arithmetic in ZF
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From: Ross A. Finlayson on 1 May 2005 22:16 Consider the universal quantifier, and the axioms, the non-logical axioms of ZF set theory. If "for each" set, it satisfies regularity, what is it "for each" of? Is it for each set? Does that mean for each set,of all sets? Yeah, that set is not well-founded. Is there a class of all classes in "ZF with classes"? No, there's not. .... If you pick another group noun, for a different type of collection that is actually the same thing and fooling yourself, "ZF theory of sets and classes and collections", then "ZF theory of sets, classes, collections, and groups", then there is no group of all groups. Instead, I think reason demands basically a more encompassing consideration of these primary objects of the theory or for convenience pure sets, as primary objects, and why the ur-element of any such theory is dually minimal and maximal. We see movements towards basically second-order or metatheoretical statements, and when the reason behind their consideration is to prevent or (re)solve these paradoxes, then they should be resolved finally. Offputting the resolution, or rather, a solution to these conundrums of antinomy, which itself deserves etymological deconstruction, is not a solution except in some cases where induction allows. I enjoy calling them non-logical axioms because it reads that they're illogical. Except for regularity, I don't think they are, they tend to apply, along with choice and inverse, as theorems of the null axiom theory. The null axiom or axiom-free theory basically fills some requirements of any theory of "everything." One reason we address these kinds of issues, for personal interest, is because this is discussion of the very fundament of mathematical logic, and that is of interest. Speaking for myself, I want to understand why and how these mathematical statements along the lines of "2+2=4" or "f(x)=0 is symmetric about the origin" have meaning, they bear truth, the full weight of truth. I see that some of the high technical achievements of philosophy have the same true meaning as the lowest common denominators of mathematical logic: so now I call the substrata of theory the deep foundations. As alluded to in mention of a theory of everything, that's about a theory of everything, and where all the truths of ZF can be inferred from another theory, then, "they're not done yet, doc." Ross
From: Babylonian on 1 May 2005 22:55 george wrote: > Babylonian wrote: > > > > Obviously, ZF-provability includes provability by > > > > means of LEM, > > > george wrote: > > > PuhLEEZE. Obviously, it doesn't. > > > LEM is from classical logic, not from ZF. > > > You can apply intuitionistic logic to the same > > > old ZF axiom-set. > > B > > The theory obtained by adding the axioms > > of ZF to intuitionist logic is not the theory > > of ZF sets. PuhLEEZE. > > This is silly. One cannot "add" the axioms of ZF > or of any other first-order language to "intuitionistic > logic". Intuitionistic logic IS A LOGIC. > It is NOT an axiom-set (which is what ZF is). > Axioms do NOT get ADDED TO intuitionistic logic or > classical logic or any other logic. Rather, logics > are USED to INFER (sentences from other sentences, > using rules of inference; they prove theorems from > axioms. 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. Yes, and I did that intentionally because the possible models of the two theories are different enough (you know what I mean) that I didn't think anyone wants to call them "ZF-sets".
From: Babylonian on 2 May 2005 00:50 george wrote: > > > > Let's take the contrapositive of that "shared first link" > above and begin by assuming we have a proof of > (qv~q)->Ex[Px], both intuitionistically and classically. > How do we get from here to Ex[Px] ? > Classically we get there trivially by LEM, > but if that is not available, how do we discharge the qv~q > hypothesis both intuitionistically and constructively? > > The q arose out of a non-constructive proof and we begin > by asking if it provides guidance for an intuitionistic one. > The classical proof didn't produce Just Any old contradiction; > it produced one with specific relevance to the problem. > Is its relevance strong enough that ~q->Ex[Px] is provable? > If it is, then, classically, we might be able to do this by > cases instead of by LEM. > Having proved ~q->Ex[Px], you could then either > a) prove ~q, which would work both intuitionistically and > classically, or b) prove q->Ex[Px], which would not work > intuitionistically but would work classically. > > My problem with calling one of these inferior to or less > constructive than the other is that I don't see how you > can dismiss the proof as non-constructive if both > q ->Ex[Px] and ~q ->Ex[Px] are constructive. > "Intuitionistic" and "constructive" are slightly different > notions here. >From this angle of approaching the subject, it isn't easy to see, but it's going to turn out that there is a proof of q or ~q in intuitionist logic if and only if the average person will look at it and say, it's constructive. I really can't explain this part; it just so happens, somehow, that this formal criteria does coincide with survey results.
From: Paul Holbach on 2 May 2005 03:26 > Chris Menzel wrote: > If you are using ZF as a > basis for your account of quantification, then it is > provable that there > is no universal set. Hence, in that framework, > quantification does not > imply a universal set. Maybe you want to work in a > framework like NF > that does have a universal set. Fine. > The point is you just can't > assert flat out that quantification implies a universal set; > you need to > specify your background set theory for the assertion > to have any > definite purchase. "According to Cantor's Domain Principle [...] any variable presupposes the existence of a domain of variation. Thus, since in ZF there are variables ranging over all sets, the theory presupposes the collection of all sets, V, even if this set cannot be shown to exist in the theory. Consistency has been purchased at the price of excluding from it a set whose existence it is forced to presuppose. " "The Domain Principle states that quantifying presupposes a corresponding totality of quantification." "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." [Priest, Graham (2002). /Beyond the limits of thought/ (2nd ed.). Oxford: Oxford University Press. (S. 158+280+281)] Of course, Priest is a heretic ... Regards PH
From: Chris Menzel on 2 May 2005 12:32
On 2 May 2005 00:26:51 -0700, 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). This is not to say there is no conceptual difficulty in Cantorian set theory at all -- if we agree in some cases that all the things of some sort (e.g., finite sets) form a further set, why not in this case of "all sets" as well? But that seems to me to be an orthogonal issue. Chris Menzel |