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From: David Libert on 31 Dec 2009 22:45 I will add some points to my account from the parent article [0] David Libert "Recursive Cardinals" sci.logic, sci.math Dec 29, 2009 http://groups.google.com/group/sci.logic/msg/8d9476e07a2e0b11 I wrote in [0] : > So the class of all recursive singletons is partitioned into > these maximal chains, where each chain is ordered like Z. I did not mention explicitly in [0], but still true, is there are no membership relations between distinct maximals chains. (I had nade a similar comment to this in an older article before I had used the word "chains" to talk about these). So if you draw each maximal chain as a vertical line, with that viertical line representing the Z ordering on the maximal chain membership, the case of many maximal chains can be visualized as a forest of vertical lines. In [0] I wrote about making models with various numbers of maximal chains: > In > > [1] David Libert "A new definition of Cardinality" > sci.logic, sci.math Nov 23, 2009 > http://groups.google.com/group/sci.math/msg/721cb8170033cf84 > > I constructed a ZFC - regularity model with a proper class of > recursive cardinals. > > I still think that construction is ok. > > You could modify that construction, to only put in one maximal > chain of recrusvie singletons: just take the alpha as 0 instead of varying overll all > ordinals as [1] did. (In [1] I was using my old definiton of singleton towers > > So that would give a ZF - regularity model with exaclty one maximal chain, > and hence a model where your discussion above defined bracket sequences for all > recursive singletons. > > But the original [1], or variants of it making alpha vary over less than a > proper class but still over more than one value, would give models of > ZF - regularity with more than one maximal chain of recursive singletons. I went on to note in [0] that your main opening discussion earlier of bracket sequences only defined them on one maximal chain. I went on to suggest how it might be possible to extend that to more maximal chains, involving P(omega) as you had been writing: > A few more points. > > Later in you article as quoted above, you write about the possiblilty of > have P(omega) recursive cardinals. (You called them singleton towers). > > Here is a possible approach along suich lines. > > Above, you had picked a recursive singleton x, and used it to define > a bracket sequence for x's maximal chain. > > Maybe you could extend such a definition to a second maximal chain, > by picking a member rectrusive singleton y there, and making a similar > defintion. > > But now the two parts of the definition, on eahc maximal chain, > depeneded on the arbitrary choide of x or y, so its this dependency > that allows diostinct recursice singletons to get the same bracket > sequence. > > For example, the most straigtforward copy over y of what you did > to x would be to assign both <0, 1, 2, ... > . But x and y are > still distinct, and each bracket sequence also depernded on base point > in the singleton tower. > > You might try to make bracket sequences distinct by skipping numbers > in the bracket labelling inside y. > > So you might want to keep going that way, same idea with different > skips on different maximal chains to get different bracket sequences. > > So that gets you a P(omega) limit like you said. I went on in [0] to discuss some ~AC issues about carrying on the above plan: > Other issues. That outline I said just now depends on picking > a base point x or y or whatever in each maximal chain. > > My [1] models admit automophisms. If you map each recursive > singleton to its own singleton you get a non-identity bijection > of the class of all recursove singletons to itself. This can > be lifted by transginfinite rwecursion up the ranks over those > to the entire model. > > So there is no definable way to picj the x y etc from each > maximal chain. > > So at very least, the bracket sequeebce assignment will > not be deinfable without parameters. > > It depends on picking x y etc, anf then defining over > those parematers. > > But our discussiin was about ZF - regulairty, not > ZFC - regularity. > > [1] didn't dp this, but you could make a variant of [1], > similar to my other models, where you permute independly > within each maximal chain by shifitng the chain up or > down together, possibly different shifts on different chains. > > Obtain a model where there is no choice set to pick > a base point in each maximal chain. > > Given bracket sequences you can define the least recursive > singleton with leading positive bracket label. So from > bracket labelling you can recover a choice set of > recuirsuve singletons. > > So this would be a model with no bracketing notation > inside as the recent discussion on infinitely many > maximal chains. > > So the bracketing sequence won't be definable, > and in ~AC