arithmetic in ZF
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From: george on 2 May 2005 13:33 Ross A. Finlayson wrote: > Hi, > > In doing only cursory research, I'm apparently not > the first person to suggest a theory with no non- > logical axiomatization, Don't sell yourself short. > For example, and I only yesterday saw this from searching for > eliminable definitions, Owen Holden, with whom I agree, > presented a similar notion years ago, "Set Theory Without > Axioms." This was grossly mis-titled. It had axioms all over the place. Far worse, it claimed to be able to insist that xey was a wff, and yex was a wff, but (xey & yex) was NOT a wff. The kinds of things this proposed system would've had to do to actually SPECIFY what was a wff and what wasn't would've wound up simply TRANSLATING axioms into rules of a grammar. He was utterly without a clue. Unfortunately he got engaged (in that thread) by several people who wanted to flaunt how much they knew about Quine's NFU set theory ("New Foundations (with Urelements)") and wound up mapping OH's completely indefensible "restrictions on what's a WFF" to Quine's coherently defined "stratification". OH's thread thus wound up lasting a lot longer than it needed to. http://groups-beta.google.com/group/sci.math/browse_thread/thread/cb67edabc2fd84c8/ > Basically I've arrived at that there is one consistent, > complete, and concrete theory. In it, is the continuum hypothesis true or false? And Is the powerset of the union of all the sets in it in it, or not? Is it categorical or does it have differing models? If so, do both of them occur in some larger model of it? Or is it even meaningful to speak of this theory as having models at all?
From: george on 2 May 2005 13:52 george wrote: > > If you start with the first-order ZF axioms, > > you will get one theory if you close them under first-order > > classical consequence and a DIFFERENT theory if you close > > them intuitionistically. > > But to say that the classical way "is" "ZF-provability", > > while the intuitionistic way isn't, is discriminating > > against yourself. > Babylonian wrote: > Yes, and I did that intentionally because the possible > models of the two theories are different enough Oh, please! The various classical models of ZF are all "differnt enough" FROM EACH OTHER that nobody can get AROUND to caring how they might differ from the non-classical ones. As the loudest initial three examples, SOME OF THEM ARE COUNTABLE while others are not, SOME OF THEM OBEY THE AXIOM OF CHOICE, while others do not, and among those that do obey AC, SOME OF THEM SATISFY THE CONTINUUM HYPOTHESIS, while others do not. > (you know what I mean) that I didn't > think anyone wants to call them "ZF-sets". My rebuttal of that is simply that THERE IS NO SUCH THING as "the ZF-sets", classically! Classical-first- order-ZF's myriad Tarskian models are just TOO chaotically heterogeneous! The ONLY things you can LEGITIMATELY refer to as "the ZF-sets" are the sets that provably exist (in ALL classical models) of ZF! And EVEN THOSE sets get "realized" in completely DIFFERENT ways in DIFFERENT models of ZF! For example, ZF's axiom of infinity requires the existence of a certain set that's closed under successor. But in a different model, this is a DIFFERENT set! ZF also requires that this set have a powerset, and it is true of this set, in ALL models, that it is uncountable (from the "intra"-viewpoint of the model). In the countable model, however, It MUST (from the "extra"- viewpoint) be countable. My point is that having proved about a thing that it's a ZF set, UNLESS IT'S FINITE, is NOT having proved enough about it to identify "which set" it is. THAT is a matter of OPINION that VARIES from model to model. Now, whatEVER you think the class of provably-extant ZF-sets may be, it is clearly a different class of sets from the ones that provably exist from the same axioms under intuitionistic logic, but I repeat, BEFORE we go down that road, YOU OWE us some INTUITIONISTIC MODEL THEORY! CFOPL has a completeness theorem (something is provable iff it's true in ALL models), and that alone lends the notion of "ZF-set" what LITTLE coherence it has. But saying about a universe (like sets from ZF under intuitionistic logic) that it is "too different" from the "usual" ZF sets is NOT reasonable, because given any model of ZF, you can get another universe of sets that's ALSO "too different" JUST BY GOING TO ANOTHER MODEL, classically, under the SAME paradigm! The class of finite sets that provably exist IS the same under both logics.
