Cantor Confusion
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From: David Marcus on 18 Nov 2006 12:42 mueckenh(a)rz.fh-augsburg.de wrote: > David Marcus schrieb: > > > > > Indeed. If people *object* to an axiom, that is philosophy. > > > > > > But if people choose a set of axioms, that is what? > > > > > > > Everyone is welcome to choose their own axioms. > > > > > > That's mathematics? > > > > Of course. > > And if people not decide to use an axiom, that is what? > But if people decide not to use an axiom, that is philosophy? I said, "If people *object* to an axiom, that is philosophy." If you want to see what theorems you can prove using a particular set of axioms, that is mathematics. Whether your results will be interesting or useful to other people is a different question. > > > Would like to do. Please le me know which words are available in your > > > universe of discourse. > > > > I told you several times that the terminology in any modern textbook is > > fine. For some reason you do not like this answer. > > I told you the terminology used in a modern textbook to show that > finished infinity is used there. For some reason you do not like to > understand it. > > "Some mathematicians object to the Axiom of Infinity on the grounds > that a collection of objects produced by an infinite process (such as > N) should not be treated as a finished entity." Are you intentionally being stupid? That quote doesn't use "finished infinity" nor does it give a definition of "finished entity". The quote is simply a philosophical remark in a math textbook. > > What would be something that is "actually infinite"? > > Read Cantor, he can explain it better than me. Hah! So, you can't give an example. Then please stop using the words! > > > > e. An "infinite number" is a number other than the natural numbers. > > > > An "infinite number" would be a number other than a natural number. > > > Are you agreeing or disagreeing? > > I am astonished that you cannot understand simplest sentences. I only seem to have this "problem" with your sentences. I asked whether you agreed with a statement and you replied by offering a different statement. Do you disagree with the original or agree? > You seem > to have difficulties with conditional constructs. Should you ever > intend to study mathematics be prepared that such constructs will > appear quite frequently. How would you know such constructs appear frequently if someone studies mathematics? What mathematics have you studied? Where did you study it? Do you have a doctorate in Mathematics? Who was your thesis advisor? Have you published any articles in refereed mathematics journals? > > > If an actually infinite set of numbers existed, and if neighbouring > > > elements had a fixed distance from each other, then the set must > > > contain an infinite number. > > > Is that a "no" or a "yes"? > > Read again, simplified: If neighbouring elements have a fixed distance, > the answer is yes. > If neighbouring elements have not a fixed distance like the rational > numbers: the answer is no Let's try a simpler question: Does an actually infinite set exist? -- David Marcus
From: David Marcus on 18 Nov 2006 12:50 Lester Zick wrote: > On Thu, 16 Nov 2006 01:35:12 -0500, David Marcus > <DavidMarcus(a)alumdotmit.edu> wrote: > > >Virgil wrote: > >> In article <1163426009.651510.237050(a)h48g2000cwc.googlegroups.com>, > >> mueckenh(a)rz.fh-augsburg.de wrote: > >> > It is difficult to answer this question, because the expression "set" > >> > is occupied in modern mathematics by collections of elements which are > >> > actually there (you don't know what that means, imagine just a set as > >> > you know it). Such infinite sets do not exist. > >> > >> While infinite collections in any physical sense are not possible, why > >> are imaginary infinities, such as sets of numbers must be, unimaginable? > > Why are square circles unimaginable? Depends on what you mean by "unimaginable". > >For that matter, we can always switch from Platonism to formalism and > >declare the question of whether sets really exist to be a philosophical > >question. > > So is the switch from platonism to formalism a philosophical question? Yes. Platonism and formalism are philosophies of mathematics. Regardless of which you prefer (or if you prefer something else), it doesn't change which theorems are provable in which axiom systems. -- David Marcus
