Galileo's Paradox
in [Math]
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From: Lester Zick on 29 Nov 2006 13:04 On Wed, 29 Nov 2006 17:12:04 +0000 (UTC), stephen(a)nomail.com wrote: >Six wrote: [. . .] >> You suggested I conduct my argument without using the term >> 'infinity'. I am quite happy to do that. I suggest you conduct the rest of >> your argument without using the term 'cardinality'. > >Why? Cardinality has a definition in set theory. The issue is whether "cardinality" has a definition in mathematics not set theory unless you're suggesting set theory is all of mathematics. > 'infinity' does not have >a definition. Sure it does. In mathematics "infinity" is the "number of infinitesimals". Once again, Stephen, you seem to be confusing mathematics with set "theory" which isn't really even a theory at all because it can't be demonstrated true but merely represents a series of analytical techniques applied to sets which certainly don't encompass all of mathematics. > Do you really think that the two words are on an equal footing? Actually not because "infinity" a well defined mathematical concept whereas "cardinality" is only an ambiguously defined concept mathematically restricted to undemonstrable set analytical techniques. ~v~~
From: Eckard Blumschein on 29 Nov 2006 13:06 On 11/29/2006 3:56 PM, Tony Orlow wrote: > Cardinality is generalized from the simple count of finite sets to the > infinite case. In the finite case, the cardinality of a set is exactly a > natural number, a quantity. In the infinite case, cardinality becomes > something more ephemeral, Epheremal means shortlived. We have a saying: Lies live short. but it still has its roots in the count of a set. Let's rather say in Cantor's illusion of allegedly being able to count the uncountable. >> What about when there is more than one type of measure that can be >> applied to a set, or none at all? What happens then? Then perhaps a red light will indicate logical error.
From: Eckard Blumschein on 29 Nov 2006 13:16 On 11/29/2006 3:26 PM, Six wrote: > For me, anyway. I am just trying to work things out in > my own mind. Continue! > If it's a matter of comparative success or fruitfulness, I have no > complaint. I do not have the mathematics to judge. The less biased the better.
From: Lester Zick on 29 Nov 2006 13:19 On Tue, 28 Nov 2006 23:25:30 -0500, Bob Kolker <nowhere(a)nowhere.com> wrote: >Tony Orlow wrote: > >> >> That is correct, and that is where cardinality fails as a measure of >> such sets. Raw bijection determines cardinality, but measure involves a >> consideration of the actual mapping function which establishes the >> bijection. The two are not incompatible, Bob. > >Cardinality was never intended as a measure. It was intended as a count. Just out of curiosity, Bob, why is cardinality in set theory not a measure? I mean if you ask "how much gas" and get the answer "two gallons" you've certainly measured the gas. Or if you ask "how much space" and get the answer "two inches" you've certainly measured the space. It seems to me that you can obviously superimpose cardinality on questions like "how much" without having to count or match things. >Compare and contrast the following questions: How much vs. How many. ~v~~
From: Tony Orlow on 29 Nov 2006 13:36
Eckard Blumschein wrote: > On 11/29/2006 3:58 PM, Tony Orlow wrote: >> where one set contains all the >> elements of another, plus more, it can rightfully be considered a larger >> set. > > All of oo? > > Yes. All of the naturals are integers. Only half of all the integers are naturals. All of the points in (0,1] are in (0,2], but only half of all of the points in (0,2] are in (0,1]. |