details re Z-R |- PA consistent
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From: Transfer Principle on 2 Aug 2010 23:56 On Jul 28, 1:16 pm, "Ross A. Finlayson" <ross.finlay...(a)gmail.com> wrote: > I would wonder that Tony might prefer infinite sets, that are > standard, that work with his expectations of what the sets would be, > with the inverse function rule, where Tony basically has symmetry in > the going to infinite or infinitesimal, of the inverses of the > functions' images in the asymptotic, simply maintaining asymptotics. Let me continue working on making this RF-TO theory rigorous. We begin with the desiderata. I've decided to use the letter "s" for parthood instead of "c" or "e", in order to emphasize that parthood isn't identical to standard "e" or zuhair's "c." So "xsy" means "x is a structure of y." Now according to RF we want to be able to divide a real number into infinitely many parts called infinitesimals. Of course, positive infinitesimals are to be less than all the standard positive reals. So parthood should be analogous to either "<" or "<=" for reals. (Similarly, in standard theory, "<" for von Neumann ordinals is identical to "e".) So, for RF-reals at least, we expect parthood to satisfy the following properties (assuming non-strict "<="): 1) Reflexivity: Ax (xsx) 2) Antisymmetry: Axy ((xsy & ysx) -> x=y) 3) Transitivity: Axyz ((xsy & ysz) -> xsz) 4) Trichotomy: Axy (xsy v ysx) But these properties need not hold on all of V. They need only hold on R, the (RF-)reals, which might be only a part of V. It might be possible that some of these properties hold on all of V and the remainder only on R. If 1) holds, then we have zuhair Irregularity. If 1)-3) hold, then we have galathaea-tommy1729 Mereology. Let's see which of 1)-4) we want by returning to RF's post: > Thus examining set theory and a part theory, there's no atomism in the > part theory, just like no universalism in the set theory, (no e- > terminal or e-minimal element). In set theory the only set that > satisfies NOT "is element of" for any input is the empty set. In part > theory, the only set that satisfies "is structure of" for any input is > the universal set (which doesn't exist in set theory, no empty set in > part theory). Here RF describes a universal set, which we'll call "V". As I wrote earlier, RF considers the universal set of part theory to be dual to the empty set of set theory. And so we take the Empty Set Axiom of ZFC: Ex (Ay (~yex)) And so we replace each occurrenceof "~yex" with the dual formula "xsy" to obtain the following: Universal Set Axiom: Ex (Ay (~xsy)) But notice that at this point, if we instantiate this formula to V: Ay (~Vsy) it's not obvious that V is any sort of universal set. All this says is that V isn't a structure of another set, but that doesn't necessarily make V universal. But perhaps we can use 1)-4) to show that V must be universal. What we need for universality is something like Ay (ysV) There might be several ways to derive this from the given axiom plus 1)-4), but the most obvious path is to use Trichotomy, which turns out to be trivial. And so we include Trichotomy as holding on all of V. At this point we might be wondering whether VsV -- since after all, the dual formula in ZFC, "0e0," isn't provable in ZFC (assuming consistency). But RF doesn't make it clear whether he wants the universal set to be a structure of itself or not. (RF can feel free to jump in at any moment here, or for that matter TO since this is his theory, but TO is currently inactive.) This corresponds to whether we want parthood restricted to R to represent "<=" or "<". If the latter, then we should write Trichotomy to refer to a strict relation "s" to obtain: Trichotomy: Axy (xsy v ysx v x=y) From this version of Trichotomy and Universal Set, we prove: Ay (ysV v y=V) so that V isn't necessarly a structure of itself. Let's look at the duals of some of the other axioms. It appears that Extensionality is straightforward: ZF Extensionality: Axy (x=y <-> Az (zex <-> zey)) RF Extensionality: Axy (x=y <-> Az (xsz <-> ysz)) But Pairing appears to be tricky, for we have: ZF Pairing: Aab (Ex (Ay (yex <-> (y=a v y=b)))) RF Pairing? Aab (Ex (Ay (xsy <-> (y=a v y=b)))) This tells us that x is a structure of only two other sets, namely a and b. But shouldn't x be a structure of a third set as well, namely _V_, since _every_ set is supposed to be a structure of V? This problem occurred with zuhair Irregularity as well, since a set containing a and b also needs to contain a third element, namely itself. A trick that zuhair used to avoid this is to consider the ZF axiom: ZF Pairing? Aab (Ex (aex & bex)) This tells us that for sets a,b, there exists another set containing both a and b. But this set x isn't necessarily the desired set {a,b}. To obtain {a,b} from x, we must use the Separation Schema. (Notice that in standard theory, the Axiom of Infinity doesn't give us omega immediately, but instead we apply Separation to the inductive set x obtained from the axiom to obtain omega.) Now we can take the dual of this axiom to obtain: RF Pairing: Aab (Ex (xsa & xsb)) This tells us that x is a structure of both a and b, but doesn't exclude the possibility that there might be other sets of which x is also a structure (such as V). We can do the same with Powerset and Union, replacing the biconditionals with conditionals before taking the dual. But what about the Separation Schema itself? ZF Separation Schema: Aa (Ex (Ay (yex <-> (yea & phi(y))))) The dual of this schema is: RF Separation Schema? Aa (Ex (Ay (xsy <-> (asy & phi(y))))) But what if we let phi be the formula "~y=V"? This instance of the schema would give us: Aa (Ex (Ay (xsy <-> (asy & ~y=V)))) from which we derive ~xeV, thereby contradicting the universality of V! As it turns out, zuhair obtained this problem as well. One zuhair-like patch might be something like: RF Separation Schema? Aa (Ex (Ay (xsy <-> (y=V v (asy & phi(y)))))) but this might lead to problems further down the road. This post is already long enough, so I stop here. The next thing to consider would be the dual to Infinity. This axiom needs to tell us that all sets are infinitely divisible, so that the empty set 0 can't exist. One might call this axiom the "Axiom of Infinitesimal."
