. And don't come up with the > empty set. This statement is not wrong. The statement "There exists a gleeb" would be wrong. Aren't you allowed to speak about non-existing entities?
From: Sebastian Holzmann on 24 Oct 2006 06:37 mueckenh(a)rz.fh-augsburg.de <mueckenh(a)rz.fh-augsburg.de> wrote: > Sebastian Holzmann schrieb: >> Oh, I think I begin to see your problem here. But before we can speak of >> ZFC as a theory, we must first have some sort of "background set theory" >> available. And if we do not allow that background theory to "have" >> infinite sets (in some naive way), we cannot even formulate Z, because >> it consist of infinitely many sentences... > > So in principle we need the set of all sets in order to talk about > every thing including the fact that it does not exist and that we > cannot have any infinity unless we have the axiom of infinity. Yes, > some very naive (formalized?) opinion. We need something to talk about before we can talk about it. If you want, you can call the "background sets" not "sets" but "gleeb" to avoid misunderstandings. And, please, do educate yourself on what you are talking about. Thank you.
From: mueckenh on 24 Oct 2006 06:56
cbrown(a)cbrownsystems.com schrieb: > mueckenh(a)rz.fh-augsburg.de wrote: > > cbrown(a)cbrownsystems.com schrieb: > > > > > > Of course we cannot get to mega by counting. According to set theory we > > > > get to omega by the limit: > > > > > > No, according to most set theories, we "get to" omega by simply > > > assuming it exists. > > > > We get omega by assuming it exists. We get to omega by limits. > > No, I mean we get to omega by assuming omega exists. We do not get to > omega by limits; unless in some bizzare way, you think the phrase "We > get to omega by limits" is logically equivalent to the statement "We > get to omega by assuming that omega exists". Idle discussion. > > > > > > > > lim [n-->oo] {1 + 1/2^2 + 1/3^2 + ... + 1/n^2} = (pi^2)/6. > > > > > > Here, one assumes you mean using the usual metric topology; > > > > No. This formula stems from Leonhard Euler. I works without any > > topology. At that time topology did not exist. > > So, you claim that the above statement follows from no definitions at > all? Instead it follows simply because Euler said it? No because one can prove it by using mathematics. > > That observation shows but an inconsistency of set theory. > > No, it shows that if we are not specific about what we mean by "lim > n->oo", it is possible for two people to have two entirely different > definitions of "lim->oo", which result in two entirely incomatible > results. No. When the direction is given (like here: t varying from 1 to oo), then the limit is unique. > > That's why it is generally wise to state what you mean explicitly by a > statement such as "lim n->oo", in order to prevent confusion. > > > Cantor, in > > his first proof, used the notation a_oo. Later he changed oo to omega. > > And from this, we can draw what conclusions? 1) First he did not distinguish betwenn potential and actual. 2) Omega is what he meant by actual infinity. > > > > > > > For example, if we restrict ourself to the rationals only, then the > > > sequence you mention above regarding pi^2/6 "converges" in some sense, > > > > It is called Cauchy-convergence. > > > > Is that what Euler called it? That is what present-day mathematics calls a sequence which satisfies Cauchy's criterion but has probably not limit. > > > > not representable as numbers. > > When Euler stated that (something) = pi^2/6, do you think he considered > pi to be a number? Do you think he considered that pi could be divided > by the number 6? Do you think that he felt that he was justified in the > conclusion that pi^2/6 = pi*(pi/6)? He certainly considered it a number, because as far as I know he did not know about the finite contents of the universe. > > Conversely, do you think Euler would have considered that beauty could > be divided by the number 6? Do you think he would have felt that it > followed that beauty^2/6 = beauty*(beauty/6)? I don't know about Euler. But Pythagoas did so: Everything is number. > > Yes. That, however, does not yield a decimal representation as needed > > in Cantor's argument. > > I am not claiming that it does or does not have any relevance to > Cantor's diagonal argument. But that is just important for us. > > Al > > representations which are not actually infinite, i.e. have not omega > > digits, are representations of rational numbers. > > > Somewhat inelegantly stated; but sure. Of course, it assumes that omega > exists. Fine. > > > > > > Of course that assumes that you have /some/ axioms in mind: otherwise > > > > > it is simply not a mathematical question whether your statements > > > > > actually follow, one from the other, in your argument. It is instead an > > > > > argument of philosophy. > > > > > > > > True mathematical questions can be answered by physical experiments. > > > > > > I find this philosophical stance quite puzzling, and extremely > > > unsatisfying in its circularity; since generally physical experiments > > > rely on deductions made using some pre-existing mathematical model. > > > > Simple physical experiments like counting III and II together yield the > > foundations of mathematics. Advanced physical experiments may use > > advanced mathematics. > > And which "advanced mathematics" are these physical experiments to rely > on? The ones that give the results we expect in the first place? That mathematics which is compatible with physics. Yes. No set theory. It is a process of discovery where one is based upon the other and the other is based upon the one. > > > > > > > > For matheology you need belief or axioms. > > > > > > In order to get out of bed in the morning you need beliefs and > > > assumptions - you need to believe, for example, that the floor will > > > support your weight, and that you will not instead plunge into a fiery > > > hell. > > > > > > In order to have a reasonable mathematical discussion, we need to > > > /agree/ on what we are talking about. Those agreements are codified by > > > axioms; > > > > That is the present, deplorable opinion of the average mathematicians. > > What is the alternative? That we all blindly stumble about, > misunderstanding each other's statements; because for example what you > call a real number is different than what I call a real number? That is but a definition. > > I think instead you simply want all other mathematicians to naturally > agree with your statements, whether those statements are logical or > not. The statements must be realistic. If I hear that it needs proof that an ever increasing positive function is larger than zero, then I see that your kind og mathemaics is no mathematics. > > > I wouldn't neccessarily call ZFC - AoI "finite set theory". > > > > It is a theory without the actual infinite, a theory without omega. > > No, it is a theory in which one cannot /prove/ that there exists a set > satisfying the properties given for omega. > > It is also a theory in which it one cannot /prove/ that there exists no > set satisfying the properties given for omega. One can prove the nonexistence, because there are no sets possible which cannot be constructed from the empty set. Otherwise you could also assert that the set of all sets was in your theory. Just in oder to avoid that, the axioms were made. > > > One > > may execute any operation. The finite domain will never be left. Which > > of the r |