Galileo's Paradox
in [Math]
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From: Lester Zick on 4 Dec 2006 14:04 On 4 Dec 2006 09:31:45 -0800, imaginatorium(a)despammed.com wrote: > >Lester Zick wrote: >> On 4 Dec 2006 04:31:30 -0800, imaginatorium(a)despammed.com wrote: >> >> > >> >Lester Zick wrote: >> >> On Sat, 02 Dec 2006 13:26:13 -0500, Tony Orlow <tony(a)lightlink.com> >> >> 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: >> > >> ><snip all that> >> > >> >> Well, Tony, I don't really think any of this matters. These just seem >> >> to be word strings having nothing to do with the matters at hand. >> > >> >Well, indeed. Yet curiously it manages to be entertaining from time to >> >time. >> >> And at least in my case more than a little true. >> >> But come now, Brian, a little question in zen math to brighten the day >> as Bob seems unwilling to fess up: in modern math does set "theory" >> represent all of mathematics as Bob seems to think... > >What is the difference between "set theory" and "set 'theory'"? "Set theory" is not a theory because it can't be proven true which is why I qualify it as "set 'theory'" just to be polite to numbskulls who insist on using the term "set theory". >Most of present-day mathematics is founded on set theory; but this is >only a matter of convenience. So your answer is "no"? >> and does it have a >> pre emptive prerogative with respect to the definition of terms like >> "cardinality" and so on in mathematics? > >Uhm "pre-emptive prerogative"? No idea. Don't think so - you, or >rather, anyone with rather more inclination to mathematical ideas than >you possess is perfectly free to build a different foundation. Has to >be good though, and 99.999...% of crank stuff just doesn't cut the >mustard. [spot joke] Well see here's the problem, Brian. You segue right from complaints about set "theory" ignorance to complaints about mathematical ignorance. So it's kinda tough to tell what you're complaining about as you admit set "theory" does not constitute all of mathematics yet you and others seem to use the terms interchangeably. So seemingly there must be other areas of mathematics with which you are not conversant which you're not qualified to assess the rigor of others. >> .... I'm inclined to consider that >> set "theory" is really only a parochial group of analytical techniques >> which are inexhaustive at best and problematic in any event and not >> really a theory at all. > >I wouldn't worry about it. Your "inclination to consider" reminds me of >an old joke about Big Ben and the Leaning Tower of Pisa, which is >probably out of place here - nothing you say will have much effect, so >feel pretty free. I shall indeed considering your own degree in fourth order philosophy. ~v~~
From: MoeBlee on 4 Dec 2006 14:07 Eckard Blumschein wrote: > On 12/1/2006 7:04 PM, MoeBlee wrote: > > > > > The Cartesian plane is the set of ordered pairs of real numbers. > > Doesn't coordinate transform e.g. into circular coordinates require > fictitious real numbers? Here engineers might have a better feeling for > the categorical difference between number and continuum. What are fictitious real numbers? Anyway, I'm not sure what bearing your question has on the fact that RxR is the set of ordered pairs of real numbers. MoeBlee
From: Eckard Blumschein on 4 Dec 2006 14:09 On 12/1/2006 5:39 PM, Tony Orlow 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? Do you really have not enough power of abstraction as to comprehend that there is no number of reals in an intervall. There is merely the property that there are already infinitely many rationals but uncountably much of reals. The difference between both cases is quite simple: The rationals are thought addressable while the reals are considered fictitious, i.e. not at all addressable.
From: Eckard Blumschein on 4 Dec 2006 14:22 On 12/1/2006 4:56 PM, Tony Orlow wrote: > Eckard Blumschein wrote: >> In case of two finite heaps of size a and b of numbers, a=b/2 implies a<b. > Generalize where possible. Why is this not true in the infinite case? 2*oo is not larger than oo. Infinity is not a quantum but a quality. > >> In case of a=oo and b=oo, we may have a=b/2 while a is not smaller than >> b but simply not comparable: oo = oo/2. > > So, you adhere, then, to the tenets of imaginary alephs and the creed of > transfinitology? No. I am not even aware of it.
From: Eckard Blumschein on 4 Dec 2006 14:28
On 12/1/2006 4:50 PM, Tony Orlow wrote: > In other words, there are > fundamental concepts of addition and subtraction which are lost in the > infinite case, Virtually all operations: oo/a=oo, oo*oo = oo, oo^oo=oo, .... > It seems absurd to claim, for instance, that there is no justification > for wanting to see results like, for instance, that the even naturals > are a set half the size of the naturals, or any other of an infinite > number of subset relations not reflected in the numbers. Well, it demands the power of abstraction Galilei was in possession of. |