Cantor Confusion
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From: Randy Poe on 26 Oct 2006 14:05 On Oct 25, 3:37 pm, Lester Zick <dontbot...(a)nowhere.net> wrote: > On 24 Oct 2006 17:58:22 -0700, "Randy Poe" <poespam-t...(a)yahoo.com> > wrote: > > > > > > >Lester Zick wrote: > >> On 23 Oct 2006 21:22:10 -0700, "Randy Poe" <poespam-t...(a)yahoo.com> > >> wrote: > > >> >Lester Zick wrote: > >> >> On 23 Oct 2006 08:48:07 -0700, "Randy Poe" <poespam-t...(a)yahoo.com> > >> >> wrote: > > >> >> >Lester Zick wrote: > >> >> >> On Mon, 23 Oct 2006 02:00:06 GMT, "Dik T. Winter" <Dik.Win...(a)cwi.nl> > >> >> >> wrote: > > >> >> >> >In article <1161518242.756958.103...(a)h48g2000cwc.googlegroups.com> mueck...(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? > > >> >> The concept? > > >> >> >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? > > >> >> Self contradictory predicates defining the concept in the definition. > >> >> Ex: "squircles are square circles" "x is an even, odd" "gleeb is a > >> >> finite integer divisible by 0". > > >> >All of those are perfectly valid definitions. Just because it > >> >doesn't exist doesn't mean the concept can't have a name. > > >> Self contradictions can have names. They're just false definitions > > >There's that weird phrase again. > False definitions or self contradictions? "False definition". > Are self contradictions false? No. Definitions merely create a shorthand for a set of conditions. Anything which meets those conditions can be called by that shorthand name. A definition of such a shorthand is not an assertion of anything, even that the set of such objects is non-empty. So calling it "self-contradictory" is also meaningless. "Self-contradictory" means A & ~A, where A is some assertion. A definition has no assertions. > If so then definitions which contain them must also be false. No, that is incorrect. Consider two sets A and B. Do you have any problem with my defining "intersection of A and B". Is the concept of such an intersection "self-contradictory" or "false"? I hope you can accept that the symbol "A intersect B" can be defined as "the set of objects which are members of both A and B". Any problem with that? Is this definition true, false, or would you agree with me that it's just a shorthand for a longer phrase? Now suppose A = {2,3,4} and B = {5,6,7}. So the intersection is empty. Has the definition of intersection just become false? > >It's like saying that if you're defining an alphabet, some of > >your new squiggles can be "wrong". > "It's kinda like this; it's kinda like that." Don't you ever get tired > of Socratic dialectic? If squiggles can contain self contradiction > they could be false as well. How could a letter in an alphabet do that? > >No, giving it a name makes it neither true nor false. > And giving a definition a name doesn't make it self contradictory > either. Huh? A definition is the process of giving a name. > It's the definition not the name which can be self > contradictory. The definition is the name. > >> As for validity I'd appreciate it if you could explain the difference > >> between "valid" "correct" and "true". > > >"Valid" usually means correctly formed according to some > >rules. It doesn't imply either "true" or "false".And if the rules permit self contradiction? > > >For instance, x < 7 is a valid inequality, but obviously > >depending on the value of x it can be either true or false. > So x=>7 wouldn't be self contradictory of x<7? No, not self-contradictory. "Self" implies one statement. You have added a second statement to mine. I wrote statement P: x < 7 You ask if "not P" (x >=7) is self-contradictory of P. The answer is no. "not P" is the negation of P, but it doesn't make P self-contradictory. P still remains the single statement x < 7. - Randy
From: Virgil on 26 Oct 2006 14:25 In article <1161849556.886415.219440(a)h48g2000cwc.googlegroups.com>, mueckenh(a)rz.fh-augsburg.de wrote: > Tony Orlow schrieb: > > > > > 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? > > > > > > Regards, WM > > > > > > > Inductive subdivision of the unit continuum? We certainly seem to be > > able to specify, or approximate arbitrarily closely, some values with > > infinite strings of digits. It seems obvious that any finite interval in > > the continuum has more than any finite number of points within it. So, > > isn't that an actually infinite set, albeit with linear finite measure > > and bounds? > > > Sorry Tony, you are in error. We cannot approximate sqrt(2) arbitrarily > close. When "Mueckenh" says "we", he is only speaking for himself.
From: Virgil on 26 Oct 2006 14:31 In article <1161849774.467231.152300(a)m7g2000cwm.googlegroups.com>, mueckenh(a)rz.fh-augsburg.de wrote: > I proved that there are not more real numbers than a countable set has > elements. A claim of proof is not a proof. > I proved it in a manner which everybody with moderate > mathematical knowledge can understand. to sure I agree with that because all his prrof to date require acceptance of assumptions that not everybody finds necessary, or desireable. > And, what is important, in a > manner completely independent of your special language. But not independent of "Mueckenh"'s special assumptions. > This means that > any theory stating the uncountablility of the reals is erroneous. Only if one accepts "Mueckenh"'s axiom system, which many do not. There > may be hundreds of different theories and different languages > expressing and "proving" this error, including your pet-theory. I am > not going to investigate them all in detail as I am not trying to learn > anything about creationist sects. I am not going to learn all their > languages, as I am not going to learn how to compute astrological > horoscopes. It is clear and proven *from outside* that their results > are wrong and, therefore, uninteresting for me. It is equally clear that "Mueckenh"'s religious faith in his own infallability is misguided.
From: Virgil on 26 Oct 2006 14:34 In article <1161849844.227002.76280(a)i3g2000cwc.googlegroups.com>, mueckenh(a)rz.fh-augsburg.de wrote: > MoeBlee schrieb: > > > 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'. > > You are so much caught inside your theory that you are unable to look > at it from outside. "Mueckenh" is much caught inside his own religious theory that he is unable to look at it from outside.
From: Virgil on 26 Oct 2006 14:37
In article <1161850954.273653.209490(a)m7g2000cwm.googlegroups.com>, mueckenh(a)rz.fh-augsburg.de wrote: > David Marcus schrieb: > > > > Sorry, I do not know what you state of knowledge is. > > > > I already mentioned several books that you could use. > > It is not me who always faisl to understand. We beg to differ. > > > > > {2,4,6,...} means > > > obviously "all natural numbers". That is the usual notation in > > > mathematics. > > > > What happened to 1, 3, and 5? They aren't natural numbers? > > I forgot "even". Was it that difficult to recognize? We have to go by what you say, not by what you intended to say. > > > > In standard mathematics, a finite set of natural numbers has a largest > > > > element. > > > > > > Please prove that a set of elements consisting of 100 bits has a > > > largest element. Please prove that that predicate defines a set. |