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Topic: WM's argument disassembled - If you are his student, please PLEASE comment here
Replies: 5   Last Post: Jul 25, 2014 12:21 PM

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Ben Bacarisse

Posts: 1,972
Registered: 7/4/07
Re: WM's argument disassembled - If you are his student, please PLEASE comment here
Posted: Jul 22, 2014 11:09 AM
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PotatoSauce <> writes:

> On Tuesday, July 22, 2014 9:26:43 AM UTC-4, Ben Bacarisse wrote:
>> PotatoSauce writes:
>> > Here's the example again.
>> >
>> > R_n = n by 1/n rectangle. If you fix one corner of the rectangle and
>> > make n->oo, this geometric object turns into a 1-dimensional ray.
>> >
>> > Area(R_n) = 1
>> >
>> > Ask your professor what
>> >
>> > lim n-> oo Area(R_n) = 1 means.

>> It means that, for every finite or "given" n, the area is n * 1/n = 1.
>> I don't see how the limit would trouble him. I do see your point, but
>> since he does not think that unbounded lines are finite, there's no
>> problem. He thinks, remember, that all *actual* lines are finite.


> I guess you have become an expert in WM's mind.
> But I am not sure if I understood your comment about the n by 1/n
> rectangles. My point is that
> 1. All n by 1/n rectangles of area 1.
> 2. All lines, finite or not, has area 0.
> 3. As n->oo, the rectangle becomes a line (or a ray).
> To me this is perfectly analogous to WM's situation.
> 1.* All sets N\N_n has cardinality |N|.
> 2.* N\N has cardinality 0.
> 3.* n->oo N\N_n is the empty set.

But this is not his position. He rejects as meaningless all set
sequence limits. Yes, I know you concluded that WM accepts 3* based on
other things he's said, but that's a very dangerous thing to do. To
paraphrase Lewis Carroll, WM can believe a dozen contradictory things
before breakfast. Set limits are (wrongly) thought by him to rely on
completed infinity, so any problems that you can demonstrate using a
set sequence limit (like the analogy above) will be an argument WM can
use to bash them.

> So logically speaking, if WM has a problem with cardinality based on
> 1*, 2*, 3*, it seems like he should also have a problem with Areas,
> Volumes, and other concepts as well.
> Unless... your point is that WM wouldn't be troubled because [n by
> 1/n] re does not approach "an actual" line?

Yes, because it would be an actual unbounded line, so it might approach
it but it never becomes it. He might have a model of limits that allow
the limit to be a real thing "attained", provided it is not unbounded in
some way, but even that is in doubt. He might, for example, accept that
the sequence of squares with sides 1-1/n *is* the unit square "in the
limit", or he might not.

Ages ago I had an exchange with him about whether 1/3 = 0.333... =
sum{n->oo}[3*10^-n]. It was not clear whether the equality sign meant,
to him, what it means to you and me. WM may, in fact, be a classical
0.999... = 1 denialist, except for the fact that his book states the
above equality for 1/3, so he needs to work in a defence of that at the
same time that he denies it. That's where the magic preface of the book
comes in. It means he can claim that he does not really accept as true
anything in the book that he subsequently would like to repudiate.

Do you know the phrase "trying to nail a jelly to the wall"?


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