Re: Epistemology 201: The Science of Science

stephen_at_nomail.com
Date: 03/08/05 

Date: 8 Mar 2005 02:40:23 GMT

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

: <stephen@nomail.com> wrote in message
: news:d09vvs$2dm3$1@msunews.cl.msu.edu...
:> In sci.math Allan C Cybulskie <allan.c.cybulskie@yahoo.ca> wrote:
:> : I don't want to reply to all the posts right now, but really need to
: reply
:> : to this part of this here ...
:>
:> : Here is the definition (also helpfully left above) that you gave for
: "proper
:> : subset":
:>
:> :>>If A is a proper subset of B than B contains
:> :> :> EXACTLY THE SAME elements as A plus some more
:>
:> : If that is the definition of proper subset, then what it says is that B
: must
:> : have a larger number of elements than A since it has exactly the same
:> : elements (thus, the same number of elements) as A PLUS SOME MORE, as you
:> : said. Yes, it says nothing about bijection but note that for the
: infinite
:> : sets we have been talking about the bijection approach says that a
: proper
:> : subset has the same number of elements as the superset. THAT is a
:> : contradiction between the two definitions.
:>
:> 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.

: 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.

: 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. You apparently think infinity+1 is bogus,
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. So what is your definition of
the "relative number of elements" for infinite sets?

Stephen

Relevant Pages

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  • Re: Epistemology 201: The Science of Science
    ... :> In sci.math Allan C Cybulskie wrote: ... :>: If that is the definition of proper subset, then what it says is that B ... infinity, but then we try to get rid of the infinity which is the only thing ...
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