In article <email@example.com>, "DavidW" <firstname.lastname@example.org> writes: >quasi wrote: >> On Fri, 9 Mar 2012 20:42:53 +1100, "DavidW" <email@example.com> wrote:
>> But natural numbers are not elements of the physical world. >> They are conceptual entities, as are sets of such numbers. >> Mathematics allows for the conceptualization of objects >> such as the set of all natural numbers. > >But it's an impossible concept. There can be no such thing as "all" if there >are infinitely many.
I contend that there is such a thing as the set of all natural numbers. I refer to this set as "N".
Since you contend that there is no such thing as an infinite set, there are obviously natural numbers that are not in "N". Please name three of them.
> You might be able to, for example, write a simple >iterative formula to define the set, but you can never complete the iterations.
You don't have to "complete the iterations" in order to reason about the elements of the set. For instance, I can work out that all elements of the set of even numbers are divisible by two, without writing down all of them -- or any of them.
> There's no >such thing as all of them to contain.
If there's not "all" of them, there must only be "some" of them. Who decides which ones there are and which ones there aren't?
>> In Geometry, we can have 2 points as close to each other as >> we please. >> >> In the real world, distinct objects can get close but not >> arbitrarily close. >> >> In Geometry, we regard a line as containing infinitely many >> points. > >I have no problem with any of these. All they require is imagination to >conceive. But the very concept of containing "all" of something of which there >are infinitely many is self-contradictory.
Does your imagination allow a line segment to include all of its points?
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