Uncountability of the Rationals?
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From: Jesse F. Hughes on 30 Jun 2010 08:15 Transfer Principle <lwalke3(a)lausd.net> writes: > On Jun 28, 5:42 am, "Jesse F. Hughes" <je...(a)phiwumbda.org> wrote: >> Transfer Principle <lwal...(a)lausd.net> writes: >> > In 2010, TO admits that these lead to inconsistency, >> > since they can lead to tav being both finite and >> > infinite or both in N+ and not in N+. And so, we avoid >> > the inconsistency by noting that tav is _not_ a number >> > (natural, nonstandard, or otherwise). >> You grant Tony too much. >> Recall the induction principle? That if phi is any formula not >> involving Tav, >> ( (E n in N+)phi(n) & (An)(phi(n) -> phi(n+1)) ) -> phi(tav)? >> Let phi(x) stand for "x is a number". Since the "fact" that tav is not >> a number plays so much work in Tony's reasoning, it must be the case >> that "x is a number" is expressible in Tony's (non-existent) theory. >> Work with me here, Walker. What follows from this? > > A contradiction? I suppose that's the answer that Hughes is > looking for. Well, I was just looking for "tav is a number." That's enough to kill Tony's defense that tav is not a number. > So now we must ask ourselves the following questions: > > 1) Does this schema truly represent TO's current beliefs? > 2) Does this schema truly lead to a contradiction? > 3) Is TO intentionally trying to create an inconsistent theory? Surely, no one thinks that Tony is trying to create an inconsistent theory, but so what? The question really is whether Tony's theory *is* inconsistent and whether Tony can recognize that fact. > So let's consider these questions in turn: > > 1) When I first came up with this schema, it was based on what > I was told was TO's beliefs. But this was based on what TO had > posted back in 2005, not 2010. (See my response to Tribble for > more info on the 2005 vs. 2010 issue.) So does the schema > represent TO's 2010 beliefs? > > Assuming the answer is "yes," we can move on to: > > 2) Earlier, we mentioned some of the instances of this schema > and their results: > > -- The schema proves that tav is a number. > -- The schema proves that tav is finite. > > This may sound like a contradiction, but so far it isn't, since > we haven't proved that tav is infinite yet. > > -- For each natural number n, the schema proves that n is in tav. > > This may sound like a contradiction, but there's a loophole. > > We think back to Ross Finlayson and his attempted proof that > ZFC is inconsistent, as follows: ZFC proves that the set R of > real numbers is uncountable, but by Lowenheim-Skolem, R has a > countable model. RF claims that therefore ZFC is inconsistent. > > Of course, the usual response to RF is that _outside_ the model, > R is countable, but _inside_ the model, R is uncountable. Thus, > RF has not proved ~Con(ZFC). > > So perhaps it's possible that tav can contain all the natural > numbers, yet still be finite. _Outside_ a model of the theory, > tav is infinite, but _inside_ the model, tav is finite. And this > would be the loophole that we seek. This loophole is surely not satisfactory to Tony. It would require that his theory proves tav is the same size as some number n. [...] > But nonetheless, I still _refuse_ to believe that _any_ theory > which refutes properties a)-d) from Tribble's post, or proves > (b) from Chandler's post, must be inconsistent, just because > the theory based on TO's posting so far is inconsistent. I don't recall exactly what (a)-(d) are, but likely you're right that there is some theory in which they are all false -- but this theory may not be much of a *set* theory. In any case, the proof is in the pudding. > At this point, what I'd like to do is find a theory which > proves as many of Chandler's (b) and the negations of a)-d) as > possible without introducing an inconsistency. If we can do so, > then perhaps the resulting theory would be one that TO is > willing to accept, yet avoids the inconsistency that was found > in TO's current theory. > > This is what I seek to do with the post in which I mention > axioms for the relations "<=" and "~=". If the source of the > inconsistency is this object called "tav," then let's first > consider comparing sets via Chandler's (b) without worrying > about any object called "tav." So, you're abandoning Tony's theory (for now)? -- Jesse F. Hughes "Right now I'm above the margin of error. I do exist." -- Presidential candidate Dennis Kucinich, Sept. 2007
