From lojban-beginners+bncCML0xpmUARCCqcnkBBoEDPAn6g@googlegroups.com Thu Sep 16 10:32:19 2010 Received: from mail-qw0-f61.google.com ([209.85.216.61]) by chain.digitalkingdom.org with esmtp (Exim 4.72) (envelope-from ) id 1OwIJj-000489-7L; Thu, 16 Sep 2010 10:32:19 -0700 Received: by qwi4 with SMTP id 4sf2524220qwi.16 for ; Thu, 16 Sep 2010 10:32:08 -0700 (PDT) DKIM-Signature: v=1; a=rsa-sha256; c=relaxed/relaxed; d=googlegroups.com; s=beta; h=domainkey-signature:received:x-beenthere:received:received:received :received:received-spf:received:mime-version:received:received :in-reply-to:references:date:message-id:subject:from:to :x-original-sender:x-original-authentication-results:reply-to :precedence:mailing-list:list-id:list-post:list-help:list-archive :sender:list-subscribe:list-unsubscribe:content-type; bh=ep/6M69Pf5YxnIxLqjx72+pdEgllWGv8SIHJjhL9Zo0=; b=yKY/gekUpWi/byI6ZneXW0btIHQzfJh8Cnx+3x0PQtKA1McvFRYlNFt87K+ZhdNoIP BmCgaN0rUeBOYSPqgU3+mCB+6pxOJFVS1KEZodiHogF85jrCnoJJKqVzIeRl+Y3KY5vR cz9AQC5C/yvoByqIkfiGK0XqnWnqIwWJFxx70= DomainKey-Signature: a=rsa-sha1; c=nofws; d=googlegroups.com; s=beta; h=x-beenthere:received-spf:mime-version:in-reply-to:references:date :message-id:subject:from:to:x-original-sender :x-original-authentication-results:reply-to:precedence:mailing-list :list-id:list-post:list-help:list-archive:sender:list-subscribe :list-unsubscribe:content-type; b=dCDRFt3VXev54FjOafaFc6XouL1e1ayooGttW1m2tqUv8vsCfgUYYJ/5+KvCmA7J9d Wq1snRIlYNsSmZwHGTqo6zqRQGVWv2cgXni6yecReT+FTG8ORzYVDSbK/pbBcfEEvr5h BxlP+U7RLK6oyo8D61Jivb9bFpU8h4Wo7whsg= Received: by 10.224.69.146 with SMTP id z18mr366925qai.47.1284658306070; Thu, 16 Sep 2010 10:31:46 -0700 (PDT) X-BeenThere: lojban-beginners@googlegroups.com Received: by 10.224.66.155 with SMTP id n27ls485086qai.5.p; Thu, 16 Sep 2010 10:31:45 -0700 (PDT) Received: by 10.224.37.14 with SMTP id v14mr260171qad.3.1284658304959; Thu, 16 Sep 2010 10:31:44 -0700 (PDT) Received: by 10.224.37.14 with SMTP id v14mr260170qad.3.1284658304916; Thu, 16 Sep 2010 10:31:44 -0700 (PDT) Received: from mail-qy0-f177.google.com (mail-qy0-f177.google.com [209.85.216.177]) by gmr-mx.google.com with ESMTP id mz6si1569254qcb.5.2010.09.16.10.31.43; Thu, 16 Sep 2010 10:31:43 -0700 (PDT) Received-SPF: pass (google.com: domain of blindbravado@gmail.com designates 209.85.216.177 as permitted sender) client-ip=209.85.216.177; Received: by mail-qy0-f177.google.com with SMTP id 34so2243291qyk.1 for ; Thu, 16 Sep 2010 10:31:43 -0700 (PDT) MIME-Version: 1.0 Received: by 10.229.73.132 with SMTP id q4mr734048qcj.132.1284658303754; Thu, 16 Sep 2010 10:31:43 -0700 (PDT) Received: by 10.229.101.208 with HTTP; Thu, 16 Sep 2010 10:31:43 -0700 (PDT) In-Reply-To: References: <201009152248.22472.phma@phma.optus.nu> Date: Thu, 16 Sep 2010 13:31:43 -0400 Message-ID: Subject: Re: [lojban-beginners] Cauchy sequences From: Ian Johnson To: lojban-beginners@googlegroups.com X-Original-Sender: blindbravado@gmail.com X-Original-Authentication-Results: gmr-mx.google.com; spf=pass (google.com: domain of blindbravado@gmail.com designates 209.85.216.177 as permitted sender) smtp.mail=blindbravado@gmail.com; dkim=pass (test mode) header.i=@gmail.com Reply-To: lojban-beginners@googlegroups.com Precedence: list Mailing-list: list lojban-beginners@googlegroups.com; contact lojban-beginners+owners@googlegroups.com List-ID: List-Post: , List-Help: , List-Archive: Sender: lojban-beginners@googlegroups.com List-Subscribe: , List-Unsubscribe: , Content-Type: multipart/alternative; boundary=0016364ee3f2da60a8049063d1ca --0016364ee3f2da60a8049063d1ca Content-Type: text/plain; charset=ISO-8859-1 Now you're past my analysis level; are systems in which Cauchy sequences don't converge to a limit necessarily not Archimedean? Regardless, in the sentence "{x_k} is a Cauchy sequence representing a real number x", how many predicates would you use? It's sounding like you would use two. -- mi'o bazu klama ti tu zi'o On Thu, Sep 16, 2010 at 2:55 AM, Ross Ogilvie wrote: > But Cauchy sequences don't necessarily converge to a limit. I would say > that convergence of a sequence to a