arithmetic in ZF
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From: Babylonian on 3 May 2005 11:08 Very well then; this application of the BHK interpretation is too gappy to convince anyone of anything. Truth is, I don't care much about ZF anyway; type theory is a better way to explore intuitionist model theory.
From: Ross A. Finlayson on 3 May 2005 21:50 What do you mean, "heretic"? That's pretty modern, in just barely introducing myself to Priest's views, that's some interesting stuff. I think the "dialetheism", or characteristics of paraconsistency, applies to the ur-element, where that is basically "dually self intraconsistent", and not inconsistent. Please present a well-ordering of the reals, and reread the thread "On Well-Ordering(s) and Sets Dense in the Reals, Infinity". How am I supposed to compete, i.e. to disagree and proffer my own better alternative, with world-renowned and widely published and respected logical luminaries like "Georg Cantor" or "Graham Priest", and you? Uh oh, a 3'rd dan karateka. I'm interested in that. Coat Jacket Grappling: streetfighting in suits. http://www.st-andrews.ac.uk/academic/philosophy/gp.html I'll tell you how: very directly, in terms of logic. So, like I was saying, quantification over sets implies a universal set. Immanuel Kant agrees, as do Cantor, and Priest. Stop fooling yourself. If that does not sit well with you, then I encourage you to address the other points there that illustrate ZF's inconsistency, basically because of irregularity. If "Not Con(ZF)", that is, ZF is inconsistent, then all the forcing results based upon "Con(ZF)" would reflect that. Ross
From: Ross A. Finlayson on 3 May 2005 22:01 Hi, Basically GCH, but only because Aleph_0 = omega, Aleph_1 = omega+1, etcetera, Aleph_n = omega + n. The Generalized Continuum Hypothesis does not say much of anything about the cardinality of the continuum, which has been shown to be equivalent to Aleph_1, Aleph_2, etcetera, and Aleph_0, via EF and a suitable definition of the real numbers, all it says is that there isn't a cardinality between Aleph_n and Aleph_n+1 and that for n in N Aleph_n exists. I say not GCH, for a variety of reasons. Among them, infinite sets are equivalent. As Hausdorff once noted, A countable union of countable sets may be... uncountable. That's a counterexample to the opposite. In the similar way that I claim transfinite cardinal numbers are meaningless, where they have some utility but reflect inconsistent assumptions, I think a model, in terms of a theory is meaningless. That's because I think a model is to a theory as a class is to a set. They're only brought into the discussion because you're doing set theory wrong. Ross
From: Ross A. Finlayson on 4 May 2005 06:57 Hi Graham, Dr. Priest, On sci.logic we are discussing arithmetic in ZF and got to discussion of the Domain Principle. Your book "Beyond the Limits of Thought" was quoted, I wonder if you might have something to say, on sci.logic on the thread "arithmetic in ZF". http://groups-beta.google.com/group/sci.logic Where I'm coming from, I'm an amateur logician who advocates a theory free of non-logical axioms, and think that that theory can thus be Goedelianly complete, and the axioms of set structure of ZFC minus the regularity axiom are theorems of the Null Axiom or Axiom-Free theory, which is a theory with sets, numbers, or physical or geometric objects as primary objects, at once. Basically it has an ur-element that is dually minimal and maximal, I've gotten to calling it "dually-self-intraconsistent". In scanning some few words of yours written on the Internet, and about dialetheism, it's basically about Janus' introspection. I have the singular ur-element, which is as well a set and the proper class, where there can be only one proper class, being at once the root of the Liar, the Russell set, infinity, the empty set, Kant's Ding-an-Sich and Hegel's Being and Nothing, and the void from which all springs. Particularly for the Liar and Russell, and parallelly for Cantor/Burali-Forti/apeiron, I've discussed that on sci.math and sci.logic for some years. I think infinite sets are equivalent, I show that, basically with ubiquitous ordinals or naturals in the cumulative hierarchy, and work on some analytical tools that have to do with bijections between the natural integers and unit interval of the real numbers, with nonstandard real numbers that have atomic infinitesimal iota-values, indubitably, as a logical consequence of their structural consequence, for the normal ordering of the positive reals being its natural well-ordering. I don't only promote a theory with zero non-logical (or proper) axioms, I promote that it's first-order logic and that it's the only true theory. This is basically towards Deep Foundations, and to some extent a theory of everything. I hope this serves as a decent introduction, my name's Ross, Ross A. Finlayson, USA, I post this to you via e-mail and onto that thread on the sci.logic discussion forum. I'm interested in your opinion, and would be made happy to receive a reply, publicly or privately. Thank you, Ross Finlayson
From: george on 5 May 2005 14:43
> > Alan Smaill wrote: > >> .. providing an intuitionist/constructive proof of > >> qv~q, which may go via a proof of q, or of ~q, or indeed > >> of qv~q where neither q nor ~q is provable, I replied on behalf of someone else: > > "Babylonian" is later going to say that this is impossible, > > that it is a meta-theorem about provability under IPC that > > pvq is provable iff p is provable or q is provable. AS> > OK for propositional logic with no non-logical axioms; Well, that's boring; every interesting theory DOES have some axioms. Your core point here is surely that in the case of an axiom-set under IPC, one COULD have an axiom of the form q v ~q. Then q v ~q is (trivially) provable even if neither q nor ~q is. > but that doesn't mean it follows for set > theory in constructive FOL, > which is I take it what is at issue. Again, in that context, one HAS non-logical axioms. So all one has to do to achieve this is have an axiom of the form P -> (qvr). Then, one (constructively, intuitionstically) proves p, and voila, again, one has an intuitionistic proof of the disjunction without being able to prove either disjunct. Unfortunately I am having trouble trying to involve you and Babylonian in the same conversation. In ZF, the axiom that produces this most easily is called "pairing"; one formulation is AxyEzAw[ wez <-> (w=x v w=y) ]; if one wants to actually name the Ez (and one usually does) then this is re-spellable as Axyw[ we{x,y}<-> (w=x v w=y) ]. In either case, the point is, the fact that a disjunction occurs as the conclusion of an axiom means you can prove it by proving the hypothesis of the axiom AS OPPOSED to a disjunct. In set theory, any set-existence axiom where the set-being- postulated-to-exist is finite will produce this kind of disjunction (being an element of a finite set is equivalent to satisfying a disjunction of equalities to its elements). |