From: MoeBlee on 30 Jun 2010 11:35 On Jun 29, 9:50 pm, Transfer Principle <lwal...(a)lausd.net> wrote: > On Jun 28, 5:42 am, "Jesse F. Hughes" <je...(a)phiwumbda.org> wrote: > > > Transfer Principle <lwal...(a)lausd.net> writes: > > > In 2010, TO admits that these lead to inconsistency, > > > since they can lead to tav being both finite and > > > infinite or both in N+ and not in N+. And so, we avoid > > > the inconsistency by noting that tav is _not_ a number > > > (natural, nonstandard, or otherwise). > > You grant Tony too much. > > Recall the induction principle? That if phi is any formula not > > involving Tav, > > ( (E n in N+)phi(n) & (An)(phi(n) -> phi(n+1)) ) -> phi(tav)? > > Let phi(x) stand for "x is a number". Since the "fact" that tav is not > > a number plays so much work in Tony's reasoning, it must be the case > > that "x is a number" is expressible in Tony's (non-existent) theory. > > Work with me here, Walker. What follows from this? > > A contradiction? I suppose that's the answer that Hughes is > looking for. > > So now we must ask ourselves the following questions: > > 1) Does this schema truly represent TO's current beliefs? > 2) Does this schema truly lead to a contradiction? > 3) Is TO intentionally trying to create an inconsistent theory? > > So let's consider these questions in turn: > > 1) When I first came up with this schema, it was based on what > I was told was TO's beliefs. But this was based on what TO had > posted back in 2005, not 2010. (See my response to Tribble for > more info on the 2005 vs. 2010 issue.) So does the schema > represent TO's 2010 beliefs? > > Assuming the answer is "yes," we can move on to: > > 2) Earlier, we mentioned some of the instances of this schema > and their results: > > -- The schema proves that tav is a number. > -- The schema proves that tav is finite. > > This may sound like a contradiction, but so far it isn't, since > we haven't proved that tav is infinite yet. > > -- For each natural number n, the schema proves that n is in tav. > > This may sound like a contradiction, but there's a loophole. > > We think back to Ross Finlayson and his attempted proof that > ZFC is inconsistent, as follows: ZFC proves that the set R of > real numbers is uncountable, but by Lowenheim-Skolem, R has a > countable model. RF claims that therefore ZFC is inconsistent. > > Of course, the usual response to RF is that _outside_ the model, > R is countable, but _inside_ the model, R is uncountable. Thus, > RF has not proved ~Con(ZFC). > > So perhaps it's possible that tav can contain all the natural > numbers, yet still be finite. _Outside_ a model of the theory, > tav is infinite, but _inside_ the model, tav is finite. And this > would be the loophole that we seek. > > (In an old thread, there was discussion of a theory that is > consistent as long as ZFC itself is, which proves the existence > of a set which, from outside a model of the theory, contains all > the naturals, all the reals, and even all the _ordinals_, yet > inside the model, the set is finite!) > > But based on a current discussion between MoeBlee and Charlie-Boo > in another thread, this loophole might not work. Once again, we > have proved from this schema that: > > -- For each natural number n, it is provable that n is in tav. > > According to MoeBlee, we have: > > "(1): |- (allX)P(X) > (2*): For all variables 'x', we have |- Px > (1) iff (2*)" To be more neat, I would write that as: (1) |- AxPx (2*) for all variables 'x', we have |- Px (1) iff (2*) Unless I wrote it incorrectly, that is just an instance of the principles of universal generalization and universal instantiation. MoeBlee