cases it might not even exist in the model as > a set. So in term of the forest visualisation above, the laat quoted permutation model is making independ vertical shifts in each vertical line. This makes ~AC troubles to pick a base point from each vertical line, and jence as noted trouble to have sets in the constructed model assigning bracket sequences to infinitely many veritcal lines. The above is the background to this present article. I am writing this now to note an additional related point I had not thought of when writng [0]. Above I was permuting vertically in each verrical line. Actually not an arbitrary permutation: a vertical shift respecting E membership. My new point is we can also permute horizonally, permuting one maximal chain to another. The [1] models with more than one maximal chain have such automorphisms. And if there are infinitely many maxamal chain we can make a permutation model based on such permutations. If we do that with full permutations on omega many maximal chain we get an amorphouse set of maximal chains. In general, we could copy other permutation groups and actions onto the maximal chains, and make the set of maximal chains be as various sets made in permutation models. Return to the case of full permuations on omega, ie an amorphous set of maximal chains, for definteness. In the resulting contructed ~AC model, there is no injection of infinitely many maximal chains into P(omega), by the usual sorts of permutation arguments. (Ie in the finite support model, any function from an infinite subset of the amorphous set into P(omega) must have finite range). So this makes an additional difficulty for extending the definition of bracket sequences to infinitely many maximal chains. I earlier in [0] discussed vertical permutations. I just introduced horizontal ones. We can combione them anyway way we please. We could put both into one model. Or we could put either one in and leave out the other. If both are left out, we construct a ZFC - regularity model, and the current round of difficulties disappear. We could make a set (though undefinable as discussed above) assigning unique P(omega) bracket sequences to to the members of #P(omega) or fewer many maximal chains. But the permutations if present make 2 distinct problems for the construction. To pick a base point in each maximal chain, and to assign uniqye P(omega) differences across maximal chains. In particular, if we use horizontal permutations and no vertical ones, we get a model which has a choice set picking a base point for each maximal chain, yet we still can't have a set assigning bracket sequences to infinitely many maximal chains by the trouble to assign P(omega) elements to chains. That concludes my comments about difficulties for bracket sequences. I will note further, I had said above to make the horizontal permutations correspond to other ~AC permutation constructions, and so get many kinds of models like this. I will note further this is ever more gereral for all ~AC type sets. Given any ZF + ~AC model and any set A in the model we could redo a variant of the [1] construction, with that ~AC model replacing the ZFC base model from [1]. Redo the construction so alpha defining ther various maximal chains varies over A members, instead of all ordinals. So obtain a ZF - regularity model where there maximal chains are isomorphic to that A. So any way that AC can fail involving a set A can be copied over to maxinal chains. -- David Libert ah170(a)FreeNet.Carleton.CA
From: zuhair on 1 Jan 2010 02:49 On Dec 31 2009, 10:45 pm, ah...(a)FreeNet.Carleton.CA (David Libert) wrote: > I will add some points to my account from the parent article > > [0] David Libert "Recursive Cardinals" > sci.logic, sci.math Dec 29, 2009 > http://groups.google.com/group/sci.logic/msg/8d9476e07a2e0b11 > > I wrote in [0] : > > > So the class of all recursive singletons is partitioned into > > these maximal chains, where each chain is ordered like Z. > > I did not mention explicitly in [0], but still true, is there > are no membership relations between distinct maximals chains. > > (I had nade a similar comment to this in an older article before > I had used the word "chains" to talk about these). > > So if you draw each maximal chain as a vertical line, with that > viertical line representing the Z ordering on the maximal chain > membership, the case of many maximal chains can be visualized > as a forest of vertical lines. > > In [0] I wrote about making models with various numbers of > maximal chains: > > > > > > > In > > > [1] David Libert "A new definition of Cardinality" > > sci.logic, sci.math Nov 23, 2009 > > http://groups.google.com/group/sci.math/msg/721cb8170033cf84 > > > I constructed a ZFC - regularity model with a proper class of > > recursive cardinals. > > > I still think that construction is ok. > > > You could modify that construction, to only put in one maximal > > chain of recrusvie