limit should be a separate predicate, > but my vocab isn't up to finding one. You could then indicate the limit of a > Cauchy sequence with a tagged place. > > Ross > > On Thu, Sep 16, 2010 at 3:18 PM, Ian Johnson wrote: > >> That makes sense, but should that really be the x2? Or should there be >> another predicate that we use to relate numbers and sequences? For example, >> we could say: >> x1 is a Cauchy sequence converging to limit x2 in metric space x3. >> >> >> mi'o bazu klama ti tu zi'o >> >> On Wed, Sep 15, 2010 at 10:48 PM, Pierre Abbat wrote: >> >>> On Wednesday 15 September 2010 20:01:06 Ian Johnson wrote: >>> > I found myself being lazy in my analysis class having to repeatedly >>> write: >>> > Let x in R. Suppose {x_k} is a Cauchy sequence representing x. >>> > I was trying to come up with a good word to use to represent this >>> clunky >>> > relation, that is: >>> > x1 is a Cauchy sequence representing the real number x2. >>> > The thing I came up with first was pretty bad, but I didn't have a >>> > dictionary on me. It was {listrkoci}. Once I got to a dictionary I >>> thought >>> > of {porsrkoci}, which seems a bit better. Does anyone have any better >>> > ideas? Maybe something that isn't a fu'ivla? >>> >>> "porsrkoci" and "pornkoci" are both good, and are different forms of the >>> same >>> word (though the Book doesn't say that different rafsi of one gismu are >>> equivalent, except in lujvo). The alternatives are a lujvo, which would >>> be >>> longish, and "kocis.zei.porsi", which is also longish. I'd go >>> with "pornkoci". >>> >>> > To clarify, this should hold, if broda is assigned to this relation: >>> > li pa ce'o li pa fi'u re ce'o li pa fi'u ci ce'o ... broda li no >>> > (sorry that I don't know a good way to say "et cetera ad infinitum" in >>> > lojban.) >>> >>> I think the place structure should be "x1 (sequence) is a Cauchy sequence >>> in >>> x1 (metric space)". I know a sequence of rational numbers which converges >>> to >>> +3 in the real numbers and to -3 in the 2-adic numbers. There are Cauchy >>> sequences of rational numbers which don't converge to any rational >>> number, >>> and there are sequences of rational numbers which are Cauchy sequences in >>> one >>> metric but not in another. >>> >>> Pierre >>> -- >>> li fi'u vu'u fi'u fi'u du li pa >>> >>> -- >>> You received this message because you are subscribed to the Google Groups >>> "Lojban Beginners" group. >>> To post to this group, send email to lojban-beginners@googlegroups.com. >>> To unsubscribe from this group, send email to >>> lojban-beginners+unsubscribe@googlegroups.com >>> . >>> For more options, visit this group at >>> http://groups.google.com/group/lojban-beginners?hl=en. >>> >>> >> -- >> You received this message because you are subscribed to the Google Groups >> "Lojban Beginners" group. >> To post to this group, send email to lojban-beginners@googlegroups.com. >> To unsubscribe from this group, send email to >> lojban-beginners+unsubscribe@googlegroups.com >> . >> For more options, visit this group at >> http://groups.google.com/group/lojban-beginners?hl=en. >> > > -- > You received this message because you are subscribed to the Google Groups > "Lojban Beginners" group. > To post to this group, send email to lojban-beginners@googlegroups.com. > To unsubscribe from this group, send email to > lojban-beginners+unsubscribe@googlegroups.com > . > For more options, visit this group at > http://groups.google.com/group/lojban-beginners?hl=en. > -- You received this message because you are subscribed to the Google Groups "Lojban Beginners" group. To post to this group, send email to lojban-beginners@googlegroups.com. To unsubscribe from this group, send email to lojban-beginners+unsubscribe@googlegroups.com. For more options, visit this group at http://groups.google.com/group/lojban-beginners?hl=en. --0016364ee3f2da60a8049063d1ca Content-Type: text/html; charset=ISO-8859-1 Content-Transfer-Encoding: quoted-printable Now you're past my analysis level; are systems in which Cauchy sequence= s don't converge to a limit necessarily not Archimedean?

