Re: Completeness & consistency, was: A sad thread
Eliah Kagan wrote: Duncan Patton a Campbell wrote: When you mathematically formalize a physical theory, you use a system that is powerful enough to formulate the theory's physical claims. That system is also going to be powerful enough to formulate all sorts of other stuff, things that have nothing to do with physics. Suppose one extends Zermelo-Frankel Set Theory (or designs a system that works like it) to do physics. I think the undecidability of the Axiom of Choice would be a poor example in this specific case because I believe the Axiom of Choice is used in topology, and therefore may be valuable in doing physics. Be careful. Axiom of choice is used all over but theoretically one could pin point results that depend on it. Very careful mathematicians will do. Some results like existence of non-measurable sets in classical Lebesgue measure theory can not be proved without Axiom of choice. (that fact is the very hard theorem in its own right) But the Continuum Hypothesis--the claim that there are no sets cardinally bigger than the set of integers but cardinally smaller than the set of reals- That is NOT true. Late Poul Cohen got a Filds medal for proving something like this (I will get myself now in to BIG trouble because of imprecise statements I am making to describe the results) Roughly, the continuum hypothesis in mathematics has the same position as the fifth euclidean postulate i.e. You could build consistent mathematics if you assume that there are no sets of bigger cardinality than alef0 and less than c. (Like assuming Euclidean axiom of parallelility you get Euclidean geometry.) You could also build consistent mathematics assuming that there are sets with cardinality bigger than alef0 and less than c roughly corresponding to the case of geometry of Lobachevsky-Bolyai when an alternative axiom of parallelility is added on the top of axioms of incidence, congruence, order, and continuity to build geometry of Lobachevsky. The further discussion is definitely out-side of the scope of this mailing list. -probably has nothing to do with physics. So your physical formalization can be used to formulate the Continuum Hypothesis, which is undecidable, but since that's not actually *about* physics, it doesn't make your system incomplete as a formalization of physics. Now, often ideas in abstract mathematics turn out to be useful in applied fields, and I would not discount the possibility that transfinite arithmetic might turn out to be applicable in physics, though I can conceive of no way that it would. But so long as there is *some* formulable question arising out of the math you use to do your physics that is itself not physically important, you can have a formal system that, as a formulation of physics, is complete and consistent. -Eliah I also notice very uncareful use of the Russell paradox on this mailing list. Godel's results are invoked very uncarefully as I noticed earlier and as the content of the above mail points out by giving little bit more details about formal systems used in mathematics. Kind Regards to Everyone Predrag
Re: Completeness & consistency, was: A sad thread
On Mon, 7 Jan 2008 20:09:47 -0500 "Eliah Kagan" <[EMAIL PROTECTED]> wrote: > For the record, I do not believe that there is necessarily no complete > and entirely correct *physical* theory "out there" to be discovered. Is it not the case that you can show that you cannot prove a system both complete and consistent? ---> doesn't mean there is no "God" just that you cannot prove it. Dhu
Re: Completeness & consistency, was: A sad thread
Reid Nichol wrote: > Point of fact, Mathematics has been proven to have the option to be > either consistent OR complete. Axioms: Empty set -- this is consistent. There is nothing there to be inconsistent. Foo and NOT-Foo -- this is complete From these two little axioms all true statements can be derived. (and all the false ones too) So what? Going from those trivial cases to the whole of 19th century math is more than fits into one mind. This is now the 21st century.
Re: Completeness & consistency, was: A sad thread
Predrag Punosevac wrote: Predrag Punosevac wrote: Ingo Schwarze wrote: Reid Nichol wrote on Mon, Jan 07, 2008 at 12:02:19AM -0800: Duncan Patton a Campbell <[EMAIL PROTECTED]> wrote: "Eliah Kagan" <[EMAIL PROTECTED]> wrote: (There are also multiple useful, mutually-inconsistent formal systems in both fields.) Provably so? I'd love an example of Math being inconsistent. Quite frankly, I'd be surprised if this is true. Eliah has beautifully demonstrated this for both Mathematics and Physics. What is flabbergasting me about such questions is that these are extremely old facts - essentially, known for more than 70 years - and many people still believe that formal science can be both complete and consistent. http://en.wikipedia.org/wiki/Nicolas_Bourbaki - nicely narrating how the attempt to transform mathematics into a single unified and consistent theory miserable failed http://wiki/G%C3%B6del%27s_incompleteness_theorem - explaining why http://en.wikipedia.org/wiki/Kurt_G%C3%B6del (1906-1978) - "One of the most significant logicians of all time, GC6el's work has had immense impact upon scientific and philosophical thinking in the 20th century, a time when many, such as Bertrand Russell, A. N. Whitehead and David Hilbert, were attempting to use logic and set theory to understand the foundations of mathematics." I would be little bit more careful about dragging the incompletness theorem into the discussion without properly understanding the statement of the theorem, its meaning, and corollaries. The connections that you are trying to make between the incompletness theorem and Burbaki project are very shallow at best and I certainly have not heard them before. Kind Regards, Predrag Department of Mathematics University of Arizona P. S. I am no expert on mathematical logic but definitely know little bit better than your average bystander. Still, many people appearantly never heard of the problems he described, even though we are now well into the 3rd millenium... Reply-To: poster set, we are *terribly* off-topic.
