Aleph_0 ^ Aleph_0
in [Math]
|
Prev: why Peano Axioms never had Frontview & Backview #291 CorrectingMath
Next: Helmholtz equation and the Maximum Principle
From: Math1723 on 12 Jan 2010 17:58 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! :-)
From: scattered on 13 Jan 2010 08:52 On Jan 12, 5:58 pm, Math1723 <anonym1...(a)aol.com> 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 I don't think so. It is a theorem of ZFC that there exists a unique ordinal alpha such that c = Aleph_alpha. Since it is unique I don't get to chose it. > 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. The operative word is "model". If ZFC is consistent then it has a model, which is a set with certain properties. In these models there will be objects corresponding to the set of real numbers, but they won't (in general) *be* the set of real numbers. Some of these models will have the property that they their set of real numbers has size Aleph_1 and some will have the property that their set of real numbers has size > Aleph_1 (I'm not being quite accurate here since cardinality *in the model* need not be the same as genuine cardinality as the Lowenheim-Skolem paradox illustrates). Since all this takes place on the level of models it doesn't tell you much if anything about how the set of real numbers actually behaves. Either there exists an uncountable set of real numbers whose size is less than the continuum or there doesn't. If there doesn't then the method of forcing doesn't allow you to magically create one, it just allows you to construct a *model* in which there is one. Models != reality, even in the universe of sets. > > > 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. I'm an unrepentant Platonist. I have good company; here is a quote from Goedel: "For if the meanings of the primitive terms of set theory are accepted as sound, it follows that the set-theoretical concepts and theorems describe some well-determined reality, in which Cantor's conjecture must be either true or false. Hence its undecidability from the axioms being assumed today can only mean that the axioms do not contain a complete description of that reality." (quoted in Penelope Maddy's book "Realism in Mathematics", which is the most sophisticated defense of a form of Platonism that I am aware of. She also published an article in two parts in the Journal of Symbolic Logic sometimes in the late 80s/ early 90s called "Believing the Axioms" which is well worth reading). > In any case, we can certainly agree that it is a thought-provoking > problem. Yes, it is. Sometimes I wish that I had pursued set theory more in graduate school. Thanks for your input! :-) -scattered
From: Leonid Lenov on 13 Jan 2010 10:28 On Jan 13, 2:52 pm, scattered <still.scatte...(a)gmail.com> wrote: > Goedel: "For if the meanings of the primitive terms of set theory > are accepted as sound, it follows that the set-theoretical concepts > and theorems describe some well-determined reality, in which Cantor's > conjecture must be either true or false. Hence its undecidability from > the axioms being assumed today can only mean that the axioms do not > contain a complete description of that reality." Beautifully said. > Maddy's book "Realism in Mathematics", which is the most sophisticated > defense of a form of Platonism that I am aware of. Now I have to have this book :). It will also give me more arguments with which to defend Platonism form all those "heretics". Why are there some people against Platonism will never be clear to me...
From: Math1723 on 13 Jan 2010 11:28 On Jan 13, 8:52 am, scattered <still.scatte...(a)gmail.com> wrote: > > The operative word is "model". If ZFC is consistent then it has a > model, which is a set with certain properties. In these models there > will be objects corresponding to the set of real numbers, but they > won't (in general) *be* the set of real numbers. This assumes that only one can *be* the real numbers. Back to my analogy with Geometry, without declaring an axiom on the Parallel Postulate, Euclidean, Elliptic and Hyperbolic Geometries are each models of this limited set of axioms. Yet, a number of propositions associated with PP remain undecidable and (expectedly) yield different results in each model. Maybe you're right that there is only one "real" set of reals, and we merely haven't come up with the appropriate axioms to decide CH. However, you are ignoring the possibility that there are more than one possible "real" sets of reals numbers (just as centuries past, geometers ignored this possibility). > Since all this takes place on the level of models it doesn't tell you > much if anything about how the set of real numbers actually behaves. It's the word "actually" that I call into question. To me, it's much like the the question "Is Geometry ACTUALLY Euclidean or Non- Euclidean"? > Either there exists an uncountable set of real numbers whose size is > less than the continuum or there doesn't. "Given a line and a point off that line, either 0, 1 or more than 1 parallel line can be draw through it." > If there doesn't then the method of forcing doesn't allow you to > magically create one, it just allows you to construct a *model* in > which there is one. Is that not exactly what Lobachevsky, Gauss, and Bolyai did? "Magically create" new geometries? Or did they merely discover them? > Models != reality, even in the universe of sets. So ... are you saying that Non-Euclidean Geometry isn't part of "reality"? And exactly what makes a certain part of Mathematics more "real" than another? Since they are all mental constructs, how is one any less "real"? > I'm an unrepentant Platonist. I have good company; here is a quote > from Goedel: "For if the meanings of the primitive terms of set theory > are accepted as sound, it follows that the set-theoretical concepts > and theorems describe some well-determined reality, in which Cantor's > conjecture must be either true or false. Hence its undecidability from > the axioms being assumed today can only mean that the axioms do not > contain a complete description of that reality." (quoted in Penelope > Maddy's book "Realism in Mathematics", which is the most sophisticated > defense of a form of Platonism that I am aware of. She also published > an article in two parts in the Journal of Symbolic Logic sometimes in > the late 80s/ early 90s called "Believing the Axioms" which is well > worth reading). Again, I am curious as to how you apply this principle to my analogy with Geometry. By the way, in case it doesn't come through, I appreciate your input here. Although I appear to be arguing one side on this matter, in truth I am divided on the question myself. The Platonist part of me agrees with everything you've said, but I have difficulty resolving this even to myself. The Formalist position seems to me still the safest approach for undecidability, lest we end up being like the small minded individuals from a couple centuries ago who berated proponents of Non-Euclidean Geometry. Thanks again.
From: scattered on 13 Jan 2010 11:30
On Jan 13, 10:28 am, Leonid Lenov <leonidle...(a)gmail.com> wrote: > On Jan 13, 2:52 pm, scattered <still.scatte...(a)gmail.com> wrote:> Goedel: "For if the meanings of the primitive terms of set theory > > are accepted as sound, it follows that the set-theoretical concepts > > and theorems describe some well-determined reality, in which Cantor's > > conjecture must be either true or false. Hence its undecidability from > > the axioms being assumed today can only mean that the axioms do not > > contain a complete description of that reality." > > Beautifully said. > > > Maddy's book "Realism in Mathematics", which is the most sophisticated > > defense of a form of Platonism that I am aware of. > > Now I have to have this book :). It will also give me more arguments > with which to defend Platonism form all those "heretics". Why are > there some people against Platonism will never be clear to me... Not that I endorse everything in her book. She defends a version of Platonism that doesn't seem completely plausible to me. I subscribe to what she calls "pre-theoretic realism" which sees statements like "there exist continuous nowhere differentiable functions" as being as unproblematics as statements like "there exist kangaroos in Australia." But then again I am not a philosopher of mathematics and am not regularly confronted with arguments to the contrary. -scattered |