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From: scattered on 13 Jan 2010 13:06 On Jan 13, 12:44 pm, Math1723 <anonym1...(a)aol.com> wrote: > On Jan 13, 11:46 am, scattered <still.scatte...(a)gmail.com> wrote: > > > [snip] > > > > Again, I am curious as to how you apply this principle to my analogy > > > with Geometry. > > > [snip] > > > The difference is that standard axiomizations of the real numbers are > > categorical - meaning that they have only one model (e.g. Dedekind- > > complete ordered fields are unique up to a field isomporhpism). > > Yes, but "Dedekind-completeness" is not itself a model-independent > concept. You can have two models of R, say M1 and M2, with reals R1 > and R2 respectively, in which R1 and R2 are not isomorphic. In M1, R1 > is necessarily M1-Dedekind-complete and in M2, and R2 is necessarily > M2-Dedekind-complete. Yet (depending on M1 and M2), neither would be > "Dedekind complete" in the other model. > > It seems that all you have done here is replace the concept of a > "real" R with the concept of "real" Dedekind Completeness. > > Nothing you have said persuades me out of thinking it's *possible* > that there are more than one concepts of real numbers, each equally > valid and real, and each with differing values for 2^Aleph_0. > > > The first four postulates of Euclid do *not* specify a categorical > > theory. What is happenening with Euclidean vs. NonEuclidean > > geometries is much more akin to what is happenening with Abelian > > vs. nonAbelian groups. It doesn't make sense to ask if *the* group > > is really abelian, since there is no "the group". Similarly it does > > not make sense to say if "the geometry" is really Euclidean - since > > there isn't any unique geometry. > > I admit my ignorance of Category Theory, so I cannot respond to your > point on that. However, a century or two ago, there were people > arguing passionately that there is indeed a unique geometry, in much > the same way you argue that there is a unique concept of R, and saying > rather ugly things about proponents of Non-Euclidean Geometry. You > say now that there isn't a unique Geometry. How can you be certain > that a hundred years from now, mathematicians will agree that R isn't > unique either. > > > But there *is* a unique set of real numbers (up to > > isomorphism - which preserves cardinality) hence it makes perfect > > sense to ask if the real numbers really satisfies CH. > > The very fact that there are models of R with differing answers to CH > seems to me to make such an assertion a bit hasty. > I guess I'm a hasty Platonist :) There *aren't* different models of the (second order) axioms of real numbers. There are different models of ZFC in which *their* versions of R behave variously, but in V itself(the universe of all sets) there is up to isomprphism only one set of real numbers. You can study models of ZFC to your hearts content - but that won't tell you (in this case) what is happening in V itself. By the way, my use of "categorical" is the model-theoretic one of having just one model up to isomorphism, and is not directly related to category theory. -scattered
From: Math1723 on 13 Jan 2010 13:28 On Jan 13, 1:06 pm, scattered <still.scatte...(a)gmail.com> wrote: > > I guess I'm a hasty Platonist :) No problem, I enjoy the discussion either way. :-) By the way, are you familiar at all with the philosophical line of mathematicians known as "Intuitionists"? Not only do they deny the Axiom of Choice, they go as far as denying the basic logical Law of the Excluded Middle. In other words, proving that ~A is false does not prove that A is true. I find this rather disconcerting and bizarre. So although I sometimes straddle between Platonism and Formalism, I know I do not come anywhere near that one. (Although Formalistically, I see Intuitionist Mathematics as nothing more than a mathematical theory simply missing a few fundamental axioms.) Thanks for the discussion.
