Uncountability of the Rationals?
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From: David R Tribble on 16 Jun 2010 17:22 Tony Orlow wrote: > Sometimes there are logical arguments themselves which don't appeal to > everyone equally. What I have been told is that the size of omega must > be larger than every natural since no natural is large enough to > express it, and so it is some infinite number, aleph_0. However, that > logical argument, as I pointed out, really just proves that aleph_0 > cannot be finite. Along with the argument that any initial segment of N > + of size x contains an xth element whose value is x, which would > imply that aleph_0 or omega is a member of N+, we arrive at a > contradiction implying that aleph_0 cannot actually exist. Which axiom > contradicts this logic? The problem is that there is no logic there. You consider finite sets of the form {1, 2, 3, ..., k}, all of which have a least and a greatest member, to have size k and to also have k as a member. Then you take an unjustified leap and claim that this implies that sets of the form {1, 2, 3, ...}, which do not have a largest member, to also have a size equal to one of their members. At the very least, you're contradicting your previous statement that Aleph_0 cannot be a member of N+. More egregiously, though, is your unjustified "implies" leap. So the question becomes, what axiom justifies your logical leap, applying a property of finite sets to infinite sets without largest members?
From: David R Tribble on 16 Jun 2010 17:26 Tony Orlow wrote: > In standard N ordinals and cardinals are the same thing, really. I > have never seen the need for any distinction. As a computer scientist, > a number is a number, whether used as a data element (cardinal) or > memory address (ordinal). Does this include the IEEE floating-point numbers +INF and -INF? You will recall that arithmetic operations on these values don't produce the same results as those on regular numbers.
From: Tony Orlow on 16 Jun 2010 17:39 On Jun 16, 1:09 pm, "Jesse F. Hughes" <je...(a)phiwumbda.org> wrote: > "Jesse F. Hughes" <je...(a)phiwumbda.org> writes: > > > So it appears to me that I don't know 1/3 is in R at all. This chain of > > reasoning does not end in a statement that I know. Can you give me a > > finite proof (or, let's be generous, anything that looks like a proof) > > that 1/3 is a real number. > > In fact, I'd be interested also in a proof that 8 is a real number. That, of course, would follow from the existence of 3. > > -- > Jesse F. Hughes > "I thought it relevant to inform that I notified the FBI a couple of > months ago about some of the math issues I've brought up here." > -- James S. Harris gives Special Agent Fox a new assignment. Is that what they were calling me about? Harris, yeah... Geeze! TOny
From: David R Tribble on 16 Jun 2010 17:51 Tony Orlow wrote: > The H-riffics are a bit of a sidebar. Besides, it occurs to me that > within R there exists one element which cannot be a child node of an > element in R, namely, 0. So, 0 can be taken to be the root case for > the tree of values, the foundation. Additionally, if one wants to > include all reals positive and negative then the H-riffics may be > redefined as: > E(0) > E(x) -> E(-2^x) ^ E(2^x) This still omits vast (uncountable) subsets of of the reals. 3, for example, is not a node in the binary tree that is E. As I pointed out to you several times, any real of the form r = f*k^m, for integers f, k, and m, and k /= 2, is not a node in the E tree. You've never addressed this. Likewise, your definition is inherently described by a countable binary tree, where each node is an H-riffic. Therefore, E can't possibly contain all the uncountable reals, nor any uncountable subset of them. Now (as Walker pointed out previously) if you want to amend the H-riffics to include all infinite paths in the tree, that's fine, as there are an uncountable number of such paths. However, once you do that, you lose the well-ordering of the tree. The countable nodes are still well-orderable, of course, but the infinite-length paths are not. (Choose any infinite path in the tree, then ask what the "next" path is; there is none.) So you can't have both an uncountable number of elements and a well-ordering on them.
From: Tony Orlow on 16 Jun 2010 17:52
On Jun 16, 1:33 pm, MoeBlee <jazzm...(a)hotmail.com> wrote: > On Jun 16, 8:15 am, Tony Orlow <t...(a)lightlink.com> wrote: > > > On Jun 15, 8:22 pm, Transfer Principle <lwal...(a)lausd.net> wrote: > > > If TO accepts all of the axioms of ZFC, but rejects the > > > theorem "there exists a cardinal (or ordinal) number," > > > then I'll agree to call TO "wrong." Still, I believe that > > > if we can show him a theory which does satisfy his > > > intuitions, he'll have less of a reason to criticize the > > > adherents of ZFC. > > What do you mean Transfer Principle "adherent of ZFC"? > > > Okay, perhaps I am "wrong" about this. I am going over the axioms of > > ZFC, and I simply don't see any reference to any primitive referring > > to ordinality or cardinality > This is egregious: > Also not among the primitives are mentions of > > subsets Wrong. "X is a subset of Y" can be expressed as "aeX -> aeY", which is used explicitly in the axiom of the power set, if not implied elswhere in the axiom set. > the empty set Implied by the other axioms. > union Axiom > intersection Implied by separation using xeG as condition phi on set F. Extend the size of the intersection to the infintie case... > pairs > ordered pairs Not axioms? > natural numbers Axiom of infinity, but without reference to its size, only its existence. > prime numbers (thank you, Aatu) No, that comes later, and has nothing to do with cardinality, except peripherally, as a tough example for Bigulosity. > metric spaces Not mentioned in ZFC, but rather, handled by topology, wherein which measure is somewhat at odds with transfinite set theory. > Banach spaces > > or ANYTHING other than > > equality (if taken among the primitives) > and > elementhood Where do the axioms mention "equality" except among identical sets ala the axiom of extensionality? > > and the variables, the sentential connectives, and the quantifiers > (and left and right parentheses, if we don't go Polish). That's all the logical foundation, about which there are only a few questions. > > MoeBlee TeeKnow |