From: Paul Holbach on 2 May 2005 16:34 > Chris Menzel wrote: > > Paul Holbach <paulholbachSPAMBAN(a)freenet.de> said: > > [Quoting Priest:] > > "For any claim of the form 'all sets are so and so' to have > > determinate sense there must be a determinate > > totality over which the > > quantifier ranges." > Well, if the idea of "determinate totality" of things of a > certain sort > is meant to imply that there is an *object* of some sort --- the > "totality" --- in addition to the things in question, this is just a > non-sequitur. All we need to make sense of quantification is for the > things quantified over to exist. There doesn't need to be a further > thing that contains them (though of course there does in our usual > mathematical *models* of quantification, which are themselves set > theoretic objects). Priest replies as follows: "The Domain Principle seems patent to me. It would seem to be an obvious and brute semantic fact that whenever there are things of a certain kind, there are a l l of those things. Yet the principle has drawn flak from some commentators. [...] Cartwright rejects the Domain Principle, which he calls 'All-in-One-Principle', for reasons that he explains as follows: 'There would appear to be every reason to think ... [the All-in-One-Principle] false. Consider what it implies: that we cannot speak of the cookies in the jar unless they constitute a set; that we cannot speak of the natural numbers unless there is a set of which they are the members; that we cannot speak of all pure sets unless there is a class having them as members. I do not mean to imply that there is no set the members of which are the cookies in the jar, nor that the natural numbers do not constitute a set, nor even that there is no class comprising the pure sets. The point is rather that the needs of quantification are already served by there being simply the cookies in the jar, the natural numbers, the pure sets; no additional objects are required. It is one thing for there to b e certain objects; it is another for there to be a s e t, or set-like object, of which those objects are the members. [...]' The passage is rhetorically persuasive, but it achieves this effect by trading on a number of important confusions. The first is between quantification and description. Cartwright says that the Domain Principle entails that 'one cannot speak of the cookies in the jar unless they constitute a set'. It does not. Speaking of the cookies in the jar does not quantify over the cookies. It refers to them by means of a definite description. (It is a plural description, but this does not affect the matter.) Clearly, to speak of the so and so(s) requires us to presuppose no more than the existence of the so and so(s). The passage also holds up as implausible the thought that one can speak of all (pure) sets without there being a set of (pure) sets. This i s a consequence of the Domain Principle. But once one separates it from the case of descriptions, its supposed implausibility is much harder to see. [...] For any claim of the form 'all sets are so and so' to have determinate sense there must be a determinate totality over which the quantifier ranges. It would clearly be wrong to suppose that this totality is a set satisfying the axioms of Zermelo-Fraenkel set theory, or of some other theory of sets; but that there is a well-defined totality seems to me undeniable. Moreover, it is clearly a totality that we can think of as a single thing, since we can legitimately refer to it as that totality: the totality of all sets. The passage continues by saying that 'the needs of quantification are already served by there being simply the cookies in the jar'. This brings us to the second confusion in the passage. Loosely, it is a confusion between sense and reference. Whar are 'the needs of quantification'? First of all, the variable of the quantifier must have values. These are the cookies in the jar themselves: cookie_1, cookie_2, cookie_3, ... . This, indeed, has nothing to do with totalities of any kind. The totality is presupposed as soon as we talk, not of the possible references of the variables, but of the sense of the sentence containing the quantifier. This is not determinate unless the totality of the cookies is determinate. We can put the point in a Quinean fashion. The ontological commitments of the sentence concerned, the entities there must be for the sentence to be true, are precisely cookie_1, cookie_2, cookie_3, ... . But it is not the ontological commitment of the sentence that is at issue here; it is the sense of the sentence; the determinacy of this does require there to be a determinate totality of cookies. Of course, if this totality is thought of as something independent of the cookies, this may again sound implausible. If you are talking about certain things, and if all existences are distinct, then invoking the existence of another entity would seem de trop. This is the rhetorical strategy employed in the last sentence quoted, 'It is one thing for there to b e certain objects; it is another for there to be a s e t, or set-like object, of which those objects are the members'. But the set and its members are not distinct existences. There could be no set of cookies if there were no cookies--and vice versa. These are no atomic, independent, existences." [Priest, Graham (2002). /Beyond the limits of thought/ (2nd ed.). Oxford: Oxford University Press.(S. 280ff)] "For every potential infinity there is a corresponding actual infinity. Following Hallett let us call this the Domain Principle. I take it to be a formulation of the Kantian insight that totalisation is conceptually unavoidable, though stated in a much more satisfactory way than Kant ever managed to achieve." [Priest, Graham (2002). /Beyond the limits of thought/ (2nd ed.). Oxford: Oxford University Press.