The End of an Era - That of The Natural Numbers
in [Math]
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From: Nam Nguyen on 16 Jun 2010 23:29 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"? > 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"! 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)? > > 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?
From: Ross A. Finlayson on 16 Jun 2010 23:36 On Jun 16, 8:10 am, Marshall <marshall.spi...(a)gmail.com> wrote: > On Jun 15, 11:41 pm, Nam Nguyen <namducngu...(a)shaw.ca> wrote: > > > Marshall wrote: > > > On Jun 15, 11:12 pm, Nam Nguyen <namducngu...(a)shaw.ca> wrote: > > > > So what's your definition of an open formula being true in a model, > > > > to begin with? > > > > How many times do you need me to say I don't follow Shoenfield? > > > How many times do you need me to say that I'm using the convention > > > that unbound variables are implicitly universally quantified? > > > > I guess the answer to these and many other questions about > > > your comprehension is that no finite number is sufficient. > > > My hat is off to Daryl. > > > You're mistaken Marshall: whatever you're saying here > > "Whatever" I'm saying here? You mean you don't know > what it is? I guess that explain why you continue not to > understand me. > > > is just > > a poor smokescreen for you incompetence to demonstrate x=x is > > true in a structure with U being non-empty. (There are people > > who can, you know!) > > Being called incompetent by a talentless buffoon is no insult. > > Marshall When is it ever? Marshall, far from it from me to call you an incompetent, not for years did somebody drive me just to plain call them incompetent. At the time, I probably felt well incompetent. Yet, why should I so casually digress from the line, the simple tow, toe the line, Incompetence! Isn't that even worse? Again, line by line, not even an insult for a talentless buffoon, how could this be? Unbounding the non-empty domain restriction makes more infinite symbols in the alphabet. Thus, the uh, traditional theorem-proving machinery sees some retrofit, regaining the model, here that's technical, from the paraconsistent to the infraconsistent and back to basically consistent in the metamodel, acknowledging the domain restriction or residual model, and using the usual symbols with the acknowledgment they're different symbols. This is a general method in inverses, admitting the result with conditions. Excuse me, that was to over there, about the structural. Nam, far from it from me to ... .... Herb of course you here excuse my levity with Marshall and Nam without saying anytime! > > is just > > a poor smokescreen for you incompetence to demonstrate x=x is > > true in a structure with U being non-empty. (There are people > > who can, you know!) Here U is the universe? Structurally, x and some identical object x would evaluate the same under the conditions defining the identity, here two objects have as structurally basic as a bit the either 1 or 0. Then, that has a basic evaluation rule that they're literally equal, two bits, if they have the same value, both 1 or both 0. Now in some systems, how they're evaluated, and whether two bits would have the same value and be equal, includes moving, structurally, from the one to the other that they would actually be possibly considered identical (one and the same). So, two bits might have different values, but by the time they could be compared or it would matter, actually by the time it would matter, they'd be either equal or not and at the same instant, indistinguishable and interchangeable. Then, the strength in mathematical systems is that moving them around does nothing to them off the labels (in error-free systems, less the applied). That's structurally. Warm regards, Ross Finlayson
From: Nam Nguyen on 16 Jun 2010 23:46 Aatu Koskensilta wrote: > "Jesse F. Hughes" <jesse(a)phiwumbda.org> writes: > >> The formula x=x is valid for every structure (of its language). That >> is, for every structure M, every M-instance of x=x is true. > > Shoenfield requires the domain of a structure to be nonempty. Of course, > nothing in the definition of truth in a structure actually depends on > this, and if we relax the requirement we do indeed find that every > M-instance of x=x is true in a structure with empty domain. So what's in you mind the definition of x=x being true be, model theoretically speaking? Can you _explicitly_ state the definition?
From: Nam Nguyen on 17 Jun 2010 00:00 Marshall wrote: > On Jun 15, 11:41 pm, Nam Nguyen <namducngu...(a)shaw.ca> wrote: >> Marshall wrote: >>> On Jun 15, 11:12 pm, Nam Nguyen <namducngu...(a)shaw.ca> wrote: >>> > So what's your definition of an open formula being true in a model, >>> > to begin with? >>> How many times do you need me to say I don't follow Shoenfield? >>> How many times do you need me to say that I'm using the convention >>> that unbound variables are implicitly universally quantified? >>> I guess the answer to these and many other questions about >>> your comprehension is that no finite number is sufficient. >>> My hat is off to Daryl. >> You're mistaken Marshall: whatever you're saying here > > "Whatever" I'm saying here? You mean you don't know > what it is? I guess that explain why you continue not to > understand me. Of course nobody can understand you Marshall! Why? because you were asked a straightforward question above": >>> > So what's your definition of an open formula being true in a model, >>> > to begin with? and your answer was idiotically irrelevant: >>> How many times do you need me to say I don't follow Shoenfield? and at the end of the post(s) you couldn't even tell what the precise definition of x=x being true is, yet claiming it's always be true. How can people know what you're saying when what you say is typically idiotic. >> is just >> a poor smokescreen for you incompetence to demonstrate x=x is >> true in a structure with U being non-empty. (There are people >> who can, you know!) > > Being called incompetent by a talentless buffoon is no insult. You're being called incompetent because you could _not_ define what x=x being true be, yet claiming this or that about it. And it doesn't have to be Nam asking you the question: anyone could ask you for your understanding of what x=x (as a formula) being true is and you still can't answer!
From: Marshall on 17 Jun 2010 00:08
On Jun 16, 9:00 pm, Nam Nguyen <namducngu...(a)shaw.ca> wrote: > Marshall wrote: > > On Jun 15, 11:41 pm, Nam Nguyen <namducngu...(a)shaw.ca> wrote: > >> Marshall wrote: > >>> On Jun 15, 11:12 pm, Nam Nguyen <namducngu...(a)shaw.ca> wrote: > >>> > So what's your definition of an open formula being true in a model, > >>> > to begin with? > >>> How many times do you need me to say I don't follow Shoenfield? > >>> How many times do you need me to say that I'm using the convention > >>> that unbound variables are implicitly universally quantified? > >>> I guess the answer to these and many other questions about > >>> your comprehension is that no finite number is sufficient. > >>> My hat is off to Daryl. > >> You're mistaken Marshall: whatever you're saying here > > > "Whatever" I'm saying here? You mean you don't know > > what it is? I guess that explain why you continue not to > > understand me. > > Of course nobody can understand you Marshall! Pretty much everyone except you understands me just fine, Potato Chip. Marshall |