The End of an Era - That of The Natural Numbers
in [Math]
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From: Aatu Koskensilta on 17 Jun 2010 06:10 "Jesse F. Hughes" <jesse(a)phiwumbda.org> writes: > Also, to be fair, a suitably pedantic fella (Aatu?) might dislike my > choice of preposition ("in" every interpretation), but surely this > should not confuse a deep reader like Nam. I'm flattered you consider me suitably pedantic. I did indeed dislike your choice of preposition. What you meant was perfectly clear, though. -- Aatu Koskensilta (aatu.koskensilta(a)uta.fi) "Wovon man nicht sprechan kann, dar�ber muss man schweigen" - Ludwig Wittgenstein, Tractatus Logico-Philosophicus
From: Jesse F. Hughes on 17 Jun 2010 09:00 Nam Nguyen <namducnguyen(a)shaw.ca> writes: > Jesse F. Hughes wrote: >> Nam Nguyen <namducnguyen(a)shaw.ca> writes: >> >>> Jesse F. Hughes wrote: >>>> Nam Nguyen <namducnguyen(a)shaw.ca> writes: >>>> >>>>> Jesse F. Hughes wrote: >>>>>> stevendaryl3016(a)yahoo.com (Daryl McCullough) writes: >>>>>> >>>>>>> Nam Nguyen says... >>>>>>>> Marshall wrote: >>>>>>>>> I thought that *you* were the one claiming that x=x is not true in >>>>>>>>> all contexts. >>>>>>>> I'm still claiming that. What have I just said that made you think >>>>>>>> otherwise? >>>>>>> To claim that a formula in a language L is not true in all contexts >>>>>>> is to claim that there is a structure for L in which the formula is >>>>>>> false, which is to claim that there is a structure for L in which >>>>>>> the negation is true. There is no such structure. >>>>>>> >>>>>>> A structure for a language is a way of consistently assigning "true" >>>>>>> or "false" to each closed formula in the language. >>>>>> Given that Nam (allegedly) uses Shoenfield, I think you ought to stick >>>>>> to Shoenfield's terminology. An open formula is neither true nor false, >>>>>> but is instead either valid or invalid. >>>>> So obviously you implied: >>>>> >>>>> (a) x=x is "neither true nor false". >>>>> >>>>>> The point remains, of course: x=x is valid, since it is true in every >>>>>> interpretation of every structure. >>>>> In "since it is true" it looked like by "it" you meant x=x. So apparently >>>>> you meant: >>>>> >>>>> (b) x=x "is true". >>>>> >>>>> Why such a contradiction between (a) and (b)? >>>>> >>>> No contradiction. >>> Yes there is, because ... >>>> As a formula, x=x has no truth value. >>>> >>>> But each interpretation (or M-instance, in Shoenfield's terms) of x=x >>>> has a truth value. >>> in your (a) and (b) there isn't the phrase "each interpretation ... of >>> x=x". >> >> Pardon me? >> >> It's sitting right up there. "It (x=x) is true in every interpretation >> of every structure." > > Isn't that _contradictory_ to your previous x=x "is neither true nor > false"? No, though I admit that my phrasing may be confusing due to my wording. I should have written, "Every M-instance of it is true." >> Now, I didn't use Shoenfield's terminology >> exactly, since I used the term "interpretation" rather than >> "M-instance", but you should have no doubt that I meant the exact same >> thing as Shoenfield's definition of validity. Also, to be fair, a >> suitably pedantic fella (Aatu?) might dislike my choice of preposition >> ("in" every interpretation), but surely this should not confuse a deep >> reader like Nam. >> >> Nonetheless, I'll say it again, in Shoenfield's terminology. >> >> x=x is valid because, for every structure M, every M-instance of x=x is >> true. This includes the empty structure[1]. > > You do have a reading problem Jesse. The debate here is whether or not > x=x is true when U is empty. That means ultimately you have to define > what _being true for a formula_ is. So far, you only had x=x is "valid"! Yes, for Shoenfield, open formulas are neither true nor false and x=x is open. Read the book! > In addition, what is an M-instance of x=x when U is empty (hence all > 2-nary predicates including one for '=' symbol are empty)? There are no M-instances if U is empty and hence (vacuously), every M-instance of x=x is true in this case. You don't get it. I know you don't get it, because you don't know a thing about vacuous universals