details re Z-R |- PA consistent
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From: Ross A. Finlayson on 28 Jul 2010 16:16 On Jul 27, 9:01 pm, Transfer Principle <lwal...(a)lausd.net> wrote: > On Jul 24, 11:44 am, FredJeffries <fredjeffr...(a)gmail.com> wrote: > > > On Jul 19, 9:15 pm, Transfer Principle <lwal...(a)lausd.net> wrote: > > > I'd love to do this without "putting people in boxes." I'd love to > > > defend certain posters and have it be seen not as "putting > > > people in boxes," but as adhering to perfectly reasonable criteria > > > for choosing which posters to defend, since -- as I repeat -- it is > > > neither possible nor desirable to for me defend _everyone_. > > <anti-anti-Cantorian sermon> > > So you are trapped in an anti-Cantorian universe where it's all zero- > > sum games and you can't pay Paul without stealing from Peter. > > I'd love it if I could break out of this trap of zero-sum games. I'd > love it if I could pay both Peter and Paul. But the real question is, > will Peter or Paul let me? > > Case in point -- in another thread, there is a dispute between > Archimedes Plutonium and Iain Davidson, and I was called > upon to settle it. I tried to give a positive-sum response and > say that both AP and Davidson were right. But this response > was rejected, because AP wants me to say that AP is right > and Davidson is wrong, while Davidson wants me to say that > Davidson is right and AP is wrong. Furthermore, Tonio laughed > at me for giving such a response. > > So Jeffries wants me to avoid zero-sum games, but the only > response that AP, Davidson, and Tonio will accept from me is > a zero-sum response. > > So, out of curiosity, what would Jeffries recommend that I > post in this situation? I want to avoid resorting to the zero-sum > universe, but AP and Davidson are looking for a zero-sum > response from me. > > I will not post in that other thread until Jeffries gives me a > suggestion, then I'll decide whether to go along with that > suggestion or post something else there. > > > YOU are living evidence of the moral bankruptcy of the anti-Cantorian > > position, a system where whatever is not explicitly permitted is > > forbidden. > > I don't necessarily believe that everything that isn't explicitly > permitted is forbidden, but I admit that I might as well since I > defend posters who do. > > Indeed, I have pointed out several times before that one > distinguishing trait of those whom Jeffries calls "anti-Cantorian" > (and others call five-letter insults) is that they believe that > whatever isn't explicitly permitted is forbidden -- they want > more control over what can be proved in their theory. On the > other hand, those whom he calls "anti-anti-Cantorian" > believe the opposite -- that whatever isn't explicitly forbidden > is permitted. They delight in proving unexpected results from > a small set of axioms (or schemata). > > And never the twain shall meet? > > But I still believe that they can. I'd like to believe that there > can be a theory which doesn't forbid that which isn't explicitly > permitted, but still allows for, say, infinite sets that work > differently from those discovered by the three mathematicians > mentioned by Jeffries (Bolzano, Dedekind, Cantor). Indeed, > the poster Tony Orlow desires infinite sets that work differently > from standard infinite sets. 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. Often times people talk about sets of numbers, but to actually be those numbers, the existence of the sets imply the existences of all or any necessary sets that would result as a consequence of implications of their structure (and lack thereof as objects) in all systems. Set theory is useful where the only descriptive elements are "is same as" and "is element of", consider a different theory with "is different than" and "is structure of", same theory. Yet, the defined terms are opposite. In set theory creation starts with nothing where with this anti-set theory (part theory) then the domain of discourse is nothing, or as to rather, a fundamental term would be the structure that is not formed from anything, it is still the empty set, the only set that results as a consequence of there being a unique set for which all the other sets satisfy both predicates in negation. 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). Then having both those theories and conveniently using either with regards to then using methods of symmetry and exhaustion, they are complementary and reflective. Also with simply defining how various theories work together constructively and deconstructively, this yields shorter proofs (with the doubled language size). Then, still these work easily with standard modern mathematics, even formalists and even verbose formalists can recognize that many of the expectation's of Tony's mathematical structures as he describes them are readily shown consistent with normal expected treatments of analysis using standard modern foundations. If there's a theory where anything not expressly forbidden that it's allowed, still for there to be anything else it's the combined theory, a constructivist has for the main everything is allowed only and exactly because it is, forbidden things don't exist. That raises a point about Goedel's theory, that because theories strong enough to have the natural integers in useful inductive systems be incomplete, that there "are" other properties about the objects. This is where first Godel proved completeness having both completeness and incompleteness theorems for general theories. (Kurt Godel.) Goedel wrote completeness theorems, with for example the Hilbert programme of axiomatizing mathematics, then most characteristically the incompleteness theorems. Still, there are the completeness theorems first. (Goedel, constructivist.) Then in the part theory coincidentally with the set theory, at least then partitioning and collecting have as simple a measure of interchangeable information where it's generally provable why proofs of various systems are longer or shorter than others or the same than in other symbol-theoretic terms, in information coding, compared to machine implementation (forward implementation). Warm regards, Ross Finlayson
