Galileo's Paradox
in [Math]
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From: Lester Zick on 5 Dec 2006 16:51 On 4 Dec 2006 17:57:54 -0800, cbrown(a)cbrownsystems.com wrote: > >Lester Zick wrote: >> On 4 Dec 2006 11:29:33 -0800, cbrown(a)cbrownsystems.com wrote: >> >> > >> >Lester Zick wrote: >> >> On 3 Dec 2006 11:22:56 -0800, cbrown(a)cbrownsystems.com wrote: >> >> >> >> >Tony Orlow wrote: >> >> >> cbrown(a)cbrownsystems.com wrote: >> >> >> > Tony Orlow wrote: >> >> >> > >> >> >> > <snip> >> >> >> > >> >> >> >> Anyway, ala Leibniz, each object IS the set of properties which it >> >> >> >> possesses, so any two objects with the exact same set of properties are >> >> >> >> the same object. >> >> >> > >> >> >> > But this begs the question: what do we mean, exactly, by a SET of >> >> >> > properties? What exactly are we trying to say when we say "This set of >> >> >> > properties is the same as this other set of properties"? >> >> >> > >> >> >> > Cheers - Chas >> >> >> > >> >> >> >> >> >> Well, what we really mean is that there is a set of universal >> >> >> properties, each of which is a set of values, and that each object is >> >> >> defined by a set of values, one from each set of property values, such >> >> >> that any two distinct objects differ in at least one property value. Was >> >> >> that specific enough? >> >> > >> >> >Well, you used the term "set" four times in your above definition of >> >> >what we mean by a "set". That's why I said "this begs the question, >> >> >what do we mean, exactly, by a set of properties?". >> >> > >> >> >There's something that we intuitively seem to think of as a "set"; but >> >> >unless such a thing is carefully defined, we end up with the >> >> >contradictions of naive set theory: >> >> >> >> I think the more basic question is whether non naive sophisticated set >> >> "theory" represents all of mathematics... >> > >> >Of course it doesn't; "all of mathematics" is an extremely broad range >> >of discourse. >> >> So when mathematikers conflate mathematical ignorance with set >> "theory" ignorance they are being extremely overly broad? >> > >Not all of Italian cooking involves sauteeing things in olive oil; >however it is somewhat bizzare for someone to claim to be a >knowledgeable Italian cook without knowing how to sautee things in >olive oil. Then you undoubtedly qualify as an Italian cook and set mathematiker in your spare time. I'm just trying to ascertain the basis for your disdain of Italian cooks who don't choose to practice what you preach. >> >> and whether it has any pre >> >> emptive prerogative to the definition of terms such as "cardinality" >> >> etc. in mathematics generally? >> >> >> > >> >The reason why "cardinality" has the general definition you refer to is >> >because it is generally useful to have a term with that definition. We >> >could call it anything; but "cardinality" is the word used for this >> >term, for mostly historical reasons. >> >> Oh I don't much care what we call it. Mathematikers are the ones who >> insist definitions are only abbreviations. And I agree empirical >> utility is a cachet of modern mathematical significance. I just don't >> agree that empirical definitions are particularly true. >> > >So be it then. Each to their own. Fine. I'll practice mathematics and you can practice set mathematics. ~v~~
From: Lester Zick on 5 Dec 2006 16:54 On Tue, 5 Dec 2006 02:18:54 -0500, David Marcus <DavidMarcus(a)alumdotmit.edu> wrote: >Lester Zick wrote: >> On Fri, 01 Dec 2006 11:39:57 -0500, Tony Orlow <tony(a)lightlink.com> >> wrote: >> >> >Lester Zick wrote: >> >> On Thu, 30 Nov 2006 12:08:03 +0100, Eckard Blumschein >> >> <blumschein(a)et.uni-magdeburg.de> wrote: >> >> >> >>> On 11/29/2006 8:13 PM, Bob Kolker wrote: >> >>>> Tony Orlow wrote: >> >>>>> Uncountable simply means requiring infinite strings to index the >> >>>>> elements of the set. That doesn't mean the set is not linearly ordered, >> >>>>> or that there exist any such strings which do not have a successor. >> >>>> Uncountable means infinite but not of the same cardinality as the >> >>>> integers. For example the set of real numbers. It is an infinite set, >> >>>> but it cannot be put into one to one correspondence with the set of >> >>>> integers. >> >>> Uncountable means: Counting is impossible. This property obviously >> >>> belongs to fictitious elements of continuum. There is simply