Is ZF + CH + not-CC consistent?
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From: David Libert on 20 Nov 2009 00:19 Herman Jurjus (hjmotz(a)hetnet.nl) writes: > David Hartley wrote: >> In message <hdusrq$udh$1(a)news.eternal-september.org>, Herman Jurjus >> <hjmotz(a)hetnet.nl> writes >>> Subject says it (CC stands for 'countable choice'). >>> >>> More generally, the question is: 'how much' of AxC is implied by CH? >>> >>> (As AC is equivalent to the trichotomy of cardinals, and CH imply >>> trichotomy for sets not larger than P(N), someone might expect that >>> ZF+CH perhaps implies some weak forms of choice.) >> >> First, how do you state CH in ZF without AC? Is it: >> there is no cardinal k such that aleph_0 < k < c, >> there is no uncountable cardinal less than c, >> c = aleph_1, >> or every uncountable cardinal is greater than or equal to c >> >> >> At first glance, none of the first three rules out infinite, >> Dedekind-finite sets, so you may not have trichotomy for sets not larger >> than P(N). (All but the first obviously give it for sets smaller than or >> comparable to P(N).) > > I was in a terrible hurry when I wrote that post - sorry for that. > Forget about all the baloney. > > The question is: > Is it known whether CH implies CC, DC, or any other weak forms of AC? > > -- > Cheers, > Herman Jurjus On Nov 18 Herman Rubin posted to this thread about Fraenkel-Mostowski models of ZFU + CH + not-CC : http://groups.google.com/group/sci.logic/msg/0f956ac2e786a6d3 Herman's article appeared in google groups but not at my posting site, so I am doing this as a followup article to the one above. Herman wrote about these Fraenkel Mostowsku models copying over to corresponding Cohen models. I note that in fact such a transfer is indeed possible, by the Jech Sochar Transfer Theorem. I wrote about these models generally and that teransfer theorem in [1] David Libert "Cohen symmetric choiceless ZF models" sci.logic July 6, 2000 http://groups.google.com/group/sci.logic/msg/b4271c2585d2f1e5 In _Set Theory and the Continuum Hypothesis_ , Paul Cohen gives a ZF model with a countable sequence of 2 element sets having no choice function. Copy this over to an F-M model, and use Jech Sochar to put that sequence at high rank, with the universe at rank reals and below same as the ground model (start from CH ground model). Basically: you can make high rank sets be very different from low rank sets. Some other related points of discussion. The original question above was for CH. But for a related question, consider GCH. In the absence of AC as an assumption, there are even different reasonable versions of what GCH might mean. Fior a couple of these we can get results. One GCH variation not assuming AC to start is for every set A there are no sets of cardinality strictly between #A and #P(A). (Make a reasonable ~AC generalization of cardinality to all sets: for example Scott's trick). In _Set Theory and the Continuum Hypothesis_ , Cohen goves a proof that ZF proves that GCH version implies AC. Another reading of GCH could be that for all von Neumann cardinals, that is all initial ordinals, in other words the cardinalities of well-orderable sets, the powerset has cardinality the successor cardinal. I think I have found a proof that ZF proves this statement -> AC. So that's GCH. Regarding CH again, above were ~AC models with ~AC at high rank. But we could also ask about a local version of AC : can the reals be well-ordered. This comes down to the question, as was raised by David Hartley indirectlty quoted above, which version of CH we use in the absence of AC. One possibility, as David raised is c = aleph_1. So for this version, of course the reals are well-orderable. Another version was there are no cardinal k such that aleph_0 < k < c . I found online in Kanamori's book _The Higher Infinite_ that Solovay's famous model with everything Lebesgue measurable also satisifes that every uncountable set of reals has the perfect set property. A perfect set is defined to be a closed set with no isolated points. The perfect set property is that the set has a non-empty perfectsubset. Every perfect set has cardinality c. So Solovay's model has no cardinal k s.t. aleph_0 < k < c . Also, if the reals are well-orderable, we can doagonalize to contruct an uncountable set of reals without the perfect set property. So the reals are not well-orderable in Solvay's model. (We already knew that: if the reals are well-orderable then we can do the usual contruction of a non-measurable set in ZF, so from everything being Lebesgue measurable we already knew the reals are not well-orderable). Solovay's model started with an inaccessible cardinal in the ground model, and collapsed it by forcing to aleph_1. Kanamori in the book mentions there was a theorom of Specker in ZF : if every uncountable set of reals has the perfect set property and aleph_1 is regular, then aleph_1 is inaccessible in L. In Solovay's model aleph_1 is regular. (Even though in some ~AC models aleph_1 might not be regular, in Solovay's particular model it is). So Solovay really did need the inaccessible. I think I found a proof of the Specker result, which gives a bit more information than the statement. As noted above if every uncountable set of reals has the perfect set property, then every uncountable set of reals is c size, and the reals are not well-orderable. I think the proof actually shows these conclusions are sufficient. So if every uncountable set of reals is c size and the reals are not well-orderable and aleph_1 is regular, then aleph_1 is inaccessible in L. So this is saying there can be the no k with aleph_0 < k < c reading of CH with the reals not well-orderable, but this is in both directions has the level of an inaccessible cardinal. Also, AD gets all this. Every uncountable set has the perfect set property and the reals are not well-orderable. But AD is at a level way beyond one inaccessible. The other possibility David raised above to phrase CH without assuming AC is every uncountable cardinal is greater than or equal to c. I don't know how this version works out. I don't know if Solovay's model satisfies this. And I don't even know if the Jech Sochar models above satisfy this. So that version is all unsettled for me for now. -- David Libert ah170(a)FreeNet.Carleton.CA