Re: Completeness & consistency, was: A sad thread
Predrag Punosevac wrote: Ingo Schwarze wrote: Reid Nichol wrote on Mon, Jan 07, 2008 at 12:02:19AM -0800: Duncan Patton a Campbell <[EMAIL PROTECTED]> wrote: "Eliah Kagan" <[EMAIL PROTECTED]> wrote: (There are also multiple useful, mutually-inconsistent formal systems in both fields.) Provably so? I'd love an example of Math being inconsistent. Quite frankly, I'd be surprised if this is true. Eliah has beautifully demonstrated this for both Mathematics and Physics. What is flabbergasting me about such questions is that these are extremely old facts - essentially, known for more than 70 years - and many people still believe that formal science can be both complete and consistent. http://en.wikipedia.org/wiki/Nicolas_Bourbaki - nicely narrating how the attempt to transform mathematics into a single unified and consistent theory miserable failed http://wiki/G%C3%B6del%27s_incompleteness_theorem - explaining why http://en.wikipedia.org/wiki/Kurt_G%C3%B6del (1906-1978) - "One of the most significant logicians of all time, GC6el's work has had immense impact upon scientific and philosophical thinking in the 20th century, a time when many, such as Bertrand Russell, A. N. Whitehead and David Hilbert, were attempting to use logic and set theory to understand the foundations of mathematics." I would be little bit more careful about dragging the incompletness theorem into the discussion without properly understanding the statement of the theorem, its meaning, and corollaries. The connections that you are trying to make between the incompletness theorem and Burbaki project are very shallow at best and I certainly have not heard them before. Kind Regards, Predrag Department of Mathematics University of Arizona P. S. I am no expert on mathematical logic but definitely know little bit better than your average bystander. Still, many people appearantly never heard of the problems he described, even though we are now well into the 3rd millenium... Reply-To: poster set, we are *terribly* off-topic.
Re: Completeness & consistency, was: A sad thread
Setting the record straight because I can't in good conscience have such nonsense sitting in public not refuted. You haven't pointed to an instance of an inconsistency in Mathematics. Which, I'll point out, was what I explicitly asked for. Basically, you're referencing a choice in Mathematics that we have, that we can go for either consistent OR complete. And you seem to be saying that Mathematics is neither? You don't seem to understand the issues involved and/or have incomplete knowledge/understanding of the history of Mathematics. "What is flabbergasting me" is that you haven't a clue and/or lack the attention to detail to answer questions that were explicitly asked. Point of fact, Mathematics has been proven to have the option to be either consistent OR complete. From what I've learned, we've chosen to be consistent. Which, IMO, was a very very wise decision. If you don't agree, point to a specific instance of an inconsistency in modern Mathematics. Eliah Kagan wrote: """ Tony Abernethy's example of non-Euclidean geometries being inconsistent with Euclidean geometry is a good one. """ This is so very wrong it isn't even funny. You deserve to be ridiculed publicly into oblivion for making such nonsensical statements. I mean seriously, Euclidean geometry assumes a perfectly flat plain whereas non-Eucliden geometry does not. Do you think they'll go in different directions? Do you think that it is even remotely reasonable to compare the conclusions after such a divergence without considering limiting cases? Though a couple of the statements you make after the above statement are reasonable, you take it in a direction and make conclusions that aren't (meaningless?!?!?). This mixture of reasonable with unreasonable, including such logic makes such statements erroneously compelling, which is very dangerous for those learning this stuff for the first time. Please stay away from making any statements on the foundations of Mathematics in the future as you seem to be at least partially ill equipped to speak on this topic. In other words, you have enough knowledge and speak well enough to convince students/others and perhaps yourself, but at the same time, lack the necessary knowledge/logic to come to reasonable conclusions. regards, Reid --- Ingo Schwarze <[EMAIL PROTECTED]> wrote: > Reid Nichol wrote on Mon, Jan 07, 2008 at 12:02:19AM -0800: > > Duncan Patton a Campbell <[EMAIL PROTECTED]> wrote: > >> "Eliah Kagan" <[EMAIL PROTECTED]> wrote: > >> > >>> (There are also multiple useful, > >>> mutually-inconsistent formal systems in both fields.) > >> > >> Provably so? > > > > I'd love an example of Math being inconsistent. > > Quite frankly, I'd be surprised