Uncountability of the Rationals?
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From: Brian Chandler on 17 Jun 2010 01:52 Transfer Principle wrote: > On Jun 15, 11:25 pm, Brian Chandler <imaginator...(a)despammed.com> > wrote: > > In a truly extraordinary moment, Transfer Principle wrote: > > > On Jun 15, 9:18 pm, Brian Chandler <imaginator...(a)despammed.com> > > > wrote: > > <snip bit about Conway's ONAG> > > > So yes, sqrt(omega) is a surreal. But the big difference > > > between surreals and TO's numbers is that the latter is > > > supposed to be a sort of set _size_ (Bigulosity). It > > > makes no sense to state that the set of perfect squares > > > has the surreal sqrt(omega) as its set size. > > Have you gone barking mad? Don't you remember the Agreement? You > > *cannot* say "makes no sense". You *must* say "... mumble mumble, > > working in a different theory". > > Yes, I admit that Chandler has caught me red-handed. "Red-handed" makes you sound guilty of something. I was actually rather filled with joy at the possibility that you could reject something, instead of your normal auto-response. > And so since I made an error regarding the "Agreement," > I must admit and correct my error. It is possible that > indeed Chandler is working in a different theory -- > one in which surreals can be applied to set sizes. Oh dear. You have missed the point again. I am not "working" in any "theory". Tony is babbling, and I am poking him with a stick from time to time, because it appeals to my baser instincts. But no-one has actually suggested using surreals as set sizes. Tony uses words like "omega" without a clue what they mean; he certainly doesn't understand Conway's numbers -- after all he remains convinced (as far as we can tell) that Conway can see just like Tony that Set Theory is a Crock, and what looks to us like a set-theoretic basis for constructing the surreals is actually just a nod to the establishment. > And so let us attempt to do exactly that. So far, we > already have the following set sizes No, _we_ don't. Perhaps Tony believes he has, but none of his stuff is rigorous enough for anyone else to check. (since we really > are discussing surreals here, we can use Conway's > name for the unit infinity "omega," rather than > Chandler's name "tav"). No. Conway (who _is_ actually using normal set theory) does not have a "unit infinity". This is entirely an Orlovian "concept". Conway uses 'omega' to mean exactly what it normally means. I am trying to persuade Tony to use different names for his ideas. <snip> > > Until Chandler and TO state otherwise, we are only > assigning surreals to subsets of standard N+. Um, please don't bracket me with Tony -- we are not working together, we are just poker and pokee. Brian Chandler
From: Aatu Koskensilta on 17 Jun 2010 06:21 "Jesse F. Hughes" <jesse(a)phiwumbda.org> writes: > I think 8 is Julius Caesar. Ah, so Caesar /isn't/ a prime after all! -- Aatu Koskensilta (aatu.koskensilta(a)uta.fi) "Wovon man nicht sprechan kann, dar�ber muss man schweigen" - Ludwig Wittgenstein, Tractatus Logico-Philosophicus
From: Tony Orlow on 17 Jun 2010 08:23 On Jun 16, 5:51 pm, David R Tribble <da...(a)tribble.com> wrote: > 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. It is an uncountable number of iterations down the tree. Unless, of course, you think there are only countably infinite segments oof this tree, in which case it would apprear to be some kind of well order, not havbing any infinite descending sequences. > > 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. I have as I just did. Can you think of any sequence or finitely- branched structure that could include all the reals and yet not have two that were infinitely distant from each other within that structure? I can't. > > 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. Not by a countable tree, no, but by an uncountable one, yes. > > 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.) If there is an infinite path from 2 to 3, then there is a next one after that, as Transfer pointed out. > > So you can't have both an uncountable number of elements > and a well-ordering on them. Then you believe there is no well ordering on the reals, and you must therefore reject the Axiom of Choice? Tony
