New essay on Goedel's Incompleteness Theorem
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From: Nam Nguyen on 29 Sep 2009 22:33 Aatu Koskensilta wrote: > Nam Nguyen <namducnguyen(a)shaw.ca> writes: > >> There might have been more than one points that I might have >> "adamantly insisted" in the past. So unless you spell it out I >> couldn't know what we're really talking about here. > > David's statement, that if T and T* have the same axioms they're the > same theory, implies that the language of a theory is determined by > its axioms. By Shoenfield, a theory is determined by axioms. So, L(T) is determined by T's axioms. Why do *you* think that's incorrect? Why do you think people say "_the_ language of a theory", as opposed to "a 1st order language"? > Surely you recall the fun debate about that from a while > back. > Yes. I remember it now. Your guys were wrong!
From: Nam Nguyen on 30 Sep 2009 00:51 Scott H wrote: > On Sep 29, 12:26 pm, Aatu Koskensilta <aatu.koskensi...(a)uta.fi> wrote: >> You will of course look for discussion wherever you like. Peculiar >> outbursts like this are best avoided. > > I did not write in the tone of an outburst, though this world > collectively seems to try to give them to me. It comes as a pinch in > my heart, followed by auditory hallucinations and, eventually, > retching. I'd venture to say that over five years, my suffering has > been comparable to being shot one hundred times. > > And yet people still try to pull my strings, thinking they haven't > gone too far. The beauty a of mathematical truth doesn't require a mass of people to appreciate it. If - but only if - one sees a mathematics/reasoning truth, one doesn't really need anybody else to appreciate it. But such a truth got to be genuinely correct; and so far you've not demonstrated so.
From: Jesse F. Hughes on 30 Sep 2009 06:14 Nam Nguyen <namducnguyen(a)shaw.ca> writes: > Aatu Koskensilta wrote: >> Nam Nguyen <namducnguyen(a)shaw.ca> writes: >> >>> There might have been more than one points that I might have >>> "adamantly insisted" in the past. So unless you spell it out I >>> couldn't know what we're really talking about here. >> >> David's statement, that if T and T* have the same axioms they're the >> same theory, implies that the language of a theory is determined by >> its axioms. > > By Shoenfield, a theory is determined by axioms. So, L(T) is determined > by T's axioms. Oh, good Lord, not that again! Shoenfield does *not* claim L(T) is determined by its axioms. Here's the quote, once again: "We consider a language to be completely specified when its symbols and formulas are specified. This makes a language a purely syntactical object. Of course, most of our languages will have a meaning (or several meanings); but the meaning is not considered to be part of the language. We shall designate the language of a formal system F by L(F)." "The next part of a formal system consists of its /axioms/. Our only requirement on these is that each axiom shall be a formula of the language of the formal system." [p. 4 in my copy of Shoenfield] The first part of a formal system is the language. The *next* part is the axioms. Clearly, then, the axioms do not determine the language. -- "Mathematicians are rather important in the infrastructures of many organizations that protect civilization. I've determined that they are a consistent security risk, and seem to have other agendas, other loyalties beyond loyalty to their respective nations." -- James Harris
From: Jack Campin - bogus address on 30 Sep 2009 06:26 >> Nobody in their right mind cares about Google Groups ratings. > Only insane people care what others think of them? That doesn't > seem right. You'd have to be out of your mind to want to read Usenet via Google, and no other server pays any attention to their rating system - the rating information is not shared. The rest of us don't even see it. ==== j a c k at c a m p i n . m e . u k === <http://www.campin.me.uk> ==== Jack Campin, 11 Third St, Newtongrange EH22 4PU, Scotland == mob 07800 739 557 CD-ROMs and free stuff: Scottish music, food intolerance, and Mac logic fonts ****** I killfile Google posts - email me if you want to be whitelisted ******
From: Jesse F. Hughes on 30 Sep 2009 06:40
Marshall <marshall.spight(a)gmail.com> writes: > On Sep 29, 9:26 am, Aatu Koskensilta <aatu.koskensi...(a)uta.fi> wrote: >> >> Nobody in their right mind cares about Google Groups ratings. > > Only insane people care what others think of them? That doesn't > seem right. You have some sort of delusion that Google's rating system accurately reflects general opinion about your character? I gotta say, *that's* crazy. -- "[Y]ou never understood the real role of mathematicians. The position is one of great responsibility and power. [...] You people have no concept of what it means to be an actual mathematician versus pretending to be one, dreaming you understand." -- James S. Harris |