From: Herman Jurjus on 20 Nov 2009 07:51 David Libert wrote: > Herman Jurjus (hjmotz(a)hetnet.nl) writes: >> The question is: >> Is it known whether CH implies CC, DC, or any other weak forms of AC? >> >> -- >> Cheers, >> Herman Jurjus > > > On Nov 18 Herman Rubin posted to this thread about Fraenkel-Mostowski > models of ZFU + CH + not-CC : > > http://groups.google.com/group/sci.logic/msg/0f956ac2e786a6d3 Yes, found! > Herman's article appeared in google groups but not at my posting site, > so I am doing this as a followup article to the one above. Thanks very much to all respondents! (And I'm glad that I included sci.math.) > I found online in Kanamori's book _The Higher Infinite_ > that Solovay's famous model with everything Lebesgue measurable > also satisifes that every uncountable set of reals has the perfect > set property. Online? Where? BTW, is there an online account of Solovay's construction? Perhaps in Kanamori's book? -- Cheers, Herman Jurjus
From: David Libert on 20 Nov 2009 15:22 Herman Jurjus (hjmotz(a)hetnet.nl) writes: > David Libert wrote: >> Herman Jurjus (hjmotz(a)hetnet.nl) writes: > >>> The question is: >>> Is it known whether CH implies CC, DC, or any other weak forms of AC? >>> >>> -- >>> Cheers, >>> Herman Jurjus >> >> >> On Nov 18 Herman Rubin posted to this thread about Fraenkel-Mostowski >> models of ZFU + CH + not-CC : >> >> http://groups.google.com/group/sci.logic/msg/0f956ac2e786a6d3 > > Yes, found! > >> Herman's article appeared in google groups but not at my posting site, >> so I am doing this as a followup article to the one above. > > Thanks very much to all respondents! > (And I'm glad that I included sci.math.) > >> I found online in Kanamori's book _The Higher Infinite_ >> that Solovay's famous model with everything Lebesgue measurable >> also satisifes that every uncountable set of reals has the perfect >> set property. > > Online? Where? Just now, I didn't find the exact page I looked at last time, but found a similar one: http://books.google.com/books?id=Yctm6yuclBsC&pg=PA135&lpg=PA135&dq=perfect+set+property+AD&source=bl&ots=anVaazZ30F&sig=4kE1gk0NjKHSaTzyzYa6r5GAW64&hl=en&ei=AfYGS-KdMMSFnAeI3tnACw&sa=X&oi=book_result&ct=result&resnum=1&ved=0CAsQ6AEwADgK#v=onepage&q=perfect set property AD&f=false This is a sample from Kaanmori's book. Unfortuantely it is only a sample, to entice you to buy the book. It does include discussion of Solovay's and Speckers results though. But not in detail. I found that by googling: perfect set property AD . > BTW, is there an online account of Solovay's construction? > Perhaps in Kanamori's book? I don't know offhand. From what I saw of the samples, it looks like Kanamori will have more discussion of Solovay's proof. > -- > Cheers, > Herman Jurjus -- David Libert ah170(a)FreeNet.Carleton.CA
From: David Hartley on 20 Nov 2009 15:30 In message <he58t1$6oi$1(a)theodyn.ncf.ca>, David Libert <ah170(a)FreeNet.Carleton.CA> writes >The other possibility David raised above to phrase CH without assuming >AC is every uncountable cardinal is greater than or equal to c. > > I don't know how this version works out. I don't know if Solovay's >model satisfies this. And I don't even know if the Jech Sochar models >above satisfy this. > > So that version is all unsettled for me for now. I threw that one in as a way of getting trichotomy for all cardinals not larger than c, which Herman had mentioned (though perhaps he really meant 'smaller than or equal to c'). I too don't know anything about how it works out (not that I know much about the other versions either). -- David Hartley
From: Gc on 20 Nov 2009 23:29
On 20 marras, 14:51, Herman Jurjus <hjm...(a)hetnet.nl> wrote: > David Libert wrote: > > Herman Jurjus (hjm...(a)hetnet.nl) writes: > >> The question is: > >> Is it known whether CH implies CC, DC, or any other weak forms of AC? > > >> -- > >> Cheers, > >> Herman Jurjus > > > On Nov 18 Herman Rubin posted to this thread about Fraenkel-Mostowski > > models of ZFU + CH + not-CC : > > > http://groups.google.com/group/sci.logic/msg/0f956ac2e786a6d3 > > Yes, found! > > > Herman's article appeared in google groups but not at my posting site, > > so I am doing this as a followup article to the one above. > > Thanks very much to all respondents! > (And I'm glad that I included sci.math.) > > > I found online in Kanamori's book _The Higher Infinite_ > > that Solovay's famous model with everything Lebesgue measurable > > also satisifes that every uncountable set of reals has the perfect > > set property. > > Online? Where? > > BTW, is there an online account of Solovay's construction? I remember that G.A Edgar (Or D.Renfro) posted a link to the original paper of Solovay in sci.math a month or couple of months ago. I tried to find the link on the google group archives, but they seem to be totally useless now. > Perhaps in Kanamori's book? > > -- > Cheers, > Herman Jurjus |