From: Jesse F. Hughes on 28 Jul 2010 16:38 "Ross A. Finlayson" <ross.finlayson(a)gmail.com> writes: > 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. You know, I think we all wonder just that. -- Jesse F. Hughes "My experience indicates that the people who post on this newsgroup are about at the level of a 10 year old in the year 2060." -- More wisdom from James Harris, time traveler
From: Transfer Principle on 28 Jul 2010 23:08 On Jul 28, 10:18 am, FredJeffries <fredjeffr...(a)gmail.com> wrote: > On Jul 27, 9:01 pm, Transfer Principle <lwal...(a)lausd.net> wrote: > > But I still believe that they can. I'd like to believe that there > > can be a theory which doesn't forbid that which isn't explicitly > > permitted, but still allows for, say, infinite sets that work > > differently from those discovered by the three mathematicians > > mentioned by Jeffries (Bolzano, Dedekind, Cantor). Indeed, > > the poster Tony Orlow desires infinite sets that work differently > > from standard infinite sets. > I got the "whatever is not explicitly permitted is forbidden" vs. > "whatever is not explicitly forbidden is permitted" distinction from > our friend Alexander Yessenin-Volpin. cf. his "On the Logic of the > Moral Sciences"http://yessenin-volpin.org/onthelogic.pdf > at page 11, his "principle of liberalism" vs "principle of despotism" Interesting link. But this reference to Y-V reminds us that since Y-V is a finitist held in high esteem here, the sci.math finitists can improve their standing if their arguments could resemble Y-V's. In particular, keeping Y-V's "principle of liberalism" in mind, we recall that Y-V can't answer "yes" to "Is 10^500^500^500 a natural number?" neither will he answer "no." (I'm not sure how Y-V would respond to the question "Is pi a natural number?" I guess that he'd answer "no" as quickly as he can answer "yes" to "Is 4 a natural number?" since pi < 4.) On the other hand, WM will answer "no" to the numberhood of either 10^500^500^500 itself or some value less than 10^500^500^500. Perhaps combining Y-V's beliefs with Jeffries's reference to complexity, it would be better for WM, when asked "Is n a natural number" for some value of n, to wait an interval of time proportional to the complexity of n before answering "yes." Then WM's ideas would no longer be "despotic," and yet there would exist natural numbers n such that the universe would end before WM can answer "yes" to their numberhood. As for AP, recall that to him 10^500^500^500 is still a number, but it's an _infinite_ number. Thus AP's beliefs, at least with regards to numberhood, aren't despotic either. > Fred Jeffries is IN FAVOR of clear, unambiguous mathematical > definitions!!!!! While an argument can be made that WM is _opposed_ to clear, unambiguous mathematical definitions. And never the twain shall meet.
From: Transfer Principle on 28 Jul 2010 23:47 On Jul 28, 11:05 am, FredJeffries <fredjeffr...(a)gmail.com> wrote: > On Jul 27, 8:21 pm, Transfer Principle <lwal...(a)lausd.net> wrote: > > One analogy that I often bring up is: > > Large Cardinal : Mainstream :: Mainstream : Finitism > > that is to say, large cardinal theories extend the mainstream > > theory by adding inaccessibles, etc., in the same way that the > > mainstream theory extends the finitist theories by adding > > infinite sets. > I would say, rather, that a large cardinal axiom RESTRICTS the > mainstream theory in that it declares that a certain cardinal > definitely exists rather than leaving the existence an open question. But in that case, one could argue that the Axiom of Choice is "restrictive," since although it permits all sets to have choice functions and wellorders, all vector spaces to have bases, subsets of R to be nonmeasurable, etc., it restricts in that it prevents sets without choice functions from existing, nonwellorderable sets from existing, Kunen's cardinals from existing, Dedekind's cardinals from existing, and so on and so forth. Likewise, the Axiom of Infinity _restricts_ the finitist theory ZF without Infinity in that it declares that a certain cardinal (namely omega) definitely exists rather than leaving the existence an open question. Indeed, almost _every_ axiom could be called "restrictive." And so the only theory without a "restrictive" axiom would be the Null Axiom theory of RF! But then we'd quickly run in circles regarding semantics rather than the main issue, namely which posters to defend. This debate has come up in previous threads, and one way to settle it is instead of calling axioms "permissive" and "restrictive," we note that some axioms begin with an existential quantifier while others begin with a universal quantifier. It's often stated that standard ZFC contains only one axiom that begins with an existential quantifier -- namely Infinity. (These posters apply Separation Schema to Infinity to obtain the empty set.) And so this axiom is regularly singled out by posters who don't want to declare that the type of set mentioned