Re: 1-1/2+1/3-1/4+1/5-1/6+1/7
- From: MoeBlee <jazzmobe@xxxxxxxxxxx>
- Date: Thu, 31 Jan 2008 15:05:30 -0800 (PST)
On Jan 31, 2:45 pm, lwal...@xxxxxxxxx wrote:
On Jan 30, 11:55 am, Virgil <Vir...@xxxxxxx> wrote:
In article <c476e$47a037cd$82a1e228$10...@xxxxxxxxxxxxxxxx>,
Han de Bruijn <Han.deBru...@xxxxxxxxxxxxxx> wrote:
Aatu Koskensilta wrote:
How are they the same? That a set A is closed under the successor
function means that if n is in A, n+1 is in A. What would "applying
the successor function an infinite number of times" mean?
Here we go again, with Aatu objecting to HdB's infinite iteration of
the successor function.
But, in the other active long Cantor thread ("A consideration
concerning
the diagonal argument of G. Cantor"), Aatu had this to say (written on
the 30th of Jan., 6:15 PM Greenwich time):
Aatu:
Borel determinacy does not imply full replacement. The proof relies on
_iterating the powerset operation omega_1 times._
[emphasis mine]
I don't know the details of this particular matter, so I can't
comment; this is up to Aatu. (I say this to set up what I say later in
this post.)
So Aatu objects to HdB iterating the successor function, or the
function
x |-> x^2+c, infinitely many times, yet Aatu _himself_ is very happy
to
iterate the powerset operation, not just infinitely many times, but
_uncountably_ many times.
I see _no_ difference between HdB iterating successor or x^2+c
countably many times and Aatu iterating the powerset uncountably
many times.
Not only that, but isn't the cumulative hierarchy of sets defined by
infinite iteration of the powerset function, so that V_alpha is the
result of taking power set of the empty set exactly alpha times?
That is DEFINED by transfinite recursion.
So
anyone who would object to HdB's infinite iterations must also
object to the cumulative hierarchy of sets.
Wrong, per my above remark.
The truth is, the defenders of ZFC often hold their opponents to
higher standards of rigor than they hold themselves. Only because
HdB is trying to advocate finitism and/or intuitionism are Aatu and
the other standard mathematicians objecting to HdB's use of
infinite compositions of successor and x^2+c.
I don't know who all is included in "the other standard
mathematicians", but my own criticisms of Han de Bruijn, even as I am
not a mathematician, do NOT require him to hold to greater rigor than
I require of myself. If you claim otherwise, then please state exactly
what mathematics I have stated that is not rigorous (or that I could
not make fully rigorous given time to present a systematic treatment),
especially less rigorous than de Bruijn's undefined terminology.
Moreover, just what form of intuitionism does Han de Bruijn advocate?
Also, a while ago you made another and wild generalization about
posters here, and I asked you to name even a SINGLE poster who holds
to the view you mentioned, though still I've not seen you do that.
MoeBlee
.
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