From: sci.math on 30 Jun 2010 20:13 On Jun 3, 9:20 pm, Tim Little <t...(a)little-possums.net> wrote: > On 2010-06-03, Tony Orlow <t...(a)lightlink.com> wrote: > > > One might think there were something like aleph_0^2 rationals, but > > that's not standard theory. > > That's perfectly standard theory. It's just that aleph_0^2 = aleph_0.. > > - Tim Here you go: Thanks, Martin (Meami.org) // #define VAR(x) ((x) << 1) // #define NOT(x) ((x) ^ 1) // void visit(int v, const graph &g, // vector<int> &ord, vector<int> &num, int k) { // if (num[v] >= 0) return; // num[v] = k; // for (int i = 0; i < g[v].size(); ++i) // visit(g[v][i].dst, g, ord, num, k); // ord.push_back(v); // } // typedef pair<int,int> clause; // bool two_satisfiability(int m, const vector<clause> &cs) { // int n = m * 2; // m positive vars and m negative vars // graph g(n), h(n); // for (int i = 0; i < cs.size(); ++i) { // int u = cs[i].first, v = cs[i].second; // g[NOT(u)].push_back( edge(NOT(u), v) ); // g[NOT(v)].push_back( edge(NOT(v), u) ); // h[v].push_back( edge(v, NOT(u)) ); // h[u].push_back( edge(u, NOT(v)) ); // } // vector<int> num(n, -1), ord, dro; // for (int i = 0; i < n; ++i) // visit(i, g, ord, num, i); // reverse(ord.begin(), ord.end()); // fill(num.begin(), num.end(), -1); // for (int i = 0; i < n; ++i) // visit(ord[i], h, dro, num, i); // for (int i = 0; i < n; ++i) // if (num[i] == num[NOT(i)]) // return false; // return true; // }
From: Transfer Principle on 2 Jul 2010 21:43 On Jun 30, 12:08 pm, David R Tribble <da...(a)tribble.com> wrote: > David R Tribble wrote: > > I think that TO should have the same freedom to reject a)-d) > > that Tribble has to accept a)-d). And if we want to look for > > theories in which the negations of a)-d) are provable (or at > > least as many of the negations as possible without leading > > to inconsistency), then we should be able to do so without > > five-letter insults. > But no one (except perhaps Zuhair) has offered anything resembling > a cogent, coherent, consistent theory. I'd argue that Srinivasan, in the current thread that he shares with Charlie-Boo, offers the theory NBG-Infinity+"D=0." > [Again, when you contradict me, please provide actual posted > examples.] So Tribble wants actual posted examples. Then here you go. Srinivasan, 28th of June, approximately 5AM Greenwich time: "Here is one possible approach. Consider the theory F that I had defined in another post, where F = ZF - Inf + D=0, Where 0 is the null set and D is defined as D = {x: An(x not in P_n(0))} Here P_n(0) is power set operation iterated n times on 0, and P_0(0) = P(0), P_1(0)=P(P(0)), etc. Clearly only hereditarily finite sets can exist in models of F." Srinivasan, 30th of June, approximately 5AM Greenwich time: "you will find I *have* worked precisely in NBG-Infinity. So D is a proper class when infinite sets exist, and the proposition D=0 is undecidable in NBG-Infinity." > Tell us, in all the time that you've been re-stating their theories, > have you ever come up with anything that really would stand > as an interesting alternative to standard theories? It depends on what one means by "interesting alternative to standard theories." If one demands that such alternative, say, provide for an axiomatization for the sciences, then not yet. But keep in mind that ZFC has already had a century's head start. One should not expect a post that contains all the axioms and schemata of a proposed alternative, evidence that the theory isn't trivially inconsistent, _and_ evidence that the theory provides for the sciences _all_ in one concise post that Tribble or anyone can read in 15 minutes. Theories take time to develop. But most posters who already accept full ZFC have neither the patience nor any incentive to wait as we develop the theory. They're much more likely to ignore such developmental posts. On the other hand, it's easy for them to jump on theories that are inconsistent, incoherent, and so on, with five-letter insults. But I don't believe that a poster who objects to ZFC for one reason or another should be forced to use it merely because there is no real alternative. Srinivasan is a finitist, so he opposes ZFC. TO wants sets to be strictly larger than their proper subsets, so he opposes ZFC. Since nowhere in the physical world can one prove that an infinite object exists, it can't be _ruled_out_ that there exists a theory satisfying most of Srinivasan's, or TO's, or AP's, or RF's, desiderata and still axiomatize the sciences.
