Hi Brent, Is there a reason why we only consider the 'standard" models to apply when we are considering foundation theory (or whatever you might denote what we are studying)? Have you ever looked at the Tennenbaum theorem<https://www.google.com/search?q=Tennebaum+theorem&oq=Tennebaum+theorem&aqs=chrome..69i57&sourceid=chrome&espv=210&es_sm=93&ie=UTF-8#es_sm=93&espv=210&q=Tennenbaum+theorem&spell=1> and wondered if it could be weakened to allow for computations that are "outside" of the countable recursive functions?
I suspect that the "standard" model (of arithmetic) is a type of invariant under a strong restricted group of transformations, I do not have the proper language to explain this further at this time. :_( There is more ... Just because we can prove that N x N ->N mapping can represent all possible computations we forget that the proof assumes that the quantity of resources and the number of computational steps is irrelevant. As a researcher of computer science and physics, why is the tractability of a computation given a quantity of resources not relevant in considerations of what, say, the UD* can accomplish? I believe that the quest of a universal rule or "measure" that determines "Everything" is already excluded as a possibility; have we not learned that a measure zero set is? Why not instead look at how computations can "interact" with each other, how they might "evolve", how entropy may be involved, what does it mean for truths to be finitely accessible and infinite truths to be inaccessible, etc. We look like monkeys chasing the weasel 'round the mulberry tree... On Sun, Dec 22, 2013 at 2:55 PM, meekerdb <meeke...@verizon.net> wrote: > On 12/22/2013 5:04 AM, Bruno Marchal wrote: > > > On 21 Dec 2013, at 23:28, meekerdb wrote: > > On 12/21/2013 1:26 AM, Jason Resch wrote: > > If there exists a mathematical theorem that requires >> a countable infinity of integers to represent, no finite version can exist >> of it, in other words, can its proof be found? >> > > If its shortest proof is infinitely long, or if the required axioms > needed to develop a finite proof are infinite, (or instead of infinite, so > large we could not represent them in this universe), then its proof can't > be found (by us), but there is a definite answer to the question. > > > The other possibility is that there are mutually inconsistent axioms that > can be added. As I understand it, that was the point of > http://intelligence.org/wp-content/uploads/2013/03/Christiano-et-al-Naturalistic-reflection-early-draft.pdf > A truth predicate can be defined for arithmetic, > > > In set theory, OK. But not in arithmetic. > > > That's the point of the paper, that a "truth" predicate can be defined for > arithmetic. I put "truth" in scare quotes because the predicate is really > 1-Probability(x)-->0. > > > > And in a set theory (like ZF) you cannot define a set theoretical > predicate for set theoretical truth. > > In ZF+kappa, you can define truth for ZF, but not for ZF+kappa. > (ZF+kappa can prove the consistency of ZF). > > Shortly put, no correct machine can *define* a notion of truth > sufficiently large to encompass all its possible assertions. > > Self-consistency is not provable by the consistent self (Gödel) > Self-correctness is not even definable by the consistent self (Tarski, and > also Gödel, note). > > > > but not all models or arithmetic are the same as the standard model. > > > Computationalism uses only the standard model of arithmetic, except for > indirect metamathematical use like proof of independence of axioms, or for > modeling the weird sentences of G*, like <>[]f (the consistency of > inconsistency). > > > But aren't you assuming the standard model when you refer to the > unprovable truths of arithmetic. If you allowed other models this set > would be ill defined. > > Brent > > -- > You received this message because you are subscribed to a topic in the > Google Groups "Everything List" group. > To unsubscribe from this topic, visit > https://groups.google.com/d/topic/everything-list/1NWmK1IeadI/unsubscribe. > To unsubscribe from this group and all its topics, 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/groups/opt_out. > -- Kindest Regards, Stephen Paul King Senior Researcher Mobile: (864) 567-3099 stephe...@provensecure.com http://www.provensecure.us/ “This message (including any attachments) is intended only for the use of the individual or entity to which it is addressed, and may contain information that is non-public, proprietary, privileged, confidential and exempt from disclosure under applicable law or may be constituted as attorney work product. 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