Galileo's Paradox
in [Math]
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From: Eckard Blumschein on 6 Dec 2006 11:02 On 12/5/2006 11:36 PM, Virgil wrote: >> One has to be pretty miseducated in order to need the fancy aleph_1. > > That depends on whether one accepts or rejects the continuum hypothesis. Not at all.
From: Eckard Blumschein on 6 Dec 2006 11:06 On 12/5/2006 11:37 PM, Virgil wrote: > In article <45759016.3020802(a)et.uni-magdeburg.de>, > Eckard Blumschein <blumschein(a)et.uni-magdeburg.de> wrote: > >> On 12/4/2006 11:52 PM, Virgil wrote: >> >> >> I am an electrical engineer >> > >> > Shocking! >> >> Why? > > It's what electricity does! Electricity makes your ears hearing (therefore U>100 mV) it makes your brain thinking and your heart beating Your computer needs it,...
From: Eckard Blumschein on 6 Dec 2006 11:09 On 12/5/2006 11:41 PM, Virgil wrote: > In article <457593EB.9030809(a)et.uni-magdeburg.de>, > Eckard Blumschein <blumschein(a)et.uni-magdeburg.de> wrote: > >> >> Rationals are p/q. This system cannot be improved by adding genuine >> (i.e. rational) numbers. One can merely move to the fictitious >> continuous alternative. > > All numbers are equally genuine in any meaning of "genuine" other than > EB's improper meaning of "irrational". Kronecker was correct in that irrationals are no geunine numbers. He was merely to stubborn as to admit that fictitious real numbers are sufficient. The imaginary unit is also not a number on the number line while being tremendously useful.
From: Tez on 6 Dec 2006 11:17 Eckard Blumschein wrote: > On 12/4/2006 9:49 PM, David Marcus wrote: > > >> According to my reasoning, the power set is based on all elements of a > >> set. > > > > "Based"? > > Yes. The power set algorithm does not change what mathematicians still > used to call cardinality. 2^oo=oo. "The" power set "algorithm"? Could you write out "the" algorithm here for us so that you and I can flesh out your claims in detail? > > > >> In case of an infinite set, there are nor all elements available. > > > > "Available"? > > Yes. You cannot apply the algorithm until you have all numbers. You seem to be looking at this from a computational (but more probably, a programatic) perspective. From that perspective, you are wrong. It is possible to apply algorithms to infinite data structures. See: http://en.wikipedia.org/wiki/Haskell_programming_language http://en.wikipedia.org/wiki/Lazy_evaluation http://cs.wwc.edu/KU/PR/Haskell.html In that last link, look at the section headed "Lazy Evaluation and Infinite Lists" and examine, for example, the implementation of the Sieve of Eratosthenes. > >> Nonetheless I can do so as if they would exist, and I am calling them a > >> fiction. > > > > "Exist"? "Fiction"? > > Exist means, they have their numerical address within a rational order. > Fiction means, they don't have it but it is reasonable to do so as if. "Address"? Rational "order"? If we're going to play with words, I'm going to ask: Why don't entities with numerical addresses with an irrational order "exist"? [snip] -Tez
From: Eckard Blumschein on 6 Dec 2006 11:20
On 12/5/2006 11:48 PM, Virgil wrote: > In article <457596BC.3040307(a)et.uni-magdeburg.de>, > Eckard Blumschein <blumschein(a)et.uni-magdeburg.de> wrote: > >> On 12/4/2006 10:47 PM, David Marcus wrote: > >> >> 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". >> >> I do not like such unnecessary examination. > > Then stop trying to examine others. I examined others here? I only examine students in topics of electrical engineering. BTW, Peter Dallos, a specialist for hair cells admitted to examine his students every year with the same questions. He said: No problem, since the correct answers are different each time. >> Countably infinite refers to bijection. >> Uncountable is a correcting translation of the German nonsense word >> ueberabzaehlbar = more than countable. > > I understood that "ueber" could also bean "beyond", and "beyond > counting", at least in the English meaning of "counting", is precisely > what uncountable means. While beyond counting means beyond the possibility of counting, Cantor imagined "einfaches Hinueberzaehlen" (simple continuation of counting after having reached infinity) Uebermensch meant superman >> Meant is: there is no bijection to the naturals. > > Actually, that "meaning" would make finite sets uncountable. > > A better statement would be that there is no surjection from the > naturals. > > That excludes finite sets. Agreed |