Regardl= ess, in the sentence "{x_k} is a Cauchy sequence representing a real n= umber x", how many predicates would you use? It's sounding like yo= u would use two.


--
mi'o bazu klama ti tu zi'o

On Thu, Sep 16, 2010 at 2:55 AM, Ross Ogilvie <= ;oges007@gmail.com> wrot= e:
But Cauchy sequences don't necessarily converge to a= limit. I would say that convergence of a sequence to a limit should be a s= eparate predicate, but my vocab isn't up to finding one. You could then= indicate the limit of a Cauchy sequence with a tagged place.

Ross

On Thu, Sep 16, 2010 at 3:18 PM, Ian Johnson <= ;blindbravado@g= mail.com> wrote:
=
That makes sense, but should that really be the x2? Or should there be= another predicate that we use to relate numbers and sequences? For example= , we could say:
x1 is a Cauchy sequence converging to limit x2 in= metric space x3.=A0


mi'o bazu klama ti tu zi'o
=

On Wed, Sep 15, 2010 at 10:48 PM, Pierre A= bbat <phma@phma.optus.nu> wrote:
On Wednesday= 15 September 2010 20:01:06 Ian Johnson wrote:
> I found myself being lazy in my analysis class having to repeatedly wr= ite:
> Let x in R. Suppose {x_k} is a Cauchy sequence representing x.
> I was trying to come up with a good word to use to represent this clun= ky
> relation, that is:
> x1 is a Cauchy sequence representing the real number x2.
> The thing I came up with first was pretty bad, but I didn't have a=
> dictionary on me. It was {listrkoci}. Once I got to a dictionary I tho= ught
> of {porsrkoci}, which seems a bit better. Does anyone have any better<= br> > ideas? Maybe something that isn't a fu'ivla?

"porsrkoci" and "pornkoci" are both good, and are= different forms of the same
word (though the Book doesn't say that different rafsi of one gismu are=
equivalent, except in lujvo). The alternatives are a lujvo, which would be<= br> longish, and "kocis.zei.porsi", which is also longish. I'd go=
with "pornkoci".

> To clarify, this should hold, if broda is assigned to this relation: > li pa ce'o li pa fi'u re ce'o li pa fi'u ci ce'o .= .. broda li no
> (sorry that I don't know a good way to say "et cetera ad infi= nitum" in
> lojban.)

I think the place structure should be "x1 (sequence) is a Cauchy= sequence in
x1 (metric space)". I know a sequence of rational numbers which conver= ges to
+3 in the real numbers and to -3 in the 2-adic numbers. There are Cauchy sequences of rational numbers which don't converge to any rational numb= er,
and there are sequences of rational numbers which are Cauchy sequences in o= ne
metric but not in another.

Pierre
--
li fi'u vu'u fi'u fi'u du li pa

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