Timeline for A question regarding a remark of John Horton Conway
Current License: CC BY-SA 3.0
11 events
| when toggle format | what | by | license | comment | |
|---|---|---|---|---|---|
| Aug 31, 2012 at 8:01 | comment | added | Thomas Benjamin | that is, will probably not occur | |
| Aug 31, 2012 at 7:59 | comment | added | Thomas Benjamin | @Stephan: In No $Omega$={0,1,2,3...|}. Since $omega$ is definable in No, then it exists (at least in No). Is {0,1,2,3...|} mere notation unless in some sense backed up by an Axiom of Infinity? as regards Russell's paradox, R={x: x $not a member of$ x} just cannot have "___ is a member of R" or "___ is not a member of R" predicated of it. In any case, since No delimits "everything you can define in some natural way", a Russell-type paradox involving No (unless you have one in mind) will probably occur. | |
| Aug 29, 2012 at 5:20 | comment | added | Stefan Geschke | Just two remarks. 1) Yes, some forcing extensions are constructed by adding certain sets of natural numbers. I misunderstood this sentence. But there are forcing extensions without new sets of natural numbers. 2) No cannot be constructed without any assumptions. For example, at some point you will need the fact that $\omega$ is actually an object that you can use for your further constructions and so on. As I pointed out before, if you just assume that everything that you can define in some natural way exists, then you will arrive at Russell's paradox. | |
| Aug 29, 2012 at 3:50 | comment | added | Thomas Benjamin | in the 'real' set theoretic universe of ZFC to the countable models of ZFC. I hope you find this comment intelligible | |
| Aug 29, 2012 at 3:48 | comment | added | Thomas Benjamin | Since No as defined by Conway (unless you can find some contradiction in his formulation) seems perfectly consistent (just as defining N by assuming the existence of 0 and applying a successor function s(x) first to 0 and then to s(0), s(s(0)), etc to generate N) is consistent, it seems to me at least that No is, as Ehrlich has proven, that No is the absolute arithmetic continuum and therefore should contain the versions of No relativised to models of ZFC (that is, Conway's construction carried out in ZFC) as subfields. In this manner, No acts as P(N) does | |
| Aug 29, 2012 at 3:23 | comment | added | Thomas Benjamin | @Stefan: I was referring to P(N), the power set of N which of course can be put in 1-1 correspondence with the reals (the way I worded that part of the sentence was very ambiguous--sorry). As regards my second comment, since it refers to the naturalistic account of forcing, I guess your question would translate into 'How can you access things that are outside the set theoretic universe?'. My point would be, do you have to translate all set theoretic notions into some axiomatic theory for the set theoretic notions in question to be valid? | |
| Aug 28, 2012 at 8:34 | comment | added | Stefan Geschke | Regarding your second comment, Conway defines No without explicit recourse to some axiomatic framework because this is the way mathematics is done and understood by most people. This can lead to problems such as Russell's Paradox, but we feel that we know now how to avoid this kind of problem. But it is possible to carry out Conway's construction in ZFC. And if you do that, the class No clearly depends on the model that you are in. And if you are working in a small model of ZFC, how can you access things that are outside this model? | |
| Aug 28, 2012 at 8:24 | comment | added | Stefan Geschke | @Thomas: Concerning your first comment: If $M$ is a countable transitive model of set theory, then it contains all natural numbers. Do you mean real numbers? Some forcing extensions are obtained by adding new real numbers, but some are not. | |
| Aug 28, 2012 at 7:18 | comment | added | Thomas Benjamin | I asked this question of Professor Hamkins on boolesrings but he does not think this is the case since different models of ZFC can have different models of No (?--I hope I have not misinterpreted his answer). However, since Professor Conway has successfully defined No without recourse to axiomatic systems of set theory No seems to exist 'outside' these axiomatic systems and the models that interpret and realize them which again suggests that No might contain all the 'generic reals' various accounts of forcing (especially the naturalistic account of forcing) needs to form M[G]. | |
| Aug 28, 2012 at 6:58 | comment | added | Thomas Benjamin | @Stefan: Thanks. Since you know a thing or two about forcing, does No contain all reals defined by forcing 'constructions' of generic extensions of some model M of, say ZFC? I ask this question because, at least as I understand it, (to use the example of forcing not-CH) when M is a countable transitive model of ZFC, one uses the sets of natural numbers not contained in M to form the generic sets of M[G] and in analogy to this, No might contain generic sets for models of ZFC in a naturalistic account of forcing | |
| Aug 27, 2012 at 7:00 | history | answered | Stefan Geschke | CC BY-SA 3.0 |