singletons: just take the alpha as 0 instead of varying overll all > > ordinals as [1] did. (In [1] I was using my old definiton of singleton towers > > > So that would give a ZF - regularity model with exaclty one maximal chain, > > and hence a model where your discussion above defined bracket sequences for all > > recursive singletons. > > > But the original [1], or variants of it making alpha vary over less than a > > proper class but still over more than one value, would give models of > > ZF - regularity with more than one maximal chain of recursive singletons. > > I went on to note in [0] that your main opening discussion earlier of > bracket sequences only defined them on one maximal chain. > > I went on to suggest how it might be possible to extend that to more > maximal chains, involving P(omega) as you had been writing: > > > > > A few more points. > > > Later in you article as quoted above, you write about the possiblilty of > > have P(omega) recursive cardinals. (You called them singleton towers). > > > Here is a possible approach along suich lines. > > > Above, you had picked a recursive singleton x, and used it to define > > a bracket sequence for x's maximal chain. > > > Maybe you could extend such a definition to a second maximal chain, > > by picking a member rectrusive singleton y there, and making a similar > > defintion. > > > But now the two parts of the definition, on eahc maximal chain, > > depeneded on the arbitrary choide of x or y, so its this dependency > > that allows diostinct recursice singletons to get the same bracket > > sequence. > > > For example, the most straigtforward copy over y of what you did > > to x would be to assign both <0, 1, 2, ... > . But x and y are > > still distinct, and each bracket sequence also depernded on base point > > in the singleton tower. > > > You might try to make bracket sequences distinct by skipping numbers > > in the bracket labelling inside y. > > > So you might want to keep going that way, same idea with different > > skips on different maximal chains to get different bracket sequences. > > > So that gets you a P(omega) limit like you said. > > I went on in [0] to discuss some ~AC issues about carrying on the > above plan: > > > > > > > Other issues. That outline I said just now depends on picking > > a base point x or y or whatever in each maximal chain. > > > My [1] models admit automophisms. If you map each recursive > > singleton to its own singleton you get a non-identity bijection > > of the class of all recursove singletons to itself. This can > > be lifted by transginfinite rwecursion up the ranks over those > > to the entire model. > > > So there is no definable way to picj the x y etc from each > > maximal chain. > > > So at very least, the bracket sequeebce assignment will > > not be deinfable without parameters. > > > It depends on picking x y etc, anf then defining over > > those parematers. > > > But our discussiin was about ZF - regulairty, not > > ZFC - regularity. > > > [1] didn't dp this, but you could make a variant of [1], > > similar to my other models, where you permute independly > > within each maximal chain by shifitng the chain up or > > down together, possibly different shifts on different chains. > > > Obtain a model where there is no choice set to pick > > a base point in each maximal chain. > > > Given bracket sequences you can define the least recursive > > singleton with leading positive bracket label. So from > > bracket labelling you can recover a choice set of > > recuirsuve singletons. > > > So this would be a model with no bracketing notation > > inside as the recent discussion on infinitely many > > maximal chains. > > > So the bracketing sequence won't be definable, > > and in ~AC cases it might not even exist in the model as > > a set. > > So in term of the forest visualisation above, the laat quoted > permutation model is making independ vertical shifts in each > vertical line. > > This makes ~AC troubles to pick a base point from each > vertical line, and jence as noted trouble to have sets in the > constructed model assigning bracket sequences to infinitely > many veritcal lines. > > The above is the background to this present article. > > I am writing this now to note an additional related point > I had not thought of when writng [0]. > > Above I was permuting vertically in each verrical line. > Actually not an arbitrary permutation: a vertical shift > respecting E membership. > > My new point is we can also permute horizonally, permuting > one maximal chain to another. > > The [1] models with more than one maximal chain have such > automorphisms. > > And if there are infinitely many maxamal chain we can make > a permutation model based on such permutations. > > If we do that with full permutations on omega many > maximal chain we get an amorphouse set of maximal chains. > > In general, we could copy other permutation groups and > actions