Re: Epistemology 201: The Science of Science

stephen_at_nomail.com
Date: 03/13/05 

Date: 13 Mar 2005 17:44:32 GMT

In sci.math Allan C Cybulskie <allan.c.cybulskie@yahoo.ca> wrote:

: <stephen@nomail.com> wrote in message
: news:d0j3an$fdl$2@msunews.cl.msu.edu...
:> In sci.math Allan C Cybulskie <allan.c.cybulskie@yahoo.ca> wrote:
:> :> No it is not. The answer to "the number of elements" is a number.
:> :> I already showed you several examples of numbers where x+a=x, even
:> :> when a is non zero. There is no contradiction.
:>
:> : And I dealt with that by pointing out the mathematical trick that it
: relies
:> : on. For example, here's another claim of the same sort:
:>
:> It is not a trick. It is a definition. You are appealing
:> to some non-mathematical law of your own invention that
:> apparently claims such a definition is inconsistent.

: Did I ever say that? No, I never did say that. All I stated is that your
: application of the definition here is a mathematical "trick", since it
: relies on the fact that if you add something to infinity, you cannot express
: the result as anything other than infinity again (or, at least, that you can
: do that) but that trick has no relation to the relative number of elements.

:>
:> : infinity + 1 = infinity + 2 is in fact a balanced equation. But if I
: try to
:> : subtract out the infinities on both sides, I get 1 = 2, which is clearly
:> : ludicrous. The reason this is a trick is that we call this a balanced
:> : equation because infinity + 1 and infinity + 2 both get treated like
:> : infinity, but then we try to get rid of the infinity which is the only
: thing
:> : that made them balanced we end up with a balanced equation.
:>
:> If you look at all the number systems that include some sort
:> of "infinity", infinity-infinity is not defined. Just
:> like the fact that x/0 is not defined. It is easy to prove
:> 1=2 if you use undefined operations.

: Yes, the argument was that you cannot describe infinity - infinity because
: of ludicrous examples like the one above. The issue is that you are
: treating things differently in the above example to get to the infinity.

: But I'd like to ask you to think for an instant: putting aside the example I
: just gave WHY can't I subtract infinity from infinity? If infinity is the
: same size or number, then it should be simple to do so. But if infinity is
: simply a number for a large amount of various sizes or numbers, then you
: cannot (obviously) subtract it out, but then your claim that "x + a = x" as
: a justification of any sort for "they have the same number of elements"
: cannot work.

Why can you not divide by 0? The point is people have defined
number systems which include an "infinity". These are useful
and necessary if you want to provide an answer to a question like
"what is the number of elements in an infinite set?". In all these
number systems infinity-infinity is not defined, for the same
reason that x/0 is not defined. And this makes sense when you
look at sets. If I take an infinite set, and remove an infinite
number of elements, the result can be the empty set, an infinite
set, or anything in between.

: In short, if the point you raised about x + a is relevant, then I should be
: able to subtract out infinities and have it work. If it doesn't work, then
: x + a seems to be irrelevant to the discussion.

:>
:> : Your argument was exactly like that. You rely on infinity + anything
:> : remaining infinity, but that does not mean that the relative number of
:> : elements cannot be said to be larger based on the definition of the set
:> : itself.
:>
:> You have to define what you mean by the relative number
:> of elements.

: The number of elements of one set relative to the other one. Not merely
: ratio, BTW.

That is not a definition. How do you calculate it? What do you
mean by "number of elements"? You need to define it in such a way
that someone can actually use it.

:> You apparently think infinity+1 is bogus,

: Not me. It's you who does that by insisting that that is just infinity
: itself. Tony actually claims that infinity + 1 is reasonable.

:> but you are somehow appealing to some undefined notion
:> of infinity/infinity. After all, to compute the relative
:> size of two sets, I need to divide the size of one set by
:> the size of the other.

: That's the ratio, not the relative number of elements. For example, if I
: start an integer set at 0 or at 1 I can say that the relative number of
: elements in the set starting a 0 is always + 1 compared to the set that
: starts at 1. This is not a ratio calculation.

So what is the relative size of the evens compared to the integers?
+infinity? What is the relative size of the rationals compared to
the integers?

Stephen

Relevant Pages

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