Is ZFC axiomatizable in FOPC?
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Группы новостей: fa.analytic-philosophy
Автор: "Seth Sharpless" <seth.sharpl...@colorado.edu>
Дата: Sun, 1 Sep 2002 07:16:01 GMT
Местное время: Вс. 1 сен 2002 10:16
Тема: Re: [analytic] Re: Is ZFC axiomatizable in FOPC?
Torkel wrote, repeating my quote of Church: >Following Carnap and others, we use the term "language" in such a Church may well have followed Carnap in this unfortunate usage! It is not >sense that for any given language there is one fixed notion of a proof >in that language. [Alonzo Church] standard logical or mathematical usage. -------------------- Well, it would once have been sufficient to quote Alonzo Church's "Mathematical Logic" on any matter of standard usage in that field; he practically created "standard" usage in that field and for many years oversaw it zealously in the review pages of _J. Symb. Logic_. But customs change, as I have found to my grief in my old age. Since you are so put off by my (and Church's) usage, I will try to employ yours, if I understand it, but before continuing this exchange (in which, please note, you have been very helpful--save for a certain compulsiveness about the words 'language' and 'theory'), let me make sure I have the new usage straight: Let's see: "languages" are defined by their primitive symbols and formation rules but not by transformation rules or rules of inference. Once one adds a transformation rule, or a method of proof, one has a "theory," and it is improper to refer to a "theory" as a "language." Is that right? Once one adds, say _modus ponens_, or simply a rule specifying that a sequence of two sentences is a "proof" if the first is an axiom and the second is identical to the first, one has got a "theory." Right? Sounds batty to me; I much prefer Church's usage, but if that is what it takes to be _au courant_...and take advantage of your kind tutoring... [Non-text portions of this message have been removed] ------------------------ Yahoo! Groups Sponsor ---------------------~--> (c) 2002 by Analytic Your use of Yahoo! Groups is subject to http://docs.yahoo.com/info/terms/ Чтобы отправлять сообщения, сначала необходимо Войти.
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Группы новостей: fa.analytic-philosophy
Автор: Torkel Franzen <tor...@sm.luth.se>
Дата: Sun, 1 Sep 2002 22:39:36 GMT
Местное время: Пн. 2 сен 2002 01:39
Тема: Re: [analytic] Re: Is ZFC axiomatizable in FOPC?
Seth says: >Let's see: "languages" are defined by their primitive symbols and It's no more and no less improper than it is to refer to 1/2 as a The drawback of your preferred terminology is that it's very unclear ------------------------ Yahoo! Groups Sponsor ---------------------~--> (c) 2002 by Analytic Your use of Yahoo! Groups is subject to http://docs.yahoo.com/info/terms/ Чтобы отправлять сообщения, сначала необходимо Войти.
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Группы новостей: fa.analytic-philosophy
Автор: "Seth Sharpless" <seth.sharpl...@colorado.edu>
Дата: Sun, 1 Sep 2002 07:15:47 GMT
Местное время: Вс. 1 сен 2002 10:15
Тема: Re: [analytic] Re: Is ZFC axiomatizable in FOPC?