like this. Thus, it's a shame that you're so smitten with the empty structure, because you have to understand vacuous universals to suss out the truth conditions in the empty model. >> >> Footnotes: >> [1] Although Shoenfield does not include the empty structure in his >> definition of structure, it is obvious that every M-instance of x=x is >> true when |M|={}. > > It'd only be obvious when you prove it in an empty 2-ary predicate-set > using strictly set membership to satisfy Tarski's concept of truth. Your > claiming "it is obvious" does NOT at all make it a proof. Where's your > proof when you haven't even defined x=x being true, to start with? Your cluelessness is impenetrable. Tain't my fault you've got Shoenfield's book right there and yet still can't figure out that x=x is neither true nor false in Shoenfield's terminology, but rather valid or invalid. M-instances of x=x are either true or false (indeed, every M-instance of x=x is true, regardless of the structure M), but not the formula itself. Once again, you are welcome to proudly proclaim that only you really address technical details, while the rest of us just diddle. You have reached a certain perfection in your stupidity, still superficially coherent but incapable of actual understanding of fairly basic points in a standard text. No one's paying me to teach you, so I don't see why I should continue this pointless exercise. I would suggest that you find a tutor to help you work through this material, except I don't think it would help. It's obvious that learning the subject on your own isn't working, but neither will being tutored unless you admit the possibility that your tutor actually knows more about your subject than you do. You won't do that. You're too certain that your own grasp is already so perfect and so deep that you declare then end of the natural numbers. You've taken an odd path to irredeemable crankdom. -- Jesse F. Hughes "The Hammer has arrived." -- James S. Harris, Feb. 14 2006
From: Jesse F. Hughes on 17 Jun 2010 09:35 "Jesse F. Hughes" <jesse(a)phiwumbda.org> writes: >> Isn't that _contradictory_ to your previous x=x "is neither true nor >> false"? > > No, though I admit that my phrasing may be confusing due to my > wording. Er, ah... Yeah. My phrasing may be confusing due to my wording. Just another case of getting to the end of a sentence and mucking it up, because I forgot what the middle of the sentence contained. Obviously, my phrasing *is* sometimes confusing, but that's due to one of three things: (1) the words I choose and the order I put them in. (2) the wording. (3) the phrasing. I hope that clears it up. -- "To solve this problem, we define a security flag, known as the 'evil' bit, in the IPv4 [RFC791] header. Benign packets have this bit set to 0; those that are used for an attack will have the bit set to 1." -- RFC 3514
From: MoeBlee on 17 Jun 2010 13:49 On Jun 16, 8:27 pm, Nam Nguyen <namducngu...(a)shaw.ca> wrote: > MoeBlee wrote: > > On Jun 15, 11:13 pm, Nam Nguyen <namducngu...(a)shaw.ca> wrote: > Yes it's a fact that he chose to narrow his definition to restrict > U to be non-empty. I wouldn't say he narrowed, since it is the usual definition. But, in the sense that his definition is narrower than some other definitions (such as in free logic, as I understand), okay. > > If one wishes to > > consider a notion of a structure for a language in which the universe > > for the stucture is empty, then one must revise Shoenfield's > > definition and recognize that one is then working with a different > > definition from Shoenfield's. > > I don't agree. If you say the universe can be non-empty, then you're working with a different definition as Shoenfield. And to work with your definition we must recognize that it is revised from Shoenfields in at least the sense that you allow empty universes. Surely, you agree with this? > Each author's textbook is apt to have some _peculiarities_ > that other authors might or might not have. Shoenfield, e.g., defined > a formula being "valid" in T instead of being "true" in T. Of course, fair enough. Except that Shoenfield's stipulation of non-empty universe isn't peculiar. It's virtually