onto the maximal chains, and make the set of maximal > chains be as various sets made in permutation models. > > Return to the case of full permuations on omega, > ie an amorphous set of maximal chains, for definteness. > > In the resulting contructed ~AC model, there is no > injection of infinitely many maximal chains into P(omega), > by the usual sorts of permutation arguments. > > (Ie in the finite support model, any function from > an infinite subset of the amorphous set into P(omega) > must have finite range). > > So this makes an additional difficulty for extending the > definition of bracket sequences to infinitely many > maximal chains. > > I earlier in [0] discussed vertical permutations. > > I just introduced horizontal ones. > > We can combione them anyway way we please. We could put > both into one model. Or we could put either one in and > leave out the other. > > If both are left out, we construct a ZFC - regularity > model, and the current round of difficulties disappear. > We could make a set (though undefinable as discussed > above) assigning unique P(omega) bracket sequences > to to the members of #P(omega) or fewer many maximal > chains. > > But the permutations if present make 2 distinct problems > for the construction. To pick a base point in each > maximal chain, and to assign uniqye P(omega) > differences across maximal chains. > > In particular, if we use horizontal permutations and > no vertical ones, we get a model which has a choice > set picking a base point for each maximal chain, yet > we still can't have a set assigning bracket sequences to > infinitely many maximal chains by the trouble to assign > P(omega) elements to chains. > > That concludes my comments about difficulties for > bracket sequences. > > I will note further, I had said above to make the > horizontal permutations correspond to other ~AC > permutation constructions, and so get many kinds > of models like this. > > I will note further this is ever more gereral for > all ~AC type sets. > > Given any ZF + ~AC model and any set A in the model > we could redo a variant of the [1] construction, with that > ~AC model replacing the ZFC base model from [1]. > > Redo the construction so alpha defining ther various > maximal chains varies over A members, instead of all > ordinals. > > So obtain a ZF - regularity model where there maximal > chains are isomorphic to that A. > > So any way that AC can fail involving a set A can be copied > over to maxinal chains. > > -- > David Libert ah...(a)FreeNet.Carleton.CA Thanks David, now matters are clearer. I shall elaborate on that also. Zuhair
From: Transfer Principle on 2 Jan 2010 23:34 On Dec 25 2009, 6:28 am, zuhair <zaljo...(a)gmail.com> wrote: > Here I have presented an example of these x-Recursive Cardinals > were x is singleton, It shows that we cannot have > uncountably many recursive singletons, at least at informal level, > here these recursive singletons shall be called "singleton towers" As I've said earlier, I wanted to avoid these set theory debate threads until February. But I've been lurking this thread for a while now, and this discussion is turning out to be rather interesting, and so I make this post now (as it may be hard to find this thread in February). In this thread, zuhair declares that he is trying to define a new type of cardinality, "recursive cardinality." We already know that many so-called "cranks" would also like to redefine cardinality as well, so perhaps zuhair's "recursive cardinality" will turn out to be more acceptable to "cranks" than standard cardinality as defined by Cantor. Now zuhair begins by defining a "singleton tower": > First, a singleton tower, or what I call a "recursive singleton" can > be defined in the following manner: > x is a singleton tower iff > x is singleton & > For all y ( y e TC(x) -> y is singleton ) & > For all y ( y e TC(x) -> ~ y e TC(y) ) > So singleton towers are: hereditarily non circular singleton > hereditarily singletons. Then zuhair attempts to construct a "recursive singleton": > So in general the singleton tower xi+1 with bracket sequence of > <i+1,i+2,i+3,....> will be the member of the singleton tower xi with > bracket sequence of <i,i+1,i+2,i+3,...