in Infinity (i.e., a successor inductive set) exists. (Notice that large cardinal axioms can begin with either type of quantifier -- compare "there exists an inaccessible cardinal" with "for each natural number n, there exists an nMahlo cardinal.") Still, one can argue that ZF (where the existence of an inaccesible is open) is more permissive than ZF+~Inaccessible (where such existence is refuted), just as ZF-Infinity (where the existence of omega is open) is more permissive than ZF-Infinity+~Infinity (where such existence is refuted). The sci.math finitist Srinivasan uses "D=0" instead of ~Infinity, but actually he uses NBG-Infinity in a _logic_ (NAFL) that prevents omega from existing (so it's the _logic_, not the set theory, that is restrictive). > (If I understand correctly) in ZFC, one neither has nor doesn't have > large cardinals. One of your hypothesized ZFC users doesn't (perhaps) > use large cardinals, but she cannot prove that they don't exist. She > also cannot avoid that there are models of ZFC in which there do exist > (some varieties of) large cardinals. Applying this to our finitist analogy, we have: (If I understand correctly) in ZFC[-Infinity], one neither has nor doesn't have [infinite sets]. One of your hypothesized [finitists] doesn't (perhaps) use [infinite sets], but she cannot prove that they don't exist. She also cannot avoid that there are models of ZFC[-Infinity] in which there do exist (some varieties of) infinite sets. This raises a question that was mentioned earlier -- is it better for a finitist to work in ZF-Infinity rather than ZF-Infinity+~Infinity? In other words, if one doesn't wish to use infinite sets, then one should just avoid both Infinity and its negation (and thus leave the existence of omega open) rather than impose its restrictive negation? (Here I'm referring to FOL, not NAFL.) 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. I must admit that I don't find ~Infinity satisfying, either. Rather than give an axiom stating what doesn't exist, I'd rather give an axiom stating what _does_ exist. This is, in fact, what I attempted to do with TO -- declare the existence of a set tav which implies that the set omega doesn't exist -- just as the mainstream doesn't have an axiom stating that Kunen's cardinal doesn't exist, but rather have an axiom, AC, which implies that Kunen's cardinal doesn't exist. For AC can be used to prove several interesting theorems, and I hoped to do the same with TO's tav.
From: Transfer Principle on 29 Jul 2010 00:10
On Jul 28, 1:16 pm, "Ross A. Finlayson" <ross.finlay...(a)gmail.com> wrote: > On Jul 27, 9:01 pm, Transfer Principle <lwal...(a)lausd.net> wrote: > > But I still believe that they can. I'd like to believe that there > > can be a theory which doesn't forbid that which isn't explicitly > > permitted, but still allows for, say, infinite sets that work > > differently from those discovered by the three mathematicians > > mentioned by Jeffries (Bolzano, Dedekind, Cantor). Indeed, > > the poster Tony Orlow desires infinite sets that work differently > > from standard infinite sets. > Set theory is useful where the only descriptive elements are "is same > as" and "is element of", consider a different theory with "is > different than" and "is structure of", same theory. Yet, the defined > terms are opposite. In set theory creation starts with nothing where > with this anti-set theory (part theory) then the domain of discourse > is nothing, or as to rather, a fundamental term would be the structure > that is not formed from anything, it is still the empty set, the only > set that results as a consequence of there being a unique set for > which all the other sets satisfy both predicates in negation. > 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). Interesting idea! Indeed, the concept of a "part theory" has been mentioned in other by the posters galathaea, tommy1729, and zuhair. Suppose we were to do as RF suggests here. Instead of the primitive "e" for "is an element of," we might introduce the primitive "c" for "is a part of." Then we can take the Empty Set Axiom and replace each instance of "yex" with "xcy" to obtain the dual axioms: Ex (Ay (~yex)) Ex (Ay (~xcy)) and this set whose existence is guaranteed by this axiom would be a universal set, which we can call V. We could try replacing all of the axioms of ZFC with their c-duals. But the resulting theory would actually be equivalent to ZFC itself -- every model of the part theory would be a model of the set theory with "c" mapped to the inverse of "e." In other words, we wouldn't really have a new theory. When galathaea and tommy1729 came up with their part theory, their theory differed from ZFC because it used something called the "flattened mereology." In short, there were added axioms asserting that parthood "c" is reflexive, antisymmetric, and transitive. The axiom asserting the existence of a universal set was written as: Ex (Ay (xcy -> x=y)) since after all, if everything is a part of the universal set V, should V itself be a part of V? Otherwise V isn't truly universal! But in galathaea-tommy1729, the empty set still exists. Perhaps the theory described by RF more closely resembles zuhair's, in which the empty set need not exist. In one of zuhair's theories, "e" is still reflexive, just as parthood is reflexive to galathaea-tommy1729. I've never thought about trying to apply parthood theory to TO in order to obtain his infinitesimals (like one zillionth) until RF mentioned it. RF himself has a theory of infinitesimals ("iota"). This is something that I find worth considering further. |