(S. 124)] Of course, what most critics generally denounce is Priest's background philosophy, his dialetheism. Neil Tennant, for example, writes: "It is not enough to give an honorific title like 'The Domain Principle' to a tendentious piece of naiveté. Right at the outset [of 'Beyond the limits of thought'] Priest announces that 'there is a totality (of all things expressible, describable, etc.) and an appropriate operation that generates an object that is both within and without the totality. I will call these situations 'Closure' and 'Transcendence', respectively' (p. 4). It is at this very point that the question of the existence of the totality should be raised. The lessons of the various paradoxes are, surely, that (1) some 'totalities' cannot exist--in particular nothing contains everything: (2) no language can express every possible proposition: (3) no theory can contain every truth. Nowhere does Priest give these non-dialetheic responses the attention they deserve." [Source: http://people.cohums.ohio-state.edu/tennant9/tennant_pb1998.pdf] Is Priest really that naive ...? Regards PH
From: Bhupinder Singh Anand on 3 May 2005 04:08 On May 2, 4:34 pm, Paul Holbach wrote: PH>> Is Priest really that naive ...? <<PH Paul === Not necessarily. It may simply reflect the possibility that, first, Priest's hereticism could be confined to reasoning within the ambit of standard interpretations of classical theory; and, second, that such interpretations are yet to definitively address the issue of whether the terms 'non-algorithmic' and 'non-constructive' are to be treated as synonymous. As I argue elsewhere, Goedel's reasoning can be taken to establish that we can constructively, and in an intuitionistically unobjectionable manner, establish that an arithmetical relation R(n) holds for any given natural number n, but that there is no algorithmic way of verifying this assertion (in other words, any Turing machine that computes R(x), treated as a Boolean function, will go into a verifiable, non-terminating, loop for some natural number n). On this view, the particular argument between Priest and Cartwright, Tennant, et al, becomes vacuous; we simply define any predicate [P] of a formal language L as well-defined if, and only if, given any element s in a domain D over which the variables of the language are well-defined under an interpretation M, there is always an effective method for determining whether P(s) holds or not in M for the interpreted predicate P. (Note that, given an interpretation M of a formal theory L, we can always - according to a reasonable reading of Priest - define the domain D as consisting of all elements that satisfy the axioms of L under any interpretation. The fact that some D-elements are dissimilar may need qualification when defining satisfaction in M but, prima facie, this should not lead to any irresolvable issues.) The existence of functions in real and complex analysis that are continuous, but not uniformly continuous, and of sequences that are convergent, but not uniformly convergent, indicates that such a definition of a well-defined predicate can be mathematically constructive, and intuitionistically unobjectionable. We can then define [P] as well-defining a mathematical object (i.e., a syntactic totality), say {x: P(x)}, in M if, and only if, there is an algorithm (effective uniform method) for identifying elements such that P(x) holds in M. In the absence of such an algorithm, we may still have that, for any s in D, it is effectively decidable whether or not P(s) holds in M, but we can no longer treat all such elements as a syntactic totality that can be referred to within, or by, the language without inviting inconsistency. On this view, the 'Domain' referred to by Priest need not be a syntactic totality in the sense of being a mathematical object as defined above. However, it can still be referred to, albeit loosely, as a semantic totality for expressive, and philosophical, purposes, so long as we do not associate any algorithmic properties with it without a formal proof. Regards, Bhup
From: Babylonian on 3 May 2005 04:32
george wrote: > > Oh, please! Since you have asked politely, I'll give you a cookie at the end of my post. > Now, whatEVER you think the class of provably-extant > ZF-sets may be, it is clearly a different class of sets > from the ones that provably exist from the same axioms > under intuitionistic logic, but I repeat, BEFORE we go > down that road, YOU OWE us some INTUITIONISTIC MODEL THEORY! Hint: it's model theory done intuitionistically. I'll show you how. > CFOPL has a completeness theorem (something is provable iff > it's true in ALL models), Intuitionist logic has the following theorem: pvq is provable if and only if p is provable or q is provable. Remember this. > But saying about > a universe (like sets from ZF under intuitionistic logic) > that it is "too different" from the "usual" ZF sets is NOT > reasonable, You may decide, when I give you the cookie I promised. > > The class of finite sets that provably exist IS the same > under both logics. Is that a constructive theorem, or a classical one? Here is a classically finite set: C={{}} if the CH is true C={} if the CH is false. I just want to know: (i) does C exist in all models of the theory generated by intuitionist logic? I say by the comprehension axiom it does. (ii) is C={}v~C={} true in all said models? I say it isn't, because CHv~CH is only provable if either CH or ~CH are provable, so there is a model where C={}v~C={} is false. That is a model where CH is neither true nor false, because ~(C={}v~C={}) is the statement that there can never be a proof of C={}v~C={}. (iii) Is the class of finite sets that provably exist the same under both logics? Use only inferences of intuitionist logic to justify your answers, or you will convince me of nothing. I see extensionality colluding with comprehension to give us: (C={} == ~C={{}}) and (~C={} == C={{}}). >From these, we obtain (C={}v~C={} == C={{}}v~C={{}}) or, if you like, (C={}vC={{}} == ~C={}v~C={{}}) Does this generate C={}v~C={}? Or C={}vC={{}}? Remember, you can't generate CHv~CH until CH or ~CH are generated. Enjoy your cookie. |