> > I think this is clear, so we have xi+1 e xi for all singleton towers > defined above. But then, as zuhair noted earlier in this thread, most standard theories, in particular ZF, prove that no recursive singleton exists. The well-known proof of the nonexistence of recursive singletons uses Foundation/Regularity, and so zuhair proceeds by working in ZF-Regularity rather than ZF. Yet I've mentioned in previous threads that Regularity alone doesn't prove the nonexistence of most illfounded sets, but usually requires additional axioms of ZF to prove. After all, the axiom of Regularity only directly prescribes that every nonempty be disjoint with one of its elements (its lone element in the case of singletons). So we know that a set x such that: x = {x} can't exist using Regularity alone, since the set x fails to be disjoint with its lone element, namely itself. But to prove the nonexistence of distinct sets x,y such that: x = {y} y = {x} Regularity alone fails, since both x and y are disjoint with their respective lone elements, namely each other. Of course, we know that it's neither x nor y, but the set: z = {x, y} that fails to be disjoint with either of its elements. Notice that this set is the _transitive closure_ of both x and y, and its existence is guaranteed by the Axiom of Pairing. In general, it's not the illfounded set itself that fails to be disjoint with all of its elements, but its transitive closure (or a subset thereof) that fails, and so we need other axioms in addition to Regularity that guarantee the existence of the transitive closure. This is why zuhair frequently refers to the transitive closure in his posts, going so far as to introduce the notation TC(x) for the transitive closure of x. So to prove that no recursive singleton exists, we must look first at its transitive closure, as zuhair does here: > Now lets take the tower were i=0, i.e. x0 with bracket sequence of > <0,1,2,3,....> > Now what is the transitive closure of x0 > This would be > TC(x0)= {x1,x2,x3,..........} > right! And we see that TC(x0) fails to be disjoint with any of its elements, since (TC(x0) intersect xi) = xi+1 for any natural number i. Thus, we can contradict the existence of x0 by showing that its existence implies that of TC(x0). The usual proof in ZF thereof begins by using the Axiom of Infinity to derive the existence of an inductive set, omega. Then we use the Replacement Schema to replace the natural number i in the set omega with xi+1. The resulting set is TC(x0). Therefore the proof that no singleton tower exists requires not just Regularity, but Infinity and Replacement Schema. So we now have choice of three theories, namely ZF-Regularity, ZF-Infinity, and ZF-Replacement Schema (in other words, the theory Z+Regularity) in which to work. In any of these three theories, the proof that no singleton tower exists falls through, and so we may attempt to work with singleton towers. Of course, zuhair chose to drop Regularity, and this is what most set theorists would do in this situation. But recall how at the start of this post, my goal was to present zuhair's recursive cardinality as acceptable to "_cranks_." And, given a choice among Regularity, Infinity, and Replacement Schema to drop, most "cranks" would choose _Infinity_ as the axiom they'd like to drop. (Note that nearly every set theoretic "crank" thread is basically an argument over the Axiom of Infinity in ZF.) Notice that the general proof in ZF that every set has a transitive closure requires Infinity. Thus, instead of ZF-Regularity, I will choose to work in ZF-Infinity for the remainder of this post. This theory should be more acceptable to "cranks," including even the ultrafinitists, since even an ultrafinitist should have no problem with the existence of _singletons_, since all of these sets xi are merely singletons! (On the other hand, sometimes I'd prefer to work with sets in which not only is xi+1 an element of xi, but a _subset_ of xi as well, so that we can mimic von Neumann ordinals. But we won't do so here.) I've attempted this approach in several previous threads, but always seem to run into trouble. The problem is that I want to look at the theory: ZF-Infinity+"a singleton tower exists" but the problem is that there appears to be no way to write "a singleton tower exists" using only FOL and the language of ZF-Infinity. For zuhair gives two definitions of his concept "singleton tower," one using transitive closures and the other using natural numbers as indices, but neither of these are permitted in ZF-Infinity. Thus, even though that there is likely a (nonstandard, of course) model of the theory ZF-Infinity in which singleton towers exist, there's no way to add an axiom prescribing there existence. (At least I haven't found a way, yet.) In another thread, someone -- possibly either of the other posters is this thread, Hartley or Libert -- attempted to give a theory of nonstandard arithmetic but noting that if Goldbach's Conjecture is undecidable in PA, then GC would be true in the standard model of PA, and so PA+~GC would give a theory of nonstandard arithmetic. The language of the theory would add an additional symbol "w" to the language of PA, and w would be treated as a nonstandard natural number and a counterexample of GC. But to me, this is unsatisfactory as we don't _know_ whether GC is decidable in PA or not. I'd prefer a statement that is _known_ to be undecidable in PA, perhaps Goodstein's Theorem, and let "w" be a counterexample to Goodstein rather than Goldbach. And so we'd like to do the same in this set theory, adding the symbol "ST" to the language of ZF-Infinity and