Galileo's Paradox
in [Math]
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From: Bob Kolker on 4 Dec 2006 15:56 Eckard Blumschein wrote: > > > 2*oo is not larger than oo. Infinity is not a quantum but a quality. But aleph-0 is a quantity. Bob Kolker
From: Bob Kolker on 4 Dec 2006 15:58 Eckard Blumschein wrote: > No wonder. Exactly this selfdelusion was the intention of dedekind. 'Twas no delusion. The Dedikind cut defining the square root of 2 is just as well defined as the successor to the integer 2. Bob Kolker
From: David Marcus on 4 Dec 2006 16:47 Eckard Blumschein wrote: > On 12/1/2006 9:59 PM, Virgil wrote: > > > Sets can have the same cardinality but different 'subsettedness' because > > different qualities are being measured. > > I do not know a German equivalent to subsettedness. Wiki did not know > subsettedness or at least subsetted either. So I have to gues what you > possibly meant. Proper subset means included but not all-including. > While I consider cardinality a rather illusory notion, I distinguish > between counted (alias finite), countable (alias potentially infinite) > and uncountable (alias fictitious). Well, the counteds are a subset of > the countables. So far, the mislead bulk of mathematicians regard the > countables (i.e. genuine numbers) a subset of the uncountables, too. > > TO>> You do? Do you mean that the addition of elements not already in a set > >> doesn't add to the size of the set in any sense? > > > > In the subsettedness sense yes, in the cardinality sense, not > > necessarily. In the sense of well-ordered subsettedness, not necessarily. > > > > TO seems to want all measures to give the same results, regardless of > > what is being measured. > > Standard mathematics may lack solid fundamentals. At least it is > understandable to me. If it is understandable to you, then convince us you understand it: Please tell us the standard definitions of "countable" and "uncountable". -- David Marcus
From: Lester Zick on 4 Dec 2006 17:23 On Mon, 04 Dec 2006 15:56:58 -0500, Bob Kolker <nowhere(a)nowhere.com> wrote: >Eckard Blumschein wrote: > >> >> >> 2*oo is not larger than oo. Infinity is not a quantum but a quality. > >But aleph-0 is a quantity. It is? So which quantity is it? I mean could you show us an aleph-0 quantity or at least prove that there is such a quantity? ~v~~
From: Lester Zick on 4 Dec 2006 17:32
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? >> 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. ~v~~ |