From: Tony Orlow on 17 Jun 2010 08:26 On Jun 16, 6:08 pm, David R Tribble <da...(a)tribble.com> wrote: > David R Tribble wrote: > >> I'll ask the question again, since you didn't bother answering > >> it before: Are there square roots r in S such that r*r is not a > >> natural in N+? If so, can you provide an example r? If not, does > >> this mean that there are more members in S than in N+?- Hide quoted text - > > Tony Orlow wrote: > > Imanswered this already. I do not disagree that there exists a > > bijection, but within any segment of R greater than measure 2 exist > > more square roots of naturals than naturals. Sure, you can find a > > member in each set corresponding to a unique member of the other. They > > are equicardinal. They are not equibigulous. > > Okay, finally, perhaps, some progress. > > Bigulosity of a set is equivalent to the relative "density" (or > perhaps "frequency") of its members when they are mapped > onto the real number line. I.e., how frequently the numbers occur > *when considered as points on the real number line*, as compared > to the frequency of the naturals (what you call "units") on the real > line. > > So it's probably fair to say that Bigulosity is more about the > numeric values of members of sets than it is about the sets > themselves. It applies the numeric values of sets, or numeric measures of sets, to segments of standard countable N+, to derive a measure of the set which goes a bit beyond cardinality or natural density. Tony
From: Tony Orlow on 17 Jun 2010 08:47
On Jun 16, 6:53 pm, MoeBlee <jazzm...(a)hotmail.com> wrote: > On Jun 16, 4:52 pm, Tony Orlow <t...(a)lightlink.com> wrote: > > > > > > > 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: > > INDEED your ignorance as you yet again post it is egregious. > > BIG NOTE: I'm replying line by line to you again, but I might not > followup to further of your foolishness, just as I did not followup > recently. At a certain point, such exchanges with you are just not > productive. I need not be a hen continually cleaning up your > confusions as when I do that, and get in yet another back and forth > with you, even MORE confusions come issuing from you. > > Here, I'll reply point-to-point, but I suggest you save yourself the > trouble of doing the same in return thus to set off yet another round > of unproductive exchange. > > All you have to do, in the meantime, is get any kind of decent book > that discusses the subject of mathematical definition. > > > > Also not among the primitives are mentions of > > > > subsets > > > Wrong. > > In the PRIMITIVES, there is NO subset symbol. Oh. What do I see in the Axiom of the Power Set? http://en.wikipedia.org/wiki/Zermelo%E2%80%93Fraenkel_set_theory http://mathworld.wolfram.com/Zermelo-FraenkelAxioms.html Of course, these can be written without the subset operator, as "S c T" is equivalent to "Ax (xeS -> xeT)". Is 'e' a primitive operator in set theory? > > > "X is a subset of Y" can be expressed as "aeX -> aeY", > > Actually as Aa(aeX -> aeY), just to be exact. Yeah, "forall" is a little superfluous. Ultimately it suffices to say "if a is an element of X then a is an element of Y". > > And, yes! subset is DEFINABLE from 'e'. > > Just as 'is an ordinal' is DEFINABLE from the mere primitives. Is it? Demonstrate as I did for you just now. > > > 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. > > Yes, exactly! > > The axioms IMPLY that there exist a set x such that Ay ~yex, as either > that is itself an axiom (with some authors) or entailed by the axiom > schema of separation (as with other authors). Then from the axiom of > extensionality we get that there exists a UNIQUE such set. THEN we > DEFINE 0 to stand for that unique such set. Or, you just say X=the empty set <-> Ax (~xeX) > > There is no primitive of the language of set theory that is "the empty > set" symbol. > > '0' is DEFINED from the primitives and as enabled by axioms that prove > that there exists a unique such set. > > And as 'is an ordinal' is DEFINABLE from the mere primitives. Please demonstrate. > > It's a long trail going through previous definitions that go back to > 'e', but still a definiens of 'is an ordinal' can be stated in terms > of the mere primitives. You only have 7 or so axioms. How many steps can it take? > > There is a formula in the PRIMITIVE language of ZFC that defines 'is > an ordinal'. (We don't usually SHOW it that way, because it's pedantic > and unnecessarily laborious to do so, but still the formula is there > and we could present if we didn't mind the drudgery of doing it.) You "mind the drudgery" of defining ordinals? How do you expect me to be convinced that you have any idea what you're talking about? > > > > union > > > Axiom > > The union axiom has NO union symbol in it. The union axiom is in the > PRIMITIVES of ZFC. No, union is defined by 'e': X U Y = {a: aeX v aeY} > > THEN we DEFINE the unary union symbol. > > Again, a operation symbol defined in terms of primitives. And 'is an > ordinal' is a locution for a predicate (technically, a predicate > symbol) defined in terms of previous defined symbols that themselves > all can be