A Reformulation of Z-Reg.2
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From: MoeBlee on 17 Jun 2010 13:02 On Jun 16, 7:03 pm, zuhair <zaljo...(a)gmail.com> wrote: > > I haven't looked over your new formulations, but last night I did > > figure out a way to formulate your compression schema so that it and > > your union axiom entail the separation(with built in extensionality) > > schema, while the separation schema (with built in extensionality) > > entails your compression schema. > > How can you do that without pairing and power??? I don't see a way to > do that. > Actually it would be rather interesting to see your proof that unique > separation with union only can prove my comprehension schema, Then my formulation must not be equivalent with yours. So you're not claiming that your unique comprehension entails your unique separation (ordinary separtion but with uniqueness quantifier)? If you're not claiming that, then that would make better sense to me from what I though you had previously claimed. MoeBlee
From: zuhair on 17 Jun 2010 15:49 On Jun 17, 12:02 pm, MoeBlee <jazzm...(a)hotmail.com> wrote: > On Jun 16, 7:03 pm, zuhair <zaljo...(a)gmail.com> wrote: > > > > I haven't looked over your new formulations, but last night I did > > > figure out a way to formulate your compression schema so that it and > > > your union axiom entail the separation(with built in extensionality) > > > schema, while the separation schema (with built in extensionality) > > > entails your compression schema. > > > How can you do that without pairing and power??? I don't see a way to > > do that. > > Actually it would be rather interesting to see your proof that unique > > separation with union only can prove my comprehension schema, > > Then my formulation must not be equivalent with yours. > > So you're not claiming that your unique comprehension entails your > unique separation (ordinary separtion but with uniqueness quantifier)? > If you're not claiming that, then that would make better sense to me > from what I though you had previously claimed. > > MoeBlee Moe I think you are confused. Let me clarify matters. Of course I am claiming that unique separation is entailed by my comprehension schema and union, and I wrote a proof of that. so I am still claiming that Z-Reg. is entailed by theory A (i.e Z-Reg. is a sub-theory of A), as I exactly wrote in this post. However what I am NOT claiming is that my comprehension schema follows from unique separation alone without pairing , union and power. My comprehension schema is NOT a subtheory of (Extensionality + Separation) , But it is a subtheory of Z-Regularity-Infinity. So I am still claiming that theory A=Z-Reg. The second issue, I never claimed that my comprehensions schema is entailed by unique separation alone, I never made this claim, it is you who made such a claim not me. I said that my comprehension schema is provable in Z-Reg.-Infinity, I never said it is provable from unique separation alone, never. The problem here Moe, is that you are going around matters, you are making some assumptions that I never said, and then you attach them to me as if I said them. I think the simplest way to understand what I wrote is to read what I wrote carefully and respond to the arguments one by one. So your argument of dependence of these axioms is simply WRONG. The formulation is their, I believe it is very clear, straightforwards, and the proof is also clear. Zuhair
From: MoeBlee on 17 Jun 2010 16:39 On Jun 17, 2:49 pm, zuhair <zaljo...(a)gmail.com> wrote: > On Jun 17, 12:02 pm, MoeBlee <jazzm...(a)hotmail.com> wrote: > > On Jun 16, 7:03 pm, zuhair <zaljo...(a)gmail.com> wrote: > > > > > I haven't looked over your new formulations, but last night I did > > > > figure out a way to formulate your compression schema so that it and > > > > your union axiom entail the separation(with built in extensionality) > > > > schema, while the separation schema (with built in extensionality) > > > > entails your compression schema. > > > > How can you do that without pairing and power??? I don't see a way to > > > do that. > > > Actually it would be rather interesting to see your proof that unique > > > separation with union only can prove my comprehension schema, > > > Then my formulation must not be equivalent with yours. > > > So you're not claiming that your unique comprehension entails your > > unique separation (ordinary separtion but with uniqueness quantifier)? > > If you're not claiming that, then that would make better sense to me > > from what I though you had previously claimed. > > Moe I think you are confused. (1) Ah, I missed your word 'only' in "your proof that unique separation with union only can prove my comprehension schema" I did not say ONLY. What I said is that unique separation (that is exactly: separation schema except uniqueness quantifier) proves MY reversion (which I haven't posted) of your unique comprehension. And it's trivial, since MY re-version is just an instance of unique separation. (2) Sorry, I meant the other direction: Do you claim unique separation (exactly: separation schema except uniqueness quantifier) with union entails unique comprehension? But you answered that below, and it's fine, since your answer is 'no'. > However what I am NOT claiming is that my comprehension schema > follows from unique separation alone without pairing , union and > power. Ah, okay. (If I recall, you were saying something to the effect in a previous(?) thread that the two axiomatizations were equivalent? But you're not saying that now, okay.) Anyway, I've not looked at your new formulations or posted my suggested simplification of your earlier version , so I can't judge at this time whether your proof of unique separation from unique comprehension and union is okay or not. I only know that I was able to make it work by revising one or your earlier versions of unique comprehension. I'll try to check your latest version this week or next week. MoeBlee