Dear Murphy,
I think you may be confusing me with a disciple of early Fodor or Chomsky. As some one who spent at least half his career as a neurobiologist, trying to find out how the brain works,let me say that I do not believe that Chomsky's theories (or Montague's grammar) describe a brain mechanism involved in the production and recognition of speech. If that is what is bothering you, I quite agree with you. I do believe, as you suggest, that such theories (oops, I'm getting sensitive about that word 'theories'--well, whatever they are) are indeed very much like the grammars used to teach English as a 2nd language. It is as if you picked up one of those grammars, regarding it not as a prescription for good behavior in the presence of the Queen, but rather as a description of social practices cumulatively known as "speaking English," and said: hmm, I could do better than this, taking care of the exceptions and the lack of precision, etc. that characterize those 2nd language "grammars" by employing the resources of a symbolic language as a metalanguage for talking about English. Much like Frege, they are trying to "mathematize" grammar. The products, Montague's theories for example, would not be very helpful teaching English as a second language, not because the undertaking is different in principle from writing a grammar for people learning English as a 2nd language, but rather because much of the metalanguage used by Montague is itself in English, and the symbolic part is very dense and hard to follow, requiring a taste for mathematical-type notation. That is why it would be a disaster to use such a "grammar" in teaching a 2nd language. Another reason why it would be a disaster is because the best way to acquire a 2nd language, as everybody knows, is not by reading a grammar, even a simple-minded grammar chock full of oversimplifications, but by immersion--use in context. Nobody is composing or deciphering sentences by reading a kind of grammar book in his head, like an inhabitant of Searle's Chinese Room. The idea is preposterous. The philosophical point I am trying to find some way of making without making an ass of myself in connection with the subtleties of set theory is that even with an ideal grammar and a lexicon in which there were no 2nd definitions, and in which every definition represented itself as an exact equivalence between definiens and definiendum, even with these perfect manuals, the language so described would, of necessity, be rife with semantic indeterminacy (rife with vagueness, ambiguity, polysemy--I mention all these forms of semantic indeterminacy to avoid having to distinguish between them). It is my "uncertainty" principle for semantics: there is some kind of fundamental metaphysical principle involved: thou shalt not escape semantic indeterminacy. You seem to be fretting about my use of the expression "truth conditions." Well, I'm under no illusion that natural languages are extensional. My comment about "truth conditions" was in a context where we were discussing languages (theories? formal systems?) of mathematics, and, of course, the languages of mathematics are extensional, but natural languages are not. I should think you and I would be on the same side in all this. Regards, Seth [Non-text portions of this message have been removed] ------------------------ Yahoo! Groups Sponsor ---------------------~--> (c) 2002 by Analytic Your use of Yahoo! Groups is subject to http://docs.yahoo.com/info/terms/ Чтобы отправлять сообщения, сначала необходимо Войти.
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Группы новостей: fa.analytic-philosophy
Автор: "mjmurphy" <4m...@rogers.com>
Дата: Mon, 2 Sep 2002 18:23:57 GMT
Местное время: Пн. 2 сен 2002 21:23
Тема: Re: [analytic] Re: Is ZFC axiomatizable in FOPC?
> Dear Murphy, Chomsky. As some one who spent at least half his career as a > I think you may be confusing me with a disciple of early Fodor or neurobiologist, trying to find out how the brain works,let me say that I do not believe that Chomsky's theories (or Montague's grammar) describe a brain mechanism involved in the production and recognition of speech. If that is what is bothering you, I quite agree with you. ----- Cheers, M.J.Murphy The shapes of things are dumb. ------------------------ Yahoo! Groups Sponsor ---------------------~--> (c) 2002 by Analytic Your use of Yahoo! Groups is subject to http://docs.yahoo.com/info/terms/ Чтобы отправлять сообщения, сначала необходимо Войти.
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Группы новостей: fa.analytic-philosophy
Автор: "Seth Sharpless" <seth.sharpl...@colorado.edu>
Дата: Tue, 3 Sep 2002 17:07:27 GMT
Местное время: Вт. 3 сен 2002 20:07
Тема: Re: [analytic] Re: Is ZFC axiomatizable in FOPC?
