On Wednesday, June 3, 2015 at 2:16:31 AM UTC-7, Bruno Marchal wrote: > > > On 02 Jun 2015, at 20:10, Brian Tenneson wrote: > > Grammatical systems just might be the type of thing Tegmark is looking for > that is a framework for all mathematical structures... or at least a large > class of them. > > I am still exploring the idea of grammatical system induction. > > > I am not sure what you mean by grammatical induction. Is that not > equivalent with the omega-induction principles, like with PA? >
It is described in this document: https://docs.google.com/document/d/1amDb4Yti4egpKfcO2oLcnGAH8UpC8_tKb7ivuH3AT7A/edit?usp=sharing > > > I believe it can be used to provide an induction principle that allows > one to prove something about all sets in ZFC (or any set theory). > > > You will need transfinite induction. > > But with the usual (omega) induction, you get already Löbianity, and the > self-reference logics will not been changed with addition. > > > > Applying the general grammatical system induction to formal systems, I > believe there is a way to prove something about all theorems within a > formal system, > > > Yes, PA can prove that ZF proves things. For proving that a formal system > does not prove something, you will need strionger systems, and by > incompleteness such negative statements cannot be axiomatized in once > system. > > > > perhaps providing a little insight into truth in general. > > Also, an induction principle applies to all proofs if one wants to prove > something about all proofs in a formal system. > > The document in the first post has been updated to include all of this. > There are some words I need to change so just notice the essence... > > Any feedback is appreciated! > > > > If you assume computationalism, simple (omega) induction is enough to get > the machine psychology and theology, and to justify why machines will build > more and more induction rules, but none will get the "whole" truth, which > is beyond axiomatization and formalization (even assuming computationalism). > > You seem to try to do what the logicians have already done. You might > study the little book by Torkel Franzen on the "Inexhaustibility", which > makes rather clear the elusive character of truth. > > > I realized later that what I've done was done roughly in the 1930's. But no one has connected the notion of grammatical system to Max Tegmark's Level IV multiverse idea as far as I know. > Bruno > > > > > -- > You received this message because you are subscribed to the Google Groups > "Everything List" group. > To unsubscribe from this group and stop receiving emails from it, send an > email to everything-li...@googlegroups.com <javascript:>. > To post to this group, send email to everyth...@googlegroups.com > <javascript:>. > Visit this group at http://groups.google.com/group/everything-list. > For more options, visit https://groups.google.com/d/optout. > > > http://iridia.ulb.ac.be/~marchal/ > > > > -- You received this message because you are subscribed to the Google Groups "Everything List" group. To unsubscribe from this group and stop receiving emails from it, send an email to everything-list+unsubscr...@googlegroups.com. To post to this group, send email to everything-list@googlegroups.com. Visit this group at http://groups.google.com/group/everything-list. For more options, visit https://groups.google.com/d/optout.