too much of >> >>> them. So counting is not feasible. As long as one looks at a finite, >> >>> just potentially infinite heap of single integers, one has to do with >> >>> individuals. The set of all integers is something else. It is a fiction. >> >>> It is to be thought constituted of an uncountable amount of >> >>> non-elementary elements. Well this looks nonsensical. There is indeed a >> >>> selfcontradiction within the notion of an infinite set. >> >>> Non-elementary means not having a distinct numerical address. Element >> >>> means "exactly defined by an impossible task". >> >> >> >> You make the same mistake of assuming "infinite" means "larger than" >> >> when it only means numerically undefined. Infinites are neither large >> >> nor small; they're only undefined. Consequently there are no numerical >> >> relations or operations possible between them and finites. The reason >> >> counting is not possible is not because infinites are huge or because >> >> they form a continuum but because there is no numeric metric defined >> >> for them and counting as well as every other arithmetic relation and >> >> operation requires some kind of numeric definitional metric. >> >> >> >> ~v~~ >> > >> >Huh! So, what happens if I declare a number, Big'un, and say that that >> >is the number of reals in (0,1]? What if I say the real line is >> >homogeneous, so every unit interval contains the same number of points? >> >> If you do, Tony, then what you've defined as "points" are in point of >> fact infinitesimals not points. >> >> >And then, what if I say the positive number line is going to include >> >Big'un such unit intervals, so it has Big'un^2 reals up to Big'un? Does >> >the universe collapse, or all tautologies suddenly become false? >> >> You could do this but the result wouldn't be finite in numerical terms >> and anything you might try to do between them and finites would cause >> the universe to collapse because they just aren't there on the same >> line and have different properties because they are finitely infinite. >> >> This is the price to be paid for absurdities like the real number line >> and putting infinities on them. "Infinite" means "not finite" and you >> just can't do finite arithmetic with "not finites". > >You wrote above that "infinite" means "numerically undefined". Does >"numerically undefined" mean "not finite"? Sure. It also means "not cardinally defined". It also means "not cardinally finite". ~v~~
From: Lester Zick on 5 Dec 2006 16:57 Tony, it occurs to me there is a better way to approach this whole subject that deals both with your objections and mine except with respect to the issue of points as units of measure. So if you don't mind I'll reply with that commentary directly and later to what you write here. On Tue, 05 Dec 2006 11:53:13 -0500, Tony Orlow <tony(a)lightlink.com> wrote: >Lester Zick wrote: ~v~~
From: Virgil on 5 Dec 2006 17:10 In article <45753CB8.3080500(a)et.uni-magdeburg.de>, Eckard Blumschein <blumschein(a)et.uni-magdeburg.de> wrote: > Let's rank Tonico outside but David Marcus still inside mathematics. > > On 12/5/2006 8:55 AM, Tonico wrote: > > David Marcus wrote: > >> Eckard Blumschein wrote: > >> > >> > Reals, as indirectly defined with DA2, > >> > >> Why do you think that the diagonal argument defines the reals? > > You all know that DA2 shows by contradiction that real numbers are > uncountable. WRONG! It is not, at least in Cantor's version, a proof by contradiction. > I carefully read how Cantor made sure that the numbers > under test are real numbers. He did not use Dedekind cuts, nested > intervals or anything else. He assumed numbers with actually > indefinitely much rather than many e.g. decimals behind the decimal > point. Strictly speaking, he did not immediately show that the reals are > uncountable but that these representation like never ending decimals is > uncountable. Since Cantor's first proof was not in any way the same and IS valid for Dedekind cuts, any flaws in his second are irrelevant.
From: Virgil on 5 Dec 2006 17:12
In article <4575825A.4020304(a)et.uni-magdeburg.de>, Eckard Blumschein <blumschein(a)et.uni-magdeburg.de> wrote: > On 12/5/2006 12:18 AM, Virgil wrote: (deleted) > > In article <45747E16.6020904(a)et.uni-magdeburg.de>, > > Eckard Blumschein <blumschein(a)et.uni-magdeburg.de> wrote:(deleted) > >> > Eckard Blumschein wrote:(deleted) Turnabout! |