From: David Marcus on 18 Nov 2006 12:56 mueckenh(a)rz.fh-augsburg.de wrote: > > David Marcus schrieb: > > > mueckenh(a)rz.fh-augsburg.de wrote: > > > William Hughes schrieb: > > > > > > > The fact that something is true for all sets of the form > > > > {1,2,3,...n} where n is a finite natural number, > > > > does not mean that it is true for N. > > > > > > Oh yes, exactly that it means, because N consists of nothing else than > > > natural numbers. There are no ghosts in mathematics. > > > > How do you know this? Do you have any sort of rationale or proof? It > > seems such a silly thing to say. Consider: > > > > Each of the following sequences has a last element: > > > > 1 > > 1 2 > > 1 2 3 > > 1 2 3 4 > > 1 2 3 4 5 > > ... > > > > This sequence does not have a last element: > > > > 1 2 3 4 5 ... > > > > This last sequence has three dots on the right. None of the other > > sequences do. So, this last sequence is clearly different in some way > > from all the other sequences. > > Yes. We do not know its last element. Perhaps it can change its size. > But mathematics does not obey commands. Neither interpreted as a > command nor as a magic formula the "..." can create infinity. So, let's try a slightly different one: We "know the last element" of each of the following sequences: 1 1 2 1 2 3 1 2 3 4 1 2 3 4 5 .... We don't "know the last element" of this sequence: 1 2 3 4 5 ... Therefore, there are things that are true for 1,...,n for all n in N that are not true for N. But, I believe you said that if something is true for 1,...,n for all n in N then it is true for N. -- David Marcus
From: David Marcus on 18 Nov 2006 13:02 mueckenh(a)rz.fh-augsburg.de wrote: > David Marcus schrieb: > > > "Crank" means someone who makes up pejorative names for people whose > > language they do not understand and thinks that definitions which are > > explained in many books are "private". > > So you are a crank? I'll leave this to others to judge. > 1) You call others cranks, whose language you don't understand. Perhaps. > 2) You do not understand words used by many current text books. Very true. Although, not true of the quotes you gave from a set theory text book. > BTW: Do you really think it is a proof of your superior intellect if > every second word of yours is "sorry don't know"? Yes. A wise man knows what he doesn't know. > > How can you do mathematics without axioms? A major purpose of axioms is > > to avoid ambiguity. > > How has it been done over 4000 years? Since Euclid (and presumably for some time before), much of it has been done using axioms. As time has gone by, more and more of mathematics has become axiomatized. > > I can verify I'm using words with the same meanings as other people by > > asking them what definitions they are using, then seeing if they are the > > same as mine. > > How could you see that if you don't know what the words in their > definitions mean. An interesting question. It doesn't really pose a problem in practice, but I don't think I can give a short answer. > > We use technical terms to refer to precisely defined mathematical > > concepts. > > Why do you say "we" if you talk about mathematicians? What do you mean? -- David Marcus
From: David Marcus on 18 Nov 2006 13:13
Lester Zick wrote: > On Thu, 16 Nov 2006 02:02:49 -0500, David Marcus > <DavidMarcus(a)alumdotmit.edu> wrote: > >Lester Zick wrote: > >> "Provability" of what pray tell? If you're not concerned with proving > >> the truth of what you say in mathematics exactly when are you not > >> discussing philosophy every time you say anything in mathematics? > > > >Do you really not know the mathematical meaning of the word "prove"? If > >so, I (and others) could try to explain it to you. But, if you are just > >being argumentative, we won't bother. > > What is it you think you're proving? Does that mean you don't know the mathematical meaning of the word "prove"? It isn't the same as the English meaning. > >Please give a specific example of something that you think is absurd or > >a contradiction. I don't know what you mean by "containment of sets and > >subsets". > > Well as I recollect Stephen seems to think infinite sets are proper > subsets of themselves. Are you sure that is what Stephen thinks? A set isn't a proper subset of itself. If we have two sets A and B, then we say that A is a proper subset of B if every element in A is also in B and there is some element in B that is not in A If we take B = A, then every element in B is also in A. So, A is not a proper subset of itself. -- David Marcus |