stipulate that ST is a singleton tower. But we can't do so unless we know how to write "there exists a singleton tower" as a (first-order) axiom in the language of ZF-Infinity. Thus concludes this post for now. For February, I've already stated that I'll return to working on the problem of finding a rigorous definition of the "crank" ellipsis, and so I'll work of that before returning to "recursive cardinality" for the "cranks." (BTW, someone asked me what a "crank" ellipsis is, so let me answer that. We know that in standard analysis, some real numbers, such as 1/3, cannot be written in decimal notation with only finitely many digits, and so one often resorts to using an ellipsis, so 1/3 = 0.333... now. But some "cranks" want to extend the standard ellipsis in order to include nonstandard reals such as 0.333...3, and often numbers such as 0.999...9 as distinct from unity, with the difference being something like 0.000...1. The "crank" ellipsis problem entails finding a rigorous definition of ellipsis so that these claims are provable.)
From: David Libert on 3 Jan 2010 02:21 Transfer Principle (lwalke3(a)lausd.net) writes: > On Dec 25 2009, 6:28=A0am, zuhair <zaljo...(a)gmail.com> wrote: [Deletion] > Now zuhair begins by defining a "singleton tower": > >> First, a singleton tower, or what I call a "recursive singleton" can >> be defined in the following manner: >> x is a singleton tower =A0iff >> x is singleton & >> For all y ( y e TC(x) -> y is singleton ) & >> For all y ( y e TC(x) -> ~ y e TC(y) ) >> So singleton towers are: hereditarily non circular singleton >> hereditarily singletons. > > Then zuhair attempts to construct a "recursive singleton": > >> So in general the singleton tower xi+1 with bracket sequence of >> <i+1,i+2,i+3,....> will be the member of the singleton tower xi with >> bracket sequence of <i,i+1,i+2,i+3,...> >> I think this is clear, so we have xi+1 e xi for all singleton towers >> defined above. > > But then, as zuhair noted earlier in this thread, most > standard theories, in particular ZF, prove that no recursive > singleton exists. The well-known proof of the nonexistence of > recursive singletons uses Foundation/Regularity, and so zuhair > proceeds by working in ZF-Regularity rather than ZF. > > Yet I've mentioned in previous threads that Regularity alone > doesn't prove the nonexistence of most illfounded sets, but > usually requires additional axioms of ZF to prove. After all, > the axiom of Regularity only directly prescribes that every > nonempty be disjoint with one of its elements (its lone > element in the case of singletons). So we know that a set x > such that: > > x =3D {x} > > can't exist using Regularity alone, since the set x fails to > be disjoint with its lone element, namely itself. But to > prove the nonexistence of distinct sets x,y such that: > > x =3D {y} > y =3D {x} > > Regularity alone fails, since both x and y are disjoint with > their respective lone elements, namely each other. Of course, > we know that it's neither x nor y, but the set: > > z =3D {x, y} > > that fails to be disjoint with either of its elements. Notice > that this set is the _transitive closure_ of both x and y, > and its existence is guaranteed by the Axiom of Pairing. In > general, it's not the illfounded set itself that fails to be > disjoint with all of its elements, but its transitive closure > (or a subset thereof) that fails, and so we need other axioms > in addition to Regularity that guarantee the existence of the > transitive closure. This is why zuhair frequently refers to > the transitive closure in his posts, going so far as to > introduce the notation TC(x) for the transitive closure of x. > > So to prove that no recursive singleton exists, we must look > first at its transitive closure, as zuhair does here: > >> Now lets take the tower were i=3D0, i.e. x0 with bracket sequence of >> <0,1,2,3,....> >> Now what is the transitive closure of x0 >> This would be >> TC(x0)=3D {x1,x2,x3,..........} >> right! > > And we see that TC(x0) fails to be disjoint with any of its > elements, since (TC(x0) intersect xi) =3D xi+1 for any natural > number i. Thus, we can contradict the existence of x0 by > showing that its existence implies that of TC(x0). The usual > proof in ZF thereof begins by using the Axiom of Infinity to > derive the existence of an inductive set, omega. Then we use > the Replacement Schema to replace the natural number i in the > set omega with xi+1. The resulting set is TC(x0). > > Therefore the proof that no singleton tower exists requires > not just Regularity, but Infinity and Replacement Schema. So > we now have choice of three theories, namely ZF-Regularity, > ZF-Infinity, and ZF-Replacement Schema (in other words, the > theory Z+Regularity) in which to work. In any of these three > theories, the proof that no singleton tower exists falls > through, and so we may attempt to work with singleton towers. Yes, that's