From: David Bernier on 13 Jan 2010 13:58 Math1723 wrote: > On Jan 12, 3:17 pm, scattered <still.scatte...(a)gmail.com> wrote: >>> Assuming ^ means cardinal exponentiation, Aleph_0 ^ Aleph_0 = c where >>> c is the cardinality of the continuum. Whether or not c = Aleph_1 is >>> determined by whether you assume the Continuum Hypotheisis or not. >> You mean I get to determine how big the continuum is? I didn't realize >> I had such power. > > Actually, from a Mathematical Logic point of view, you do. It's > essentially like choosing the polarity of the Parallel Postulate when > defining your Geometry. If you want Euclidean Geometry, you choose to > accept it as a postulate, if you want a Non-Euclidean Geometry, you > accept one of its negations. > > Similarly, there are models of the real numbers in which the contimuum > is Aleph_1, there are models in which it is Aleph_2, and models in > which the continuum is weakly inaccessible. > >> More seriously, I think that it is more accurate to say that the size >> of the continuum is an open problem but that various possibilities are >> all consistent with ZFC. c is what c is. If it is in fact Aleph_1 but >> you assume that it is greater than Aleph_1 then you have made a false >> assumption, even if ZFC is not powerful enough to prove its falseness. > > I guess it's the "in fact" portion of your post I am uncertain about. > Which model of the real numbers is the "real" one? I don't know. Do > you think only one of the geometries is "real"? Do you thinbk that > the Parallel Postulate is either true (or false), but ZFC is not > powerful enough to prove its truth (or falsity)? I don't. Not only > are they equally consistent, they are equally "real" to me. > > In the same way, unless or until we embrace a new axiom which will > uniquely decide the question, our mental models of the reals is too > fuzzy, and seem to embrace a number of possibilities. > > But perhaps I am being too much of a formalist here. Putting on my > Platonist hat, your description is very well stated, and is probably > the better way to describe the situation. > > In any case, we can certainly agree that it is a thought-provoking > problem. Thanks for your input! :-) ( Does V = L ?) Not long ago, I learnt of a famous result of Dana Scott from the 1960's, I think: Thm. (Scott) If there is a measurable cardinal, then V =/= L . Cf.: < http://plato.stanford.edu/entries/set-theory/#8 > section 8, lines 4 and 5. G�del showed that in the constructible universe L, GCH and AC are true. < http://en.wikipedia.org/wiki/Constructible_universe > . Maybe V =/= L is plausible. However, given a generic subset A of the natural numbers, under V = L, A could appear at a level L_{alpha}, where alpha is a largish ordinal ... Measurable cardinals are/would be too big for me to commit to them. According to Woodin, c "is" aleph_2; then V = L is negated. With due respect to Woodin's results, I can't convince myself that V =/= L is true or probably true. David Bernier
From: Leonid Lenov on 13 Jan 2010 14:17 On Jan 13, 6:51 pm, Math1723 <anonym1...(a)aol.com> wrote: > Fair point. Then would you not concede then that since ZFC + CH and > ZFC + ~CH are each equally consistent, that therefore asking about the > "actual" truth of CH is meaningless? What does it mean for CH to be > "actually" true (or "actually" false) when each possibility equally > "exists" (as per your "consistency implies existence")? If both ZFC+CH and ZFC+!CH are consistent and you accept the view consistency implies existence, than the question if CH is "actually" true makes no sense, that is, you lose realism in truth value. Personally, I think that is not terrible thing and consider the question of existence of objects (ontology) more important. If you want to have both realism in ontology (Platonism) and realism in truth value that you might try so do so with Structuralism or something similar...
From: Leonid Lenov on 13 Jan 2010 14:28
On Jan 13, 7:28 pm, Math1723 <anonym1...(a)aol.com> wrote: > By the way, are you familiar at all with the philosophical line of > mathematicians known as "Intuitionists"? Not only do they deny the > Axiom of Choice, they go as far as denying the basic logical Law of > the Excluded Middle. While Intuitionist logic is mathematically correct it is just silly... Their "problem" is that they cannot perceive mathematics separated form the mathematician hence all the methods used, according to them, must be within human capability. Platonist has no such problems :). A typical example of their logic is this: a_2n=0 and a_{2n+1}=1 if Goldbach conjecture is true or a_{2n+1}=0 if Goldbach conjecture is false They maintain that since we do not know if GC is true or false that we cannot say: "a_n either converges or does not converge" Of course, most of us "know" that the previous sentence is true. What their "problem" is, is that they identify truthfulness with "proof- ness". |