I just had a weird thought
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From: herbzet on 20 May 2010 23:55 "Ross A. Finlayson" wrote: > herbzet wrote: > > Marshall wrote: > > > herbzet wrote: > > > > Marshall wrote: > > > > > herbzet wrote: > > > > > > > > etc., etc.! > > > > > > > > Plus �a change, eh? > > > > > > (herbzet's axiom in French) > > > > > > > "The more things change, the less they stay the same." > > > > > > Bonsoir, Marshall -- nice to hear from you. Thanks for your reply! > > > > > C'est mon plaisir. > > > > > Glad to see you back, and I hope (for entirely selfish reasons) you > > > will again be a frequent contributor. > > > > My goodness, what an agreeable remark! I'm touched! (Perhaps it's > > best not to inquire into your "entirely selfish reasons"(!)) > > > > Basically, I don't think I'm quite up to be a frequent contributor > > for awhile longer -- though I intend to come back more often in > > the not-to-distant future. I just wanted to direct people's attention > > to the startling fact of non-Boolean-algebra structures for which > > classical propositional logic is a complete theory. I was hoping > > for some discussion by the smart guys of this, but it may take some > > time for people to get used to the idea. > > > > On the subject of models of classical (propositional) logic: > > both Schroder and C.I.Lewis (inventor, with C.H.Langford, of the > > S1-5 modal logics) tried to get rid of the models of Boole's > > algebra that have more than two elements (and so make things > > nicer for true/false logic) by including the formula A = (A = 1) > > in their axiomizations for propositional logic -- an axiomization > > that is otherwise identical to that for their algebra of classes. > > Later workers have passed over this meaningless axiom in charitable > > silence. > > > > Indeed, I myself have occasionally wondered how to get rid of all > > the "extra" models ever since I started to learn about logic. Letting > > '[=]' be ascii-speak for the triple bar symbol of logical equivalence, > > my latest effort was to add the formula > > > > (A[=]B) v (B[=]C) v (C[=]A) > > > > as an axiom. "Hah!" I thought. "Not elegant, but that oughta do it!" > > thinks I -- but alas: that formula turns out to denote the universal > > class (as does any tautology interpreted as a class) when A,B,C are > > any classes, 'v' is interpreted as class union, and '[=]' is inter- > > preted (as is usual) as the complement of the symmetric difference. > > > > Then the penny finally dropped: classical propositional logic is Post > > complete -- adding *any* formula as an axiom that is not already a > > theorem makes the logic inconsistent: every formula then becomes > > provable. The "extra" models cannot be gotten rid of by adding > > new axioms, except at the cost of inconsistency (and thus having > > no models at all). > > > > What an exciting life I lead! > > > > -- > > hz > > If you find a way to complete the formulas, then adding axioms would > just make the theory still true, with an axiom that "a false axiom is > ignored" [1], or simply that the axioms retain their theoretical > structure, or rather, the fact that they are only axioms in a > conceptualization or consideration of this true, complete theory. > > Then if there's any mathematical consistency to reality in, for > example, impredicativity, and other similar notions, basically saying > that it doesn't have a complete natural standard model (and standard), > then says it doesn't. > > [1] based on the mutual consistency of the other axioms > > Nice! > > Interesting to read about the notion of the general "distributive > lattice" or what not or Marshall's n+1 power structure, from the L/S > to the Levy. It is rather the notion, what with the reals being > consistently equivalent in ZF to a wide variety of cardinals, > (trichotomy of cardinals with limit ordinals), that the associations > of sets to power structures sees the sets maintain their > distributions, for example, as well their other characteristics in > natural numerical and propositional terms, generally, and that > incoherent results are from numerical error. Hi, Ross. Glad to see your still kickin' in there. Thanks for your reply! -- hz
From: George Greene on 24 May 2010 01:19 On May 20, 6:01 pm, Aatu Koskensilta <aatu.koskensi...(a)uta.fi> wrote: > George Greene <gree...(a)email.unc.edu> writes: > > On May 11, 9:07 am, Aatu Koskensilta <aatu.koskensi...(a)uta.fi> wrote: > >> Sure. But why should you think it a physical matter? It's a > >> mathematical triviality. > > > This from the person who taught me that "finitude is not first-order > > definable" > > ?? > > Yes? I'm afraid I don't quite see the connection. OK, I cut too much. The original question was, > "Every finite set can be generated by adding one element at a time, > starting from nothing". > This seems to be true. >>> Sure. But why should you think it a physical matter? >>> It's a mathematical triviality. While it might be a dictionarial/natural-language triviality, the fact that you canNOT even SAY this in FOL must mean that it is in some Mathematical sense NON-trivial. There are several different first-order-ZF versions of the definition of "finite", but there are models of ZF under which they do not all come out THE SAME, let alone all come out RIGHT. Under ZFC, most of these collapse to equivalence (so now they all come out the same), but EVEN there, they STILL do not all "always" come out RIGHT. Trying to re-state this in logical language would begin something like, "for any set, if it is finite, then it can be generated by adding 1 element at a time". In some models of ZF, for some definitions of finite, this is not true. In PA it is sort of vacuously untrue since PA encodes finite-set- theory TO BEGIN with, and EVERY set is "finite", but (obviously) the non- standard ones, in non-standard models of PA, canNOT be built up 1 step at a time. "Well, then, those AREN'T finite", but that's NOT the point. The point is, PA thinks everything is finite; it can't tell the difference. These things may be simple to people who have studied them, but again, the mere fact that something can't be done AT ALL in FOL is a strong argument in favor of its being more (mathematically) than a "triviality". "Finite but unbounded" is arguably kind of subtle. ALWAYS having finite parts/elements while not oneself being finite is kind of NON-trivial.
From: Charlie-Boo on 29 May 2010 21:26 On May 13, 9:48 am, Marshall <marshall.spi...(a)gmail.com> wrote: > On May 12, 10:33 pm, herbzet <herb...(a)gmail.com> wrote: > > > etc., etc.! > > > Plus ça change, eh? > > (herbzet's axiom in French) > > "The more things change, the less they stay the same." > > Marshall A joke for each generation!
From: Charlie-Boo on 29 May 2010 21:31 On May 24, 1:19 am, George Greene <gree...(a)email.unc.edu> wrote: > On May 20, 6:01 pm, Aatu Koskensilta <aatu.koskensi...(a)uta.fi> wrote: > > > This from the person who taught me that "finitude is not first-order > > > definable" > > > ?? > While it might be a dictionarial/natural-language triviality, > the fact that you canNOT even SAY this in FOL must mean > that it is in some Mathematical sense NON-trivial. Can't define "It is a finite set."? ~(all X)(eY)P(Y) ^ LT(X,Y) C-B
From: Charlie-Boo on 29 May 2010 21:32
On May 29, 9:31 pm, Charlie-Boo <shymath...(a)gmail.com> wrote: > On May 24, 1:19 am, George Greene <gree...(a)email.unc.edu> wrote: > > > On May 20, 6:01 pm, Aatu Koskensilta <aatu.koskensi...(a)uta.fi> wrote: > > > > This from the person who taught me that "finitude is not first-order > > > > definable" > > > > ?? > > While it might be a dictionarial/natural-language triviality, > > the fact that you canNOT even SAY this in FOL must mean > > that it is in some Mathematical sense NON-trivial. > > Can't define "It is a finite set."? ~(all X)(eY)P(Y) ^ LT(X,Y) Or you mean a finite set in general? Who said it can't be defined? (What a great challenge!) C-B > C-B |