Murphy,
I can't remember saying "a natural language has a formal language as its rules." Perhaps I misled you by saying of Montague, Carnap, etc. that "much like Frege, they were trying to 'mathematize' grammar." But remember, a grammar is written in a metalanguage about an object language. You can use a formal language, the language of mathematics or set theory, as a metalanguage in describing the syntax and semantics of English just as you can use Latin, say, as a metalanguage to describe the syntax of English. But no one is claiming, I think, that the rules of English are the same as the rules of the mathematical metalanguage. I do not know what Dummett has said about this, but, yes, I do think of the Montague-type grammar as analogous to the kind of grammar one would use to teach a 2nd language, only much more detailed and written mostly in the language of mathematics. The metalanguage in that case--i.e., a language of mathematics--would be extensional and its connectives would be truth-functional. But it doesn't follow that the object language whose grammar it is describing need be extensional, and I think neither the 'or' nor the 'if...then...' of English is truth-functional. In saying that a language is not extensional, I mean that the truth value of the sentences of the language may depend not only on the extensions (referents) of its components but also upon their intensions--that is, that it may be necessary in the metalanguage to refer to the intensions (senses, meanings) of object language expressions as well as their extensions to explain how the components of a sentence contribute to its meaning. The metalanguage and object language may differ in many ways: For example, English is a highly tensed language. A sentence, 'Mary has auburn hair' may be true today and false tomorrow, and indeed, it is difficult to frame an English sentence in an unambiguously non-tensed way. But a Montague-type (set-theoretic)metalanguage is not tensed. That prevents a sentence-for-sentence translation, but it does not prevent the Montague-type metalanguage from specifying truth-conditions for the tensed object language sentence. It is sometimes supposed (perhaps due to too much emphasis on the "disquotational" theory of truth), that to lay down the truth conditions for an object language sentence in a metalanguage, one has to be able to _translate_ the object language sentence into the metalanguage. Not so. You cannot translate a sentence whose truth value changes from one day to the next into a language whose sentences do not vary in truth value across time. Still, you can state the truth conditions for the tensed object language sentence in a tenseless metalanguage, by explicit reference to its time and context of utterance (which is more or less what Montague showed us how to do). You seem to have some general objection to the idea that what a declarative sentence means can be expounded by stating its truth conditions. I don't think I understand your objection, but taking it at face value, I would go along with it a little way. Thus, in the previous paragraph, I explained how one could state the truth conditions of a tensed object language sentence in a tenseless metalanguage. Now, in this case, the tenseless metalanguage sentence describing the truth conditions of the tensed object language sentence is not equivalent in meaning to the latter, since the one can't be translated into the other. The metalanguage sentence describes the meaning of the object language sentence, but it doesn't _have_ that meaning itself. Thanks for the comments (and no need to worry about your tirade--it was vivifying), (and, if you are reading this, Tapper, thanks for the defense. I seem to need a lot of that lately). Seth ------------------------ Yahoo! Groups Sponsor ---------------------~--> (c) 2002 by Analytic Your use of Yahoo! Groups is subject to http://docs.yahoo.com/info/terms/ Чтобы отправлять сообщения, сначала необходимо Войти.
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Группы новостей: fa.analytic-philosophy
Автор: "larry_tapper" <larry_tap...@yahoo.com>
Дата: Tue, 3 Sep 2002 17:11:55 GMT
Местное время: Вт. 3 сен 2002 20:11
Тема: [analytic] Re: Is ZFC axiomatizable in FOPC?
--- In analytic@y..., "mjmurphy" <4mjmu@r...> wrote:
MJ, No, I don't think recognizing this equivalence commits us to any By the way, I didn't intend my use of the word "tirade" to be a Larry ------------------------ Yahoo! Groups Sponsor ---------------------~--> (c) 2002 by Analytic Your use of Yahoo! Groups is subject to http://docs.yahoo.com/info/terms/ Чтобы отправлять сообщения, сначала необходимо Войти.
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ZFC, Att: Franzen, Mortensen, Murphy, Sides, Tapper
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Группы новостей: fa.analytic-philosophy
Автор: "Seth Sharpless" <seth.sharpl...@colorado.edu>
Дата: Wed, 4 Sep 2002 03:55:23 GMT
Местное время: Ср. 4 сен 2002 06:55
Тема: [analytic] ZFC, Att: Franzen, Mortensen, Murphy, Sides, Tapper
This is addressed to those who contributed to the later parts of the thread on ZFC, concerned with the question of whether set theory is semantically determinate, or, more provocatively, whether mathematicians know what they are talking about when they talk about "sets." A substantial part of the thread was devoted to instructing me in the nuances of set theory, for which, many thanks. I am about to go offline for a while for some light canoeing (and mosquito battling) in Northern Minnesota, and in reading over the thread by way of trying to summarize it before I forgot it, I found your contributions very instructive with respect to questions that I had wondered about (and some I had not wondered about). I wanted to acknowledge that.