right. In ZF, refuting singleton towers depends on all three of regularity, infinity and replacement. As you say, by dropping any one we can then get models with singleton towers (if ZF is consistent). > Of course, zuhair chose to drop Regularity, and this is what > most set theorists would do in this situation. But recall how > at the start of this post, my goal was to present zuhair's > recursive cardinality as acceptable to "_cranks_." And, given > a choice among Regularity, Infinity, and Replacement Schema > to drop, most "cranks" would choose _Infinity_ as the axiom > they'd like to drop. (Note that nearly every set theoretic > "crank" thread is basically an argument over the Axiom of > Infinity in ZF.) Notice that the general proof in ZF that > every set has a transitive closure requires Infinity. > > Thus, instead of ZF-Regularity, I will choose to work in > ZF-Infinity for the remainder of this post. This theory > should be more acceptable to "cranks," including even the > ultrafinitists, since even an ultrafinitist should have no > problem with the existence of _singletons_, since all of > these sets xi are merely singletons! (On the other hand, > sometimes I'd prefer to work with sets in which not only is > xi+1 an element of xi, but a _subset_ of xi as well, so that > we can mimic von Neumann ordinals. But we won't do so here.) > > I've attempted this approach in several previous threads, > but always seem to run into trouble. The problem is that > I want to look at the theory: > > ZF-Infinity+"a singleton tower exists" > > but the problem is that there appears to be no way to write > "a singleton tower exists" using only FOL and the language > of ZF-Infinity. For zuhair gives two definitions of his > concept "singleton tower," one using transitive closures and > the other using natural numbers as indices, but neither of > these are permitted in ZF-Infinity. Thus, even though that > there is likely a (nonstandard, of course) model of the > theory ZF-Infinity in which singleton towers exist, there's > no way to add an axiom prescribing there existence. (At > least I haven't found a way, yet.) You are right. It is hard in this theory to even define what a singleton tower is. Zuhair have a definition above. But he was working in ZF - regularity, so he had infinity and replacement. Based on those two, he had transitive closures exist and are definable, so he could use them in his definition of singleton tower. I think I have a reasonable way to proceed in the theory ZF - infinity. In this theory we can't prove transitive closures exist as a set. It is not even so easy at first to define transitive closures as a definable class. The obvious way to define transitive closure of x, TC(x) as a class would be the class of all y such that there is a finite sequence of membership from x to y. But that depends on having a definition of finite. The usual definition of finite, with axiom of infinity available, would be having cardinality a member of omega. This doesn't work in ZF - infinity. I think I do have a reasonable definition of TC(x) as a class in ZF - infinity. It is trying to formalize the idea above: a finite chain of membership from x to y. Any first order formalization will never fully capture this full definition as seen from outside models. But I think this one is a reasonable first order approximation, just as the usual ZF definition of omega is a reasonable first order approximation to the notion of the set of finite ordinals, even though there are nonstandard models of ZF with interpretion of the omega definition including more than finite ordinals as viewed from outside the model. So here is the proposed definition of the class TC(x) in ZF - infinity. TC(x) is the class of those y such that there exists a set A such that A is linearly ordered by set inclusion & that linear ordering on A has smallest and largest elements & {x} is a member of A & {x} is the smallest member of A by set inclusion & y is a member of the largest element of A by set inclusion & every member of A other than the largest element has an immediate successor by set inclusion & every member of A other than {x} has an immediate predecessor by set inclusion & for every members B, C of A, if B is immediate predecessor of C then there is some z such that C = B union {z} and z is not member of C & for every C, D members of A if C is immediate predecessor of D and for z1, z2 such that (C = {x} and z1 = x or there exists B in A such that B is immediate predecessor of C and C = B union {z1} and z1 not member of B) & D = C union {z2} and z2 not member C then z2 member of z1 & if C is the largest member of A by set inclusion and B is the immdediate successor of C then C = B union {y} & every subset E of A has a smallest element with respect to set inclusion . The set A above will code a finite sequence of distinct members in a chain from x to y. The coding above assumes A has