From: Transfer Principle on 2 Jul 2010 22:20
On Jun 30, 5:15 am, "Jesse F. Hughes" <je...(a)phiwumbda.org> wrote: > Transfer Principle <lwal...(a)lausd.net> writes: > > This is what I seek to do with the post in which I mention > > axioms for the relations "<=" and "~=". If the source of the > > inconsistency is this object called "tav," then let's first > > consider comparing sets via Chandler's (b) without worrying > > about any object called "tav." > So, you're abandoning Tony's theory (for now)? If the theory is inconsistent, then I'm abandoning the theory, but I'm not abandoning TO's _ideas_. I still believe that a theory that satisfies at least some of TO's desiderata, but isn't trivially inconsistent, can be found. I hope that this can represent in a compromise, if it can satsify the desiderata of both TO and Hughes (i.e., that it avoid inconsistency). The "<=" and "~=" theory represents one such attempt, and also asks questions about how much of TO's ideas can be maintained while avoiding inconsistency. Here is another such attempt, inspired by the current discussion in the Srinivasan thread. Several months ago, someone (forgot who) once told Srinivasan that the theory IST proves the existence of a finite set that contains every standard set. This is the loophole to which I was referring earlier -- an object which outside a model of the theory appears to be possibly infinite, possibly uncountable, possibly even too large to be a set, yet inside the model, the same object is _finite_. In particular, such a set can contain every standard natural number (i.e., every "pofnat" in Chandler's parlance), and is thus infinite outside the model, yet the set is finite inside the model. Such a set must contain other objects besides the pofnats, which we could call nonstandard natural numbers. According to the theory, this set is "finite," but what do we mean by "finite"? The inventor of the theory, Ed Nelson, was working in ZF_C_+IST, where all definitions of finite are equivalent, but TO rejects AC. Thus, Nelson doesn't have to state which type of "finite" he means when he asserts the existence of the finite set containing all standard naturals, but what about ZF+IST? Since Dedekind finite is one of the weakest types of "finite," perhaps we can at least assert that this finite set containing all pofnats is D-finite. Then there is no bijection between the set and any of its proper subsets -- which is, of course, exactly what TO needs to establish Bigulosity. But where does "tav" fit into all of this? We could simply declare the Bigulosity of this set to be "tav." > This loophole is surely not satisfactory to Tony. It would require that > his theory proves tav is the same size as some number n. But here's the thing -- without AC, can we prove that this set has a (von Neumann ordinal as a) cardinality? I suspect not -- and thus we can give it a Bigulosity instead. Even if such a cardinality existed, it would be a _nonstandard_ natural (i.e., not a pofnat) and so tav would not be equal to any pofnat n. At this point, one might wonder why we even have to go to IST in the first place. Can't we start with ZF+~AC, show that there exists a D-finite T-infinite set (not equinumerous to any von Neumann ordinal), then simply declare it to have Bigulosity tav without IST at all? The answer is that the in ZF+~AC approach, any set that contains all the pofnats might contain _all_ the naturals, meaning that the set would be D-infinite. We need IST to prove the existence of nonstandard naturals (i.e., naturals that aren't pofnats), so that a set can contain all the pofnats without containing all the naturals and therefore possibly be a D-finite set. Indeed, IST appears to prove that such a D-finite set must exist. Will this theory work? The main problem I see is that I'm not sure what role AC plays in all of this. We have our base set, the D-finite set containing all pofnats, whose Bigulosity is declared to be tav. But is AC required to assign Bigulosities to sets other than this base set? If AC is needed at any point, then we must reject the theory as TO rejects AC. Still, the earlier theory including "<=" and "~=" might be able to satisfy TO's desiderata, depending on whether the theory is inconsistent. |