X-Received: by 2002:a05:622a:49:b0:3f9:a7d2:f86 with SMTP id y9-20020a05622a004900b003f9a7d20f86mr1858410qtw.1.1686435006805; Sat, 10 Jun 2023 15:10:06 -0700 (PDT) X-Received: by 2002:a81:af58:0:b0:56d:791:d1a4 with SMTP id x24-20020a81af58000000b0056d0791d1a4mr376759ywj.7.1686435006452; Sat, 10 Jun 2023 15:10:06 -0700 (PDT) Path: csiph.com!1.us.feeder.erje.net!3.us.feeder.erje.net!feeder.erje.net!usenet.blueworldhosting.com!diablo1.usenet.blueworldhosting.com!peer03.iad!feed-me.highwinds-media.com!news.highwinds-media.com!news-out.google.com!nntp.google.com!postnews.google.com!google-groups.googlegroups.com!not-for-mail Newsgroups: sci.logic Date: Sat, 10 Jun 2023 15:10:06 -0700 (PDT) In-Reply-To: Injection-Info: google-groups.googlegroups.com; posting-host=77.57.53.44; posting-account=UjEXBwoAAAAOk5fiB8WdHvZddFg9nJ9r NNTP-Posting-Host: 77.57.53.44 References: <1941110f-b45f-4e42-8290-9f7949082816n@googlegroups.com> <48c34278-2a72-47b1-8409-2a8b19abbb20n@googlegroups.com> <756a6232-e267-46f5-a0fd-86f5f1377f39n@googlegroups.com> <406ef698-8190-41e7-9dc7-15416defd9c8n@googlegroups.com> <33ed59fc-415f-49cb-94c9-19f53adbec46n@googlegroups.com> <38f8ea3f-6438-4295-98f5-370aefb00e5dn@googlegroups.com> <03e0a48e-7a65-4fb4-8507-1bedcd7869b5n@googlegroups.com> User-Agent: G2/1.0 MIME-Version: 1.0 Message-ID: <2cd03f75-8eca-4e4d-8a9e-2f1365ac0b4en@googlegroups.com> Subject: Re: Expressability in the notation of set theory From: Mild Shock Injection-Date: Sat, 10 Jun 2023 22:10:06 +0000 Content-Type: text/plain; charset="UTF-8" Content-Transfer-Encoding: quoted-printable X-Received-Bytes: 4672 Xref: csiph.com sci.logic:254219 To push the cart, here is a suggestion how to further proceed and replicate some results from Terrence Tao. The sentence produced when translated to first order logic will have two parameters. The set Q, for the quaternary predicate Q, and the set U for the domain U. In honor of Terrence Tao, let the sentence produce be: /* What we get when we translate the dependency=20 quantifier example from Terence Tao */ T(U, Q) Now to replicate this result here: "For every real numbers ... is true in the non-standard model of the real numbers, but false in the standard model" https://terrytao.wordpress.com/2007/08/27/printer-friendly-css-and-nonfirst= orderizability/#more-172 You can preform the following logic experiments with DC Proof. Try finding a Q0 and U_1 and U_2 such that: |- T(U1,Q0) |- ~T(U2,Q0) So the domain U1 will play the role of the non-standard model, and the domain U2 will play the role of the standard model, and we interpret "true" and "false" as provability and provable refutation FOL + Set Theory. Can you find some Q0, U1 and U2 with such a property. Then one thing should lead to the other, SOL we usually assume that there is only one U. Especially in Second Order Arithmetic the assumption is that U =3D =CF=89, which would explain why SOL would give another result than FOL + Set Theory. Mild Shock schrieb am Samstag, 10. Juni 2023 um 20:27:59 UTC+2: > Whats the domain of r1? Can you prove r1 =E2=8A=86 U x U?=20 > Please show us. Currently r1 can be anything. > Dan Christensen schrieb am Samstag, 10. Juni 2023 um 20:21:24 UTC+2:=20 > > On Saturday, June 10, 2023 at 2:06:27=E2=80=AFPM UTC-4, Mild Shock wrot= e:=20 > > > Note there is only one domain U in SOL, which appears=20 > > > first order and second order, i.e. first order as for example x e U= =20 > > > and second order as for example R =E2=8A=86 U x U.=20 > > >=20 > > > On the other hand your formula has not a single domain U.=20 > > I have only one domain, the set u here, the domain of quantification fo= r each quantifier.=20 > > ALL(a):[a in u =3D> EXIST(b):[b in u & (a,b) in r1]]=20 > >=20 > > & ALL(a):[a in u =3D> EXIST(b):[b in u & (a,b) in r21]]=20 > >=20 > > & ALL(x):ALL(x'):ALL(y):ALL(y'):[x in u & x' in u & y in u & y' in u = =3D> [(x,y) in r1 & (x',y') in r2 =3D> Q(x,x',y,y')]]=20 > > > I don't know what your formula does. Its surely not Terrence=20 > > > Taos formula.=20 > > In what way is it inconsistent with his statement:=20 > > "For every x and x=E2=80=99, there exists a y depending only on x and a= y=E2=80=99 depending only on x=E2=80=99 such that Q(x,x=E2=80=99,y,y=E2=80= =99) is true"=20 > > Dan=20 > >=20 > > Download my DC Proof 2.0 freeware at http://www.dcproof.com=20 > > Visit my Math Blog at http://www.dcproof.wordpress.com