Article View: sci.logic
Article #794Re: A Guide for Cantor Cranks
From: Transfer Principle
Date: Tue, 29 Mar 2011 20:58
Date: Tue, 29 Mar 2011 20:58
48 lines
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4547 bytes
On Mar 27, 2:46 pm, Jim Burns <burns...@osu.edu> wrote:
> On 3/27/2011 3:23 PM, Transfer Principle wrote:
> > Of course, if there really is a proof in ZF-Infinity or
> > ZF-Powerset of the existence of uncountable sets, only then
> > will I agree that Albrecht's post is "ignoramus mathematics."
> Would you be satisfied with ZF-Infinity
> or ZF-Powerset (on Albrecht's behalf) even if
> there were no naturals, integers or rationals,
> which Albrecht does seem to find useful?
I know that there's a way to construct the naturals and integers,
and possibly the rationals as well, without infinite sets.
Consider the naturals. Although the _set_ N of all naturals is an
infinite set, each individual natural is a _finite_ set. One only
needs
Empty Set (for 0), Pairing, and Union (for nu{n}) to construct the
von Neumann naturals. Thus, it's very easy to work with individual
naturals without having Infinity or Powerset.
Albrecht _does_ mention proper classes in his posts. So even though
there's no _set_ of all naturals, there might be such a proper class.
The integers? Instead of letting integers be certain equivalence
classes of ordered pairs of naturals, we let each one be just a
_single_
ordered pair. For example, for each natural number n, let (n,0) be the
positive integer +n, and (0,n) be the negative integer -n. (Note that
this
is just the first ordered pair in each traditional equivalence class.)
The rationals? Instead of letting rationalss be certain equivalence
classes of ordered pairs of rationals, we let each one be just a
_single_
ordered pair. For example, we can let (a,b), with b>0, gcd(a,b)=1, be
the rational a/b.
This should work. If necessary, we can try to prove this, but I'm
already
in the middle of a similar construction for zuhair's ND in another
thread,
so perhaps when that is done, we can attempt to prove that this
construction gives us integers and rationals in ZF-Infinity. (Or
perhaps
NBG-Infinity if we want proper classes.)
Note: by (a,b) we mean the Kuratowski ordered pair, which requires
only Pairing to construct.
Technical Details
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X-Google-Thread: 109d8a,bdbafb069b8118a7 X-Google-NewGroupId: yes X-Google-Thread: f8362,bdbafb069b8118a7 X-Google-NewGroupId: yes X-Google-Thread: 10b271,bdbafb069b8118a7 X-Google-NewGroupId: yes X-Google-Thread: fd648,af7892021029fe4a X-Google-NewGroupId: yes X-Google-Attributes: gid9ef9b79ae9,gided4ff1f581,gid54ac8f410d,giddb42311052,domainid0,public,usenet X-Google-Language: ENGLISH,ASCII Path: arc1-ca.pugleaf.net!g2news2.google.com!postnews.google.com!l2g2000prg.googlegroups.com!not-for-mail From: Transfer Principle <lwalke3@lausd.net> Newsgroups: sci.logic,sci.math,comp.theory,sci.physics,aus.science Subject: Re: A Guide for Cantor Cranks Date: Tue, 29 Mar 2011 13:58:17 -0700 (PDT) Organization: http://groups.google.com Lines: 48 Message-ID: <82025101-e1c9-4ca8-82ba-93d35db82cb6@l2g2000prg.googlegroups.com> References: <40e69d0c-eb77-4308-996e-c7dfc1a0ec1e@j13g2000pro.googlegroups.com> <92a4e0aa-6497-4357-8a6b-ecc706fa10f5@d26g2000prn.googlegroups.com> <5d2c87a8-5926-4b27-a445-3bc3ab5c98d0@u3g2000vbe.googlegroups.com> <35760f05-df0e-4c29-85c0-08f98cf10a42@s18g2000prg.googlegroups.com> <521e5cfe-5269-4d9a-ab45-5e8cf597b2d1@y31g2000vbp.googlegroups.com> <971970b6-b722-4f38-96cb-4aef5f17e96e@b22g2000prb.googlegroups.com> <d9d86644-ea52-4f53-ba2f-42d2718630c0@k7g2000yqj.googlegroups.com> <b05355b2-d80e-47ec-9c3b-43e7bc40ab32@22g2000prx.googlegroups.com> <4D8B5041.9060702@osu.edu> <936fb9f1-da8e-4533-93a1-15cf5d060cc5@k7g2000yqj.googlegroups.com> <357b5567-048f-41ad-bb72-b662b2fd0163@k22g2000yqh.googlegroups.com> <a1dcf298-c9ac-493b-8f84-aba738455b49@i35g2000prd.googlegroups.com> <imob8a$92i$1@charm.magnus.acs.ohio-state.edu> NNTP-Posting-Host: 65.208.187.194 Mime-Version: 1.0 Content-Type: text/plain; charset=ISO-8859-1 Content-Transfer-Encoding: quoted-printable X-Trace: posting.google.com 1301432298 30111 127.0.0.1 (29 Mar 2011 20:58:18 GMT) X-Complaints-To: groups-abuse@google.com NNTP-Posting-Date: Tue, 29 Mar 2011 20:58:18 +0000 (UTC) Complaints-To: groups-abuse@google.com Injection-Info: l2g2000prg.googlegroups.com; posting-host=65.208.187.194; posting-account=euF15goAAACbw3KIqEWxZHCIPUc2KPmU User-Agent: G2/1.0 X-HTTP-UserAgent: Mozilla/4.0 (compatible; MSIE 7.0; Windows NT 5.1; .NET CLR 1.1.4322; .NET CLR 2.0.50727; .NET CLR 3.0.4506.2152; .NET CLR 3.5.30729; InfoPath.2; OfficeLiveConnector.1.4; OfficeLivePatch.1.3; .NET4.0C),gzip(gfe)