Usefulness of statements contradicted by AC
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From: Marc Alcobé García on 28 Jan 2010 02:59 On 28 ene, 03:31, Newberry <newberr...(a)gmail.com> wrote: > On Jan 27, 12:45 am, Aatu Koskensilta <aatu.koskensi...(a)uta.fi> wrote: > > > Marc Alcob Garc a <malc...(a)gmail.com> writes: > > > > Could anyone provide some examples of the usefulness of these > > > investigations, specially those with applications to ZFC and its > > > extensions? > > > The obvious example is the axiom of determinacy, which holds "in the > > clarity of an inner model" given suitable large large cardinals, and its > > use in descriptive set theory. > > Could you expand on the "usefulness" of this axiom? The question is about the kind of applications Levy had in mind when he was writing the quoted paragraph, "useful" must be, in this context, understood only in the sense of "for these aplications". I am sure the axiom of determinacy satisfies this requirement. However, in my ignorance, I cannot imagine how the study of theories not satisfying AC can help to obtain a better picture of ZFC or its extensions.
From: Herman Jurjus on 28 Jan 2010 03:31 Marc Alcob� Garc�a wrote: > On 28 ene, 03:31, Newberry <newberr...(a)gmail.com> wrote: >> On Jan 27, 12:45 am, Aatu Koskensilta <aatu.koskensi...(a)uta.fi> wrote: >> >>> Marc Alcob Garc a <malc...(a)gmail.com> writes: >>>> Could anyone provide some examples of the usefulness of these >>>> investigations, specially those with applications to ZFC and its >>>> extensions? >>> The obvious example is the axiom of determinacy, which holds "in the >>> clarity of an inner model" given suitable large large cardinals, and its >>> use in descriptive set theory. >> Could you expand on the "usefulness" of this axiom? > > The question is about the kind of applications Levy had in mind when > he was writing the quoted paragraph, "useful" must be, in this > context, understood only in the sense of "for these aplications". > > I am sure the axiom of determinacy satisfies this requirement. > However, in my ignorance, I cannot imagine how the study of theories > not satisfying AC can help to obtain a better picture of ZFC or its > extensions. Can the study of p-adic analysis lead to a better understanding of ordinary analysis? Mmmyes, sort of. How? Because it makes you become more aware of where and how you need certain properties of R. AC allows us to prove certain mathematical results 'too easily'. It's at least worth an investigation on what can be done without it. -- Cheers, Herman Jurjus
From: Marc Alcobé García on 28 Jan 2010 04:44 On 28 ene, 09:31, Herman Jurjus <hjm...(a)hetnet.nl> wrote: > Marc Alcobé García wrote: > > On 28 ene, 03:31, Newberry <newberr...(a)gmail.com> wrote: > >> On Jan 27, 12:45 am, Aatu Koskensilta <aatu.koskensi...(a)uta.fi> wrote: > > >>> Marc Alcob Garc a <malc...(a)gmail.com> writes: > >>>> Could anyone provide some examples of the usefulness of these > >>>> investigations, specially those with applications to ZFC and its > >>>> extensions? > >>> The obvious example is the axiom of determinacy, which holds "in the > >>> clarity of an inner model" given suitable large large cardinals, and its > >>> use in descriptive set theory. > >> Could you expand on the "usefulness" of this axiom? > > > The question is about the kind of applications Levy had in mind when > > he was writing the quoted paragraph, "useful" must be, in this > > context, understood only in the sense of "for these aplications". > > > I am sure the axiom of determinacy satisfies this requirement. > > However, in my ignorance, I cannot imagine how the study of theories > > not satisfying AC can help to obtain a better picture of ZFC or its > > extensions. > > Can the study of p-adic analysis lead to a better understanding of > ordinary analysis? Mmmyes, sort of. How? Because it makes you become > more aware of where and how you need certain properties of R. > > AC allows us to prove certain mathematical results 'too easily'. > It's at least worth an investigation on what can be done without it. > > -- > Cheers, > Herman Jurjus- Ocultar texto de la cita - > > - Mostrar texto de la cita - This answer is not satisfactory for two reasons: 1. An investigation of what can be done without it is not exactly coincident with an investigation of what can be done with a negation of it. 2. It seems that the applications involved are only of the heuristic kind.
From: Frederick Williams on 28 Jan 2010 09:05 Marc Alcob� Garc�a wrote: > > On 28 ene, 09:31, Herman Jurjus <hjm...(a)hetnet.nl> wrote: > > Marc Alcob� Garc�a wrote: > > > On 28 ene, 03:31, Newberry <newberr...(a)gmail.com> wrote: > > >> On Jan 27, 12:45 am, Aatu Koskensilta <aatu.koskensi...(a)uta.fi> wrote: > > > > >>> Marc Alcob Garc a <malc...(a)gmail.com> writes: > > >>>> Could anyone provide some examples of the usefulness of these > > >>>> investigations, specially those with applications to ZFC and its > > >>>> extensions? > > >>> The obvious example is the axiom of determinacy, which holds "in the > > >>> clarity of an inner model" given suitable large large cardinals, and its > > >>> use in descriptive set theory. > > >> Could you expand on the "usefulness" of this axiom? > > > > > The question is about the kind of applications Levy had in mind when > > > he was writing the quoted paragraph, "useful" must be, in this > > > context, understood only in the sense of "for these aplications". > > > > > I am sure the axiom of determinacy satisfies this requirement. > > > However, in my ignorance, I cannot imagine how the study of theories > > > not satisfying AC can help to obtain a better picture of ZFC or its > > > extensions. > > > > Can the study of p-adic analysis lead to a better understanding of > > ordinary analysis? Mmmyes, sort of. How? Because it makes you become > > more aware of where and how you need certain properties of R. > > > > AC allows us to prove certain mathematical results 'too easily'. > > It's at least worth an investigation on what can be done without it. > > > > -- > > Cheers, > > Herman Jurjus- Ocultar texto de la cita - > > > > - Mostrar texto de la cita - > > This answer is not satisfactory for two reasons: > > 1. An investigation of what can be done without it is not exactly > coincident with an investigation of what can be done with a negation > of it. If all subsets of R are measurable then AC is false. > 2. It seems that the applications involved are only of the heuristic > kind. -- Mathematics is a part of physics. Physics is an experimental science, a part of natural science. Mathematics is the part of physics where experiments are cheap. (V.I. Arnold)
From: Marc Alcobé García on 28 Jan 2010 12:05
> If all subsets of R are measurable then AC is false. Do you refer to Lebesgue measurable subsets? The existence of non- Lebesgue measurable sets can be proved directly from ZFC applying Zorn's Lemma. |