traced back to a definiens in the PRIMITIVE language. Wouldn't it be easier to simply demonstrate how ordinals are logically implied by the axioms than go through all this verbiage? I begin to think you cannot do it. > > > > intersection > > > Implied by separation using xeG as condition phi on set F. Extend the > > size of the intersection to the infintie case... > > (1) We don't "extend" to infinite case, but rather, we prove all at > once (with no mention of infinte or infinite) > > Ax(x is non-empty -> E!zAy(yez <-> Av(vex -> yev))) > > Then we DEFINE > > If x nonempty -> > /\x = the unique z such Ay(yez <-> Av(vex -> yev)). > > All in the primitives ('unique' and 'nonempty) definable back to > primitives, or we could dispense with 'unique' and do it this say: > > If x nonempty -> (/\x = z <-> Ay(yez <-> Av(vex -> yev))) > If it takes you that long to define intersection then deriving ordinals from ZFC must be drudgery indeed. XnY = {a: aeX v aeY} It's that easy. > > > pairs > > > ordered pairs > > > Not axioms? > > Pairing axiom, yes. Same remarks, mutatis mutandis as above. > > But no ordered pair axiom. Rather, ordered pairs defined in terms of > pairs which are themselves defined in terms of the PRIMITIVES. The > axioms state IN PRIMITIVES that there exist sets having certain > properties (properties in terms of the PRIMITIVES), and THEN the > DEFINITION of the { } operation is given, and, as with ALL definitions > in ZFC, including 'is an ordinal' ultimately traced back to a formula > in the PRIMITIVE language. > > > > natural numbers > > > Axiom of infinity, > > SOME authors wait for the axiom of infinity. But that's not required > or essential. We can give an equivalent definition of 'is a natural > number' without the axiom of infinity and in the PRIMITIVE language. > > Indeed NO predicate needs ANY axioms at all to support a definition. > Operations (operation symbols, pedantically) require axioms to prove > the existence and uniqueness clauses, but if we have just one defined > object (such as 0) then we may use the Fregean method so that even > operations do not require appeal to axioms (other than those required > to construct at least one unique object). > > > but without reference to its size, only its > > existence. > > Correct. > > > > prime numbers (thank you, Aatu) > > > No, that comes later, and has nothing to do with cardinality, except > > peripherally, as a tough example for Bigulosity. > > I didn't say it depends on cardinality. > > I said that there is no mention in the axioms of "prime number". > Rather, we DEFINE 'is a prime number' using previously defined terms > that are in a sequence of defined terms that ultimately reach to the > primitive. The definition of 'is a prime number' can be given in the > PRIMITIVE language, if we wish to present it that way and with such a > definition EQUIVALENT in ZFC to one given with previously defined > terminology. > > EVERY definition in ZFC (from 'subset' through 'Banach space' and > beyone) has a definiens taht may be stated in the PRIMITIVE language. > ordinal' is no different in this respect from 'subset', '0', 'prime > number', 'Banach space'. > > > > metric spaces > > > Not mentioned in ZFC, but rather, handled by topology, wherein which > > measure is somewhat at odds with transfinite set theory. > > No, 'metric space' is DEFINABLE in ZFC. > > 'is a topology', 'is a topological space', 'is open in the topology', > 'is a metric space' and all the other common terms of topology have > set theoretic definitions. Thery all have definitions in the language > of ZFC such that the definiens is RIMITIVE in the language of ZFC. > > > > > > 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? > > The equality symbol is in the axiom of extensionality. It's also in > ordinary versions of the axiom schema of replacement. I didn't say > equality symbols occurs in all axioms, but rather that it does occur > among the axioms. Moreover, the equality symbol can be defined in the > language of ZFC without the equality symbol. The equality symbol doesn't even require 'e'. a=b <-> (a->b ^ b->a). It's basic logic. > > > > 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. > > Whatever your questions about the logic, you need to understand that > 'ordinal' is definable from the primitives of the language of ZFC, and > that 'subset' and all the rest that are NOT primitives are also > definable from the primitives. > > MoeBlee- Hide quoted text - > > - Show quoted text - Okay, so, rather than simply explain how ordinals are implied by the axioms you'd rather go through all of this, which is equally as tedious, and much less fruitful. Too bad. Let me know if you change your mind. Tony |