part of any ordinary definition in mathematical logic. Of course, there are defintions in which the universe may be non-empty, and we may study such things, but in ordinary mathematical logic, the universe is stipulated (i.e., part of the defintion of 'structure for a language') to be non-empty. > And the subject > of FOL reasoning would transcend each individual author's writing/posting, > which means unless the written source was so bad that _really needs_ a > revision, we could still see in the source some _common_ knowledge of > FOL we could use to further inferences, or making arguments. In a sense, yes, fair enough. In an informal sense, we may cut through inessential variations among authors to see what the basic, essential notions are that they're presenting in different ways. > Remember why we got here: > > Early on in the thread I had: > > >On a more serious note, my > > > > >>>>> In other word there's no absolute truth. > > > > only meant _within the context of FOL reasoning_ there's > > no such thing as an absolute truth of a formula! > > (to which Marshall responded that x=x is such a formula). > > Well, "_within the context of FOL reasoning_" means the subject we've been > debating is at a level above any specific peculiarities of a given author's > writing. > > So no, I don't agree Shoenfield's book needs to be rewritten, notwithstanding > he had a restriction on the universe U. I didn't say his book needs to be rewritten. I'd just like to be clear that you agree with the following: If you say the universe can be non-empty, then you're working with a different definition from Shoenfield. And to work with your definition we must recognize that it is revised from Shoenfield's in at least the sense that you allow empty universes. Saying 'yes' to that is not a big philosophical deal. Just common sense that allows us to communicate. Right? MoeBlee
From: MoeBlee on 17 Jun 2010 14:39
On Jun 17, 5:02 am, Aatu Koskensilta <aatu.koskensi...(a)uta.fi> wrote: > Nam Nguyen <namducngu...(a)shaw.ca> writes: > > So what's in you mind the definition of x=x being true be, model > > theoretically speaking? Can you _explicitly_ state the definition? > > Just read some more Shoenfield. Wait, if 'x' is a variable, then, at least by pg. 19, there is no truth value for "x=x". That is, Shoenfield, gives truth values only for closed formulas (sentences), and he bypasses defining 'satisfaction of a formula' (for formulas in general including open formulas) with a certain technique. Right? Also, Aatu, I haven't been following this thread in every detail. That said, if I'm not mistaken there is one glitch that has to be addressed in allowing a version of Shoenfield with empty domains. In Shoenfield's definition of a structure, (when truly formalized) there is a function from the set of constants into the universe: "...for each constant e of L, e_fancyA is an individual of fancyA." I take that formally to mean there is a function f from the set of constants into the universe of fancyA (universe of fancy_A = | fancyA|): f(e) = e_fancyA and e_fancyA in |fancyA| So there must exist f such that f:set_of-constants -> |fancyA| But if the universe is empty while the set of constants is non-empty then there is no such function. So, I don't see how we can have an empty universe for a structure for a language that has constants. / Meanwhile, let me see whether I understand the heart of the discussion correctly: The truth value of Ax x=x (where 'x' is a variable) is T iff the truth value of x=x _x[i] is T for every i in the set of names. (For those who don't have Shoenfield, if I recall, where P is a formula, P _x[i] is P[i|x], i.e, putting 'i' in for all free 'x'.) So if the set of names is empty: Then the function from the universe to the set of names is the empty function. And x=x _x[i] for every i in the set of names. So the truth value of Ax x=x is T. But then so is the truth value of Ax ~x=x. For ANY formula P with at most 'x' free, the truth value of Ax P is T. And that does make sense in this regard: since then for any formula P with at most 'x' free, we have the truth value of Ex P is F which makes sense for an empty universe. MoeBlee |