Galileo's Paradox
in [Math]
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From: Eckard Blumschein on 27 Nov 2006 09:32 On 11/27/2006 3:21 AM, stephen(a)nomail.com wrote: > There is no need to resolve the paradox. There exists a > one-to-correspondence between the natural numbers and the > perfect squares. The perfect squares are also a proper > subset of the natural numbers. This is not a contradiction. What is better? Being simply correct as was Galilei or being more than wrong? (Ueberfalsch)
From: Eckard Blumschein on 27 Nov 2006 09:35 On 11/27/2006 3:30 AM, Virgil wrote: > There is, in fact, a definition of infiniteness of sets (by Dedekind) > which REQUIRES the bijection of the set with a proper subset of itself. Let's recall that neither Dedekind nor someone else was able to prove Dedekind's basis assumption. Maybe, his claim was not just ad odds with Euclid but also questionable.
From: Eckard Blumschein on 27 Nov 2006 13:18 On 11/27/2006 6:44 PM, Tony Orlow wrote: > Zigzags are discontinuous, with each > measurable section definable by a pair of reals. That makes the > countable, given a finite space, and the finiteness of each segment. Maybe, your zigzags are different from the piecewise continuous ones I have in mind. To my understanding, it does not matter how small each segment is. Even the tiniest one is a continuum that cannot be resolved into discrete numbers.
From: stephen on 27 Nov 2006 14:47 Eckard Blumschein <blumschein(a)et.uni-magdeburg.de> wrote: > On 11/27/2006 3:21 AM, stephen(a)nomail.com wrote: >> There is no need to resolve the paradox. There exists a >> one-to-correspondence between the natural numbers and the >> perfect squares. The perfect squares are also a proper >> subset of the natural numbers. This is not a contradiction. > What is better? Being simply correct as was Galilei or being more than > wrong? (Ueberfalsch) Do you deny that there exists a one-to-one correspondence between the natural numbers and the perfect squares? Or do you deny that the perfect squares are a proper subset of the naturals? Stephen
From: Virgil on 27 Nov 2006 21:35
In article <456AAE04.309(a)et.uni-magdeburg.de>, Eckard Blumschein <blumschein(a)et.uni-magdeburg.de> wrote: > On 11/25/2006 1:18 AM, Virgil wrote: > > >> According to Spinoza, infinity is something that cannot be enlarged > >> (and also not exhausted). It is a quality, not a quantity. > > > > And how did Spinoza become an authority on the mathematical meanings of > > words? > > > > In English, at least, we do not stick to a strict "one word, one > > meaning" regimen, but allow the choice from several meanings to be > > determined by context. > > The distinction between two mutually excluding qualities like countable > and uncountable (by DA2) is not a play with words and meanings. > > > > > And in mathematical contexts, non-mathematical meaning are irrelevant. > > While set theory is only something inside mathematics, proponents of it > claim to speak on behalf of mathematics. A fairly large portion of mathematic can be embedded within set theory, and those who are only set theorists do not pretend to speak with authority of those parts of mathematics which cannot be. > While mathematics is only something inside science WRONG! What is valid in mathematics is quite independent of what is valid in science. At least as much of science is "inside" mathematics as the reverse. |