There was a certain defensiveness in some of the responses, understandable if the question had been put provocatively, as above, but the basic question wasn't whether the mathematician's concept of "set" was sufficiently precise for his purposes. It is my belief that semantic indeterminacy is inevitable in all discourse, but that it is no great impediment to communication. The early logicians, Frege, Russell, Carnap, etc., all hoped, in the spirit of Leibniz, to attain precision of meaning by the use of artificial languages, and as a consequence, neglected the semantic problems associated with vagueness, ambiguity, etc., considering them defects of natural languages, though these problems have been the focus of much research in recent years. I wanted to argue that they infect the artificial languages (theories?) of the mathematician too, and that they are inevitable. I think that in connection with set theory, my thesis came down in the end to a triviality (from a mathematician's point of view), for granting that a "model" is an "interpretation," even categoricity of the "intended models" would allow for different interpretations, though the differences would not be relevant so far as the mathematician is concerned. And if the "intended models" of set theory are not even categorical, each model instead being "isomorphic to V-sigma for *some* uncountable strongly inaccessible cardinal sigma," that semantic indeterminacy is even more marked. Lowenheim-Skolem just brings the problem home, for it demands of the set theorist that he make his "intentions" with respect to interpretations explicit, which, it turns out, is not easy to do. Thank you again for a very interesting and instructive discussion. Seth [Non-text portions of this message have been removed] ------------------------ Yahoo! Groups Sponsor ---------------------~--> (c) 2002 by Analytic Your use of Yahoo! Groups is subject to http://docs.yahoo.com/info/terms/ Чтобы отправлять сообщения, сначала необходимо Войти.
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Группы новостей: fa.analytic-philosophy
Автор: Torkel Franzen <tor...@sm.luth.se>
Дата: Wed, 4 Sep 2002 22:44:16 GMT
Местное время: Чт. 5 сен 2002 01:44
Тема: Re: [analytic] ZFC, Att: Franzen, Mortensen, Murphy, Sides, Tapper
Seth says: >And if the "intended models" of This openness of set theory is only significant if we are asking >Lowenheim-Skolem just brings the problem home, for it demands of the Here it is unclear what you may have in mind. ------------------------ Yahoo! Groups Sponsor ---------------------~--> (c) 2002 by Analytic Your use of Yahoo! Groups is subject to http://docs.yahoo.com/info/terms/ Чтобы отправлять сообщения, сначала необходимо Войти.
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Группы новостей: fa.analytic-philosophy
Автор: "Seth Sharpless" <seth.sharpl...@colorado.edu>
Дата: Thu, 5 Sep 2002 09:07:10 GMT
Местное время: Чт. 5 сен 2002 12:07
Тема: Re: [analytic] ZFC, Att: Franzen, Mortensen, Murphy, Sides, Tapper
Got to go, but...(if you would like to reply to this, I will be able to read the reply but unable to respond for a couple of weeks).
------Torkel wrote-------- >And if the "intended models" of This openness of set theory is only significant if we are asking ----------Torkel wrote:--------- Seth says: >Lowenheim-Skolem just brings the problem home, for it demands of the >set theorist that he make his "intentions" with respect to >interpretations explicit, which, it turns out, is not easy to do. Here it is unclear what you may have in mind. [Non-text portions of this message have been removed] ------------------------ Yahoo! Groups Sponsor ---------------------~--> (c) 2002 by Analytic Your use of Yahoo! Groups is subject to http://docs.yahoo.com/info/terms/ Чтобы отправлять сообщения, сначала необходимо Войти.
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Группы новостей: fa.analytic-philosophy
Автор: Rudolf.Driesch...@t-online.de
Дата: Fri, 6 Sep 2002 04:52:21 GMT
Местное время: Пт. 6 сен 2002 07:52
Тема: Re: [analytic] ZFC, Att: Franzen, Mortensen, Murphy, Sides, Tapper
Apologies for, just for a moment, butting in.
Seth's formulation > [...] the "intended models" of reveals, so it seems to me, a bad misconception of categoricalness. Models > set theory are not [...] categorical [...] are never (not) categorical. It is just (sets of) _sentences_, that is (sets of) certain linguistic entities, which, at best, might be categorical (meaning of course: any two models of such (sets of) sentences are isomorphic). Maybe this comment sounds a bit pedantic; maybe, it IS pedantic, and Seth Best wishes, ------------------------ Yahoo! Groups Sponsor ---------------------~--> (c) 2002 by Analytic Your use of Yahoo! Groups is subject to http://docs.yahoo.com/info/terms/ Чтобы отправлять сообщения, сначала необходимо Войти.
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