From: MoeBlee on 3 Aug 2010 10:04 On Aug 2, 8:05 pm, "Ross A. Finlayson" <ross.finlay...(a)gmail.com> wrote: > this is about ZF and Peano Arithmetic Yes, that's how I started it, but every time you post I recall that you are vastly more interesting than anything in mathematics. MoeBlee
From: MoeBlee on 3 Aug 2010 10:14 On Aug 2, 10:56 pm, Transfer Principle <lwal...(a)lausd.net> wrote: > Let me continue working on making this RF-TO theory rigorous. Turn his typings into something rigorous? How about start by turning them into something grammatical? MoeBlee
From: FredJeffries on 3 Aug 2010 15:54 On Aug 2, 8:56 pm, Transfer Principle <lwal...(a)lausd.net> wrote: > As I wrote earlier, RF considers the universal set of part > theory to be dual to the empty set of set theory. And so > we take the Empty Set Axiom of ZFC: > > Ex (Ay (~yex)) > > And so we replace each occurrenceof "~yex" with the dual > formula "xsy" to obtain the following: > > Universal Set Axiom: > Ex (Ay (~xsy)) I don't understand. If we replace "~yex" with "xsy" don't we get Ex (Ay (xsy)) ?
From: Transfer Principle on 3 Aug 2010 23:23
On Jul 29, 3:22 am, Aatu Koskensilta <aatu.koskensi...(a)uta.fi> wrote: > Transfer Principle <lwal...(a)lausd.net> writes: > > If so, then perhaps this is one way to raise the reputations of the > > sci.math finitists. We're to convince them just to leave the existence > > of infinite sets open rather than assert their non-existence, just as > > the mainstream leaves the existence of large cardinals open rather > > than assert their non-existence. > The obvious thing is just to do interesting finitist mathematics and > leave all the silly harangues against infinity out of it. OK, I sort of see what Aatu is saying here. The mainstream doesn't oppose large cardinals in the same way that the sci.math finitists oppose infinity. We already have a finitist theory, namely ZF-Infinity, and so there's no reason to argue over the theory. Just do interesting finitist mathematics, as Aatu suggests. If the finitists can show that they can do much math that's applicable to the sciences without Infinity, then they will be more successful in their arguments, instead of giving these "harangues" about how Infinity is either counterintuitive or inconsistent. > What formal > theory we might, for this or that purpose, formalize such mathematics in > is of not much consequence or interest -- and indeed we can't decide on > such formal matters until we have a well developed body of mathematical > results, principles, techniques, modes of reasoning, and so on, to > formalize. The development of mathematics simply does not consist of > people putting forth random formal theories, and fundamental questions > about foundations are very rarely purely formal. But what if someone wants to discuss, say, positive infinitesimals, which don't exist in classical analysis? As soon as they mention infinitesimals, the others in the thread will either say that they don't exist, or ask for a definition -- and they mean a _formal_ definition, one that's eliminable to _primitives_. And so, as much as such posters want simply to do interesting mathematics regarding infinitesimals without worrying about formal theories, their opponents force them to consider formal theories -- ones that prove that their infinitesimals even exist. Such is happening right now with RF. In another subthread of this thread, there is a discussion of infinitesimals. The other posters want to see rigorous formal axioms before they'll even consider the infinitesimals (as evidenced by their reluctance to accept RF's and TO's infinitesimals without them). But of course, it'll take a long time to get these axioms correct and sufficiently rigorous before one can even attempt to derive infinitesimals from them. So Aatu's advice works for finitism, but unfortunately not when dealing with brand new objects, such as RF-TO-infinitesimals. |