Some basic.questions. When you say PA, do you mean the set of all theorems entailed by the axioms of Peano arithmetic? Does this include the true (relative to PA of course) wffs that are not provable from PA alone?
How can it be that PA+con(I) can prove its own consistency because it is inconsistent? Do you mean that it is consistent relative to itself but inconsistent in the "metalanguage"? Or else how can we have it be both consistent and inconsistent? This is probably way off the subject (hope that's ok with you): isn't all mathematical truth relative to the formal system one is operating in? "all mathematical truth is relative to the formal system one is operating in" is relative to the formal system I call "rational discourse" in which "mathematical discourse" and "machine-level discourse" are sub-systems. On Mon, Jan 27, 2014 at 7:41 AM, Bruno Marchal <[email protected]> wrote: > > On 27 Jan 2014, at 16:12, Brian Tenneson wrote: > > Yes, some day a computer might be able to figure out that the set of > rationals is not equipollent to the set of real numbers. > > > A Lôbian machine like ZF can do that already. > > > > I saw somewhere that using an automated theorem prover, one of Godel's > incompleteness theorems was proved by a computer. > > > Boyer and Moore, yes, but that is not conceptuallydifferent than ZF, > except that the Boyer-Moore machine uses more efficient sort of AI path. > > Gödel discovered that PM already proves his own incompleteness theorem. > All Lôbian machine proves their own Gödel's theorem. They all prove "If I > am consistent, then I can't prove my consistency". > > > > The question I raised initially was this: will there ever be a machine or > human who can correctly answer all questions with a mathematical theme that > have answers? > > > All? No, for any machine i in the phi_i. > But that is less clear for evolving machines, whose evolution rule is not > part of the program of the machine. Of course, at each moment of her > "life", she will be incomplete, but if her evolution is enough "non > computable", or using some special oracle, it might be that the machine > will generate the infinitely many truth of arithmetic, but not in any > provable way. > > > I didn't think so in my original post but now I'm starting to wonder. > It's the existence of undecidable statements that would probably lead to > the machine or human not being able to do it in general. This reminds me > of the halting problem. > > > Those are related. Undecidable is always relative. Consistent(PA) is not > provable by PA, but is provable in two lines in the theory PA+con(PA). Of > course PA+con(PA) cannot prove con(PA+con(PA)). > > What about PA+con(I), with I = PA+con(I). It exists as we can eliminate > the occurence of I by using the Dx = "xx" method. Well, in this case > PA+con(I) can prove its own consistency, but only because it is actually > inconsistent. > > > The good news is we will never run out of mathematical territory to think > about. > > > Yes indeed, even if we confine ourselves on elementary (first order) > arithmetic. There is an infinity of surprises there. > > Bruno > > > > > On Mon, Jan 27, 2014 at 6:58 AM, Gabriel Bodeen <[email protected]>wrote: > >> FWIW, under the usual definitions, the rationals are enumerable and so >> are a smaller set than the reals. I'd suppose that if people can figure >> that out with our nifty fleshy brains, then a well-designed computer brain >> could, too. >> -Gabe >> >> >> On Friday, January 24, 2014 1:23:40 AM UTC-6, Brian Tenneson wrote: >>> >>> There are undecidable statements (about arithmetic)... There are true >>> statements lacking proof. There are also false statements about arithmetic >>> the proof of whose falsehood is impossible; not just impossible for you and >>> me but for a computer of any capacity or other forms of rational >>> processing. We'll never have a computer, then, that will work as a >>> mathematically-omniscient device. By that I mean a computer such that every >>> question that has a mathematically-oriented theme having an answer >>> truthfully can be answered by such a device. Calculators demonstrate the >>> concept but are clearly not mathematically-omniscient: you ask the >>> calculator what is 2+2 and press a button and "presto" you get an answer. >>> What I'm talking about would be questions like "is the set of rational >>> numbers equal in size to the set of real numbers", and get the correct >>> answer. So we will never have such a computer no matter what its capacities >>> are, even if computer encompasses the entire human brain. Unfortunately, >>> that means that even for humans, we will never know everything about math. >>> Unless something weird would happen and we suddenly had infinite >>> capacities; that might change the conclusions. >>> >> >> -- >> 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/xabA-SKxTHM/unsubscribe >> . >> To unsubscribe from this group and all its topics, send an email to >> [email protected]. >> To post to this group, send email to [email protected]. >> Visit this group at http://groups.google.com/group/everything-list. >> For more options, visit https://groups.google.com/groups/opt_out. >> > > > -- > 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 [email protected]. > > To post to this group, send email to [email protected]. > Visit this group at http://groups.google.com/group/everything-list. > For more options, visit https://groups.google.com/groups/opt_out. > > > http://iridia.ulb.ac.be/~marchal/ > > > > -- > 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/xabA-SKxTHM/unsubscribe. > To unsubscribe from this group and all its topics, send an email to > [email protected]. > To post to this group, send email to [email protected]. > Visit this group at http://groups.google.com/group/everything-list. > For more options, visit https://groups.google.com/groups/opt_out. > -- You received this message because you are subscribed to the Google Groups "Everything List" group. 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