if this is true. > > Eliah has beautifully demonstrated this for both Mathematics > and Physics. What is flabbergasting me about such questions > is that these are extremely old facts - essentially, known for > more than 70 years - and many people still believe that formal > science can be both complete and consistent. > > http://en.wikipedia.org/wiki/Nicolas_Bourbaki > - nicely narrating how the attempt to transform mathematics >into a single unified and consistent theory miserable failed > > http://wiki/G%C3%B6del%27s_incompleteness_theorem > - explaining why > > http://en.wikipedia.org/wiki/Kurt_G%C3%B6del (1906-1978) > - "One of the most significant logicians of all time, GC6el's work > has had immense impact upon scientific and philosophical thinking > in the 20th century, a time when many, such as Bertrand Russell, > A. N. Whitehead and David Hilbert, were attempting to use logic > and set theory to understand the foundations of mathematics." > > Still, many people appearantly never heard of the problems he > described, even though we are now well into the 3rd millenium... > > Reply-To: poster set, we are *terribly* off-topic. > Looking for last minute shopping deals? Find them fast with Yahoo! Search. http://tools.search.yahoo.com/newsearch/category.php?category=shopping
Re: Completeness & consistency, was: A sad thread
Ingo Schwarze wrote: > Eliah has beautifully demonstrated this for both Mathematics > and Physics. What is flabbergasting me about such questions > is that these are extremely old facts - essentially, known for > more than 70 years - and many people still believe that formal > science can be both complete and consistent. For the record, I do not believe that there is necessarily no complete and entirely correct *physical* theory "out there" to be discovered. Such a theory, when formalized mathematically, would have to allow well-formed undecidable statements. But those statements would not necessarily be *about* physical reality, any more than an applied system that modestly extends Zermelo-Frankel set theory (with or without the Axiom of Choice) to contain axioms about voter demographics is incomplete with respect to classification of voters due to the undecidability of the Continuum Hypothesis. In other words, the "complete physics" would actually use only part of the mathematical framework used to formalize it. It's also possible that a complete and correct theory of physics will be discovered and be accepted, and still not be formalized mathematically. Quantum Electrodynamics is probably the most successful scientific theory ever (in terms of the number, consistency, and precision of its predictions), and yet as far as I know it has still not been formalized in the mathematical sense. -Eliah
Re: Completeness & consistency, was: A sad thread
Ingo Schwarze wrote: Reid Nichol wrote on Mon, Jan 07, 2008 at 12:02:19AM -0800: Duncan Patton a Campbell <[EMAIL PROTECTED]> wrote: "Eliah Kagan" <[EMAIL PROTECTED]> wrote: (There are also multiple useful, mutually-inconsistent formal systems in both fields.) Provably so? I'd love an example of Math being inconsistent. Quite frankly, I'd be surprised if this is true. Eliah has beautifully demonstrated this for both Mathematics and Physics. What is flabbergasting me about such questions is that these are extremely old facts - essentially, known for more than 70 years - and many people still believe that formal science can be both complete and consistent. http://en.wikipedia.org/wiki/Nicolas_Bourbaki - nicely narrating how the attempt to transform mathematics into a single unified and consistent theory miserable failed http://wiki/G%C3%B6del%27s_incompleteness_theorem - explaining why http://en.wikipedia.org/wiki/Kurt_G%C3%B6del (1906-1978) - "One of the most significant logicians of all time, GC6el's work has had immense impact upon scientific and philosophical thinking in the 20th century, a time when many, such as Bertrand Russell, A. N. Whitehead and David Hilbert, were attempting to use logic and set theory to understand the foundations of mathematics." Still, many people appearantly never heard of the problems he described, even though we are now well into the 3rd millenium... Reply-To: poster set, we are *terribly* off-topic. I would be little bit more careful about dragging the incompletness theorem into the discussion without properly understanding the statement of the theorem, its meaning, and corollaries. The connections that you are trying to make between the incompletness theorem and Burbaki project are very shallow at best and I certainly have not heard them before. Kind Regards, Predrag Department of Mathematics University of Arizona P. S. I am no expert on mathematical logic but definitely know little bit better than your average bystander.