From: zuhair on 17 Jun 2010 17:16 On Jun 17, 3:39 pm, MoeBlee <jazzm...(a)hotmail.com> wrote: > On Jun 17, 2:49 pm, zuhair <zaljo...(a)gmail.com> wrote: > > > > > On Jun 17, 12:02 pm, MoeBlee <jazzm...(a)hotmail.com> wrote: > > > On Jun 16, 7:03 pm, zuhair <zaljo...(a)gmail.com> wrote: > > > > > > I haven't looked over your new formulations, but last night I did > > > > > figure out a way to formulate your compression schema so that it and > > > > > your union axiom entail the separation(with built in extensionality) > > > > > schema, while the separation schema (with built in extensionality) > > > > > entails your compression schema. > > > > > How can you do that without pairing and power??? I don't see a way to > > > > do that. > > > > Actually it would be rather interesting to see your proof that unique > > > > separation with union only can prove my comprehension schema, > > > > Then my formulation must not be equivalent with yours. > > > > So you're not claiming that your unique comprehension entails your > > > unique separation (ordinary separtion but with uniqueness quantifier)? > > > If you're not claiming that, then that would make better sense to me > > > from what I though you had previously claimed. > > > Moe I think you are confused. > > (1) Ah, I missed your word 'only' in "your proof that unique > separation with union only can prove my comprehension schema" > > I did not say ONLY. > > What I said is that unique separation (that is exactly: separation > schema except uniqueness quantifier) proves MY reversion (which I > haven't posted) of your unique comprehension. And it's trivial, since > MY re-version is just an instance of unique separation. ah I see. we'll that's a different subject. > > (2) Sorry, I meant the other direction: Do you claim unique separation > (exactly: separation schema except uniqueness quantifier) with union > entails unique comprehension? > > But you answered that below, and it's fine, since your answer is 'no'. Yes, my answer is definitely no. > > > However what I am NOT claiming is that my comprehension schema > > follows from unique separation alone without pairing , union and > > power. > > Ah, okay. > > (If I recall, you were saying something to the effect in a previous(?) > thread that the two axiomatizations were equivalent? But you're not > saying that now, okay.) No, the equivalent axiomatizations that I was talking about were about two versions of unique separation itself. So it is totally a different subject. However if you see the third of my posts to this subject here, you'll see what I mean. > > Anyway, I've not looked at your new formulations or posted my > suggested simplification of your earlier version , so I can't judge at > this time whether your proof of unique separation from unique > comprehension and union is okay or not. I only know that I was able to > make it work by revising one or your earlier versions of unique > comprehension. I'll try to check your latest version this week or next > week. Okay. > > MoeBlee
From: zuhair on 17 Jun 2010 22:20
On Jun 17, 3:39 pm, MoeBlee <jazzm...(a)hotmail.com> wrote: > On Jun 17, 2:49 pm, zuhair <zaljo...(a)gmail.com> wrote: > > > > > On Jun 17, 12:02 pm, MoeBlee <jazzm...(a)hotmail.com> wrote: > > > On Jun 16, 7:03 pm, zuhair <zaljo...(a)gmail.com> wrote: > > > > > > I haven't looked over your new formulations, but last night I did > > > > > figure out a way to formulate your compression schema so that it and > > > > > your union axiom entail the separation(with built in extensionality) > > > > > schema, while the separation schema (with built in extensionality) > > > > > entails your compression schema. > > > > > How can you do that without pairing and power??? I don't see a way to > > > > do that. > > > > Actually it would be rather interesting to see your proof that unique > > > > separation with union only can prove my comprehension schema, > > > > Then my formulation must not be equivalent with yours. > > > > So you're not claiming that your unique comprehension entails your > > > unique separation (ordinary separtion but with uniqueness quantifier)? > > > If you're not claiming that, then that would make better sense to me > > > from what I though you had previously claimed. > > > Moe I think you are confused. > > (1) Ah, I missed your word 'only' in "your proof that unique > separation with union only can prove my comprehension schema" > > I did not say ONLY. > > What I said is that unique separation (that is exactly: separation > schema except uniqueness quantifier) proves MY reversion (which I > haven't posted) of your unique comprehension. And it's trivial, since > MY re-version is just an instance of unique separation. > > (2) Sorry, I meant the other direction: Do you claim unique separation > (exactly: separation schema except uniqueness quantifier) with union > entails unique comprehension? I just wanted to clarify this direction: so I shall quote from my first post: Quote: On the other hand it is clear that the comprehension scheme of this theory is a sub-theory of Z, since (y c r \/ y c s) <-> y e P(r) U P(s), were P stands for 'power-set' Now P(r) U P(s) is a set , thus the formula in Comprehension schema would mount to: Ar As E!x Ay (y e x <-> (y e P(r) U P(s) /\ phi)) which is a sub-theory of Z-Reg. / As one can see we need Power,Boolean union,Separation and Extensionality Now Boolean union needs both Pairing and Union axioms, so we need Z-Reg.-Infinity in order to prove the comprehension schema in theory A. So the comprehensions schema is not provable from unique separation and union and infinity alone, it does require Pairing,Power and Union as well. > > But you answered that below, and it's fine, since your answer is 'no'. > > > However what I am NOT claiming is that my comprehension schema > > follows from unique separation alone without pairing , union and > > power. > > Ah, okay. > > (If I recall, you were saying something to the effect in a previous(?) > thread that the two axiomatizations were equivalent? But you're not > saying that now, okay.) > > Anyway, I've not looked at your new formulations or posted my > suggested simplification of your earlier version , so I can't judge at > this time whether your proof of unique separation from unique > comprehension and union is okay or not. I only know that I was able to > make it work by revising one or your earlier versions of unique > comprehension. I'll try to check your latest version this week or next > week. > > MoeBlee |