more than one member. If x member x we would want to use instead {{x}} as the sequence, which my coding above won't allow for. But I am working over ZF - infinity, so x member x is imposible. If you want this to work also without regularity and handle x member x, easiest is disjunct the above with disjunct : x member x and y = x . That last conjunct above about E is trying to say in first order that A is well-founded with respect to set inclusion. So a "well-founded" linear order with immediate successors and predecessors except at endpoints, this is trying to formalize A being finite. This definition is strong enough to allow proofs by induction on all TC(x) members by level of membership nesting below x. So I think this is a reasonable first order coding of TC(X). If you add back in infinity, and have the usual definition of TC(x), that theory proves the two TC(x) definitions are equivalent. So accepting for the moment this formalization of TC(x), we can now repeat Zuhair's definition of singleton tower, using this TC(x) where Zuhair's definition used TC(x). So we come to a definition in ZF - infinity of what a singleton tower is. And it is possible to construct a model of ZF - infinity + there is a singleton tower. This even with regularity in the constructed model. I did that in [1] David Libert "Axiom of infinity and the set of all hereditary finite sets" sci.logic Oct 3, 2007 http://groups.google.com/group/sci.logic/msg/7593d4adf17732b7 Note, as [1] explictly noted, this is also a model with a set x having no set TC(x). [Deletion] > And so we'd like to do the same in this set theory, adding > the symbol "ST" to the language of ZF-Infinity and stipulate > that ST is a singleton tower. But we can't do so unless we > know how to write "there exists a singleton tower" as a > (first-order) axiom in the language of ZF-Infinity. I think above was a reasonable try at that. [Deletion] -- David Libert ah170(a)FreeNet.Carleton.CA
From: Transfer Principle on 4 Jan 2010 15:00
On Jan 2, 11:21 pm, ah...(a)FreeNet.Carleton.CA (David Libert) wrote: > Transfer Principle (lwal...(a)lausd.net) writes: > > I've attempted this approach in several previous threads, > > but always seem to run into trouble. The problem is that > > I want to look at the theory: > > ZF-Infinity+"a singleton tower exists" > > but the problem is that there appears to be no way to write > > "a singleton tower exists" using only FOL and the language > > of ZF-Infinity. For zuhair gives two definitions of his > > concept "singleton tower," one using transitive closures and > > the other using natural numbers as indices, but neither of > > these are permitted in ZF-Infinity. Thus, even though that > > there is likely a (nonstandard, of course) model of the > > theory ZF-Infinity in which singleton towers exist, there's > > no way to add an axiom prescribing there existence. (At > > least I haven't found a way, yet.) > So here is the proposed definition of the class > TC(x) in ZF - infinity. > TC(x) is the class of those y such that there exists > a set A such that > A is linearly ordered by set inclusion & > that linear ordering on A has smallest and largest > elements & > {x} is a member of A & > {x} is the smallest member of A by set inclusion & > y is a member of the largest element of A by set > inclusion & > every member of A other than the largest element has > an immediate successor by set inclusion & > every member of A other than {x} has an immediate > predecessor by set inclusion & > for every members B, C of A, if B is immediate > predecessor of C then there is some z > such that C = B union {z} and z is not member > of C & > for every C, D members of A if C is immediate > predecessor of D and for z1, z2 such that > (C = {x} and z1 = x > or > there exists B in A such that B is immediate > predecessor of C and C = B union {z1} > and z1 not member of B) & > D = C union {z2} and z2 not member C > then z2 member of z1 & > if C is the largest member of A by set inclusion > and B is the immdediate successor of C then > C = B union {y} & > every subset E of A has a smallest element with > respect to set inclusion . Thanks for the information! Wow, this does seem long, but I realize that this is the cost of dropping Infinity. So how will this information in Libert's post above help us define recursive cardinality for the so-called "cranks"? The answer is, now that we have a singleton tower, we may go back to zuhair's post and proceed as he does to define his recursive cardinality. The hope is that for at least one "crank," recursive cardinality will be an acceptable alternative to the standard (Cantorian) cardinality. According to zuhair, singleton towers of uncountable "height" may be impossible. The "cranks" may appreciate this, since many of them are opposed to the notion of uncountability in the first place. So we already have at least one point of agreement between zuhair and the "cranks." |