The End of an Era - That of The Natural Numbers
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From: Nam Nguyen on 30 May 2010 01:33 William Hughes wrote: > On May 29, 6:23 pm, Nam Nguyen <namducngu...(a)shaw.ca> wrote: >> William Hughes wrote: >>> On May 29, 5:22 pm, Nam Nguyen <namducngu...(a)shaw.ca> wrote: >>>> William Hughes wrote: >>>>> On May 29, 4:03 pm, Nam Nguyen <namducngu...(a)shaw.ca> wrote: >>>>>> William Hughes wrote: >>>>>>> On May 29, 2:55 pm, Nam Nguyen <namducngu...(a)shaw.ca> wrote: >>>>>>>> But when U = {}, there's no flexibility at all >>>>>>> So your claim is that >>>>>>> There does not exist an x such that blue(x) >>>>>>> must be false? You can refer to as many mappings >>>>>>> and definitions of "truth" as you want. At the end >>>>>>> of the day if all formula are false in a model with >>>>>>> empty universe, then >>>>>>> There does not exist an x such that blue(x) >>>>>>> must be considered false. >>>>>> It must have been the case you either didn't read or wasn't >>>>>> paying attention or wasn't able to understand what I said >>>>>> about the truth preemptive characteristics of the meta statement >>>>>> B in the post. >>>>>> It doesn't matter what I "want" here: that's Tarski's definition >>>>> Ok, rephrase. >>>>> At the end of the day your claim is that, >>>>> using Tarski's defintion of truth, >>>>> all formula are false in a model with empty universe >>>>> Then >>>>> There does not exist an x such that blue(x) >>>>> must be considered false. >>>> Are you saying that >>>> "There does not exist an x such that blue(x)" >>>> is a FOL formula of L(T4)? >>> Yes. Are you claiming it is not. >> Oh. My mistake. You're right, it is: ~Ex[blue(x]). But that's a >> just FOL and therefore is false in the structure in which U = {}, >> due to B is false, correct? > > > Nope. If there is no x then > > There is no x such that x is blue > > is true. Then you and I aren't talking about the same thing and I was talking about model theoretical truth, as in: >> Would you see in now? It doesn't matter whether or not F is >> _syntactically_ tautologous or contradictory, the meta statement >> A, by definition of structure, will also depend on B. And if B is >> false by virtual of the factual U's being empty then A is false. Don't you see that if B is false then A = (b /\ C) would be false, irregardless of what C might mean?
From: Nam Nguyen on 30 May 2010 01:46 Nam Nguyen wrote: > Don't you see that if B is false then A = (b /\ C) would be false, > irregardless of what C might mean? Of course I meant "regardless of what C might mean".
From: William Hughes on 30 May 2010 07:16 On May 30, 2:33 am, Nam Nguyen <namducngu...(a)shaw.ca> wrote: > William Hughes wrote: >> If there is no x then > > > There is no x such that x is blue > > > is true. > > Then you and I aren't talking about the same thing and I was > talking about model theoretical truth, This does not matter You claim that there is some meaning you can give to "truth" that makes both There is an x such that x is blue and There is no x such that x is blue false. Whatever, your reasoning, whomever you reference, the final conclusion is silly. - William Hughes
From: Daryl McCullough on 30 May 2010 10:05 Nam Nguyen says... >Alan Smaill wrote: >> Why on earth would Shoenfield say: >> >> "In particular, for each constant e of L, e_a is an individual of >> *A*." >> >> if he meant that individuals could denote the empty set??? > >Are you saying it's forbidden in FOL that U = {} ? Such a U cannot be part of a structure for a language that has constant symbols. >> And it leads to contradictions with basic principles, >> like Tarski's recursive definition of truth, > >If you do have reasons but don't present that with technical >terminologies (such as set) then I don't know what you mean here. > > >> and important claims, such as the completeness theorem. > >Well, Completeness is a downstream subject and we've been >debating something upstream. Let's have a closure on whether >or not x=x is true in all contexts of FOL first, before >moving on to others, I'd think. There is a class of formulas which are true for every assignment in every nonempty structure (by which I mean a structure in which the domain U is nonempty), and these are often called "valid" formulas. Everyone agrees that, given this definition of valid, "x=x" is valid. Whether you want to say that that means that it is true "in all contexts of FOL" depends on whether you consider empty structures, and how you define truth for an empty structure. Define it any way you like. For the most part, NOBODY cares. When people talk about model theory, typically there is the unspoken premise ("...for nonempty structures"). There is no content to your complaints. You want to say that "x=x" is not valid, because you want to allow empty structures, and you want to have a definition of truth for an empty structure such that nothing is valid in the empty structure. Fine. Go ahead. But who cares? x=x is valid for the cases that the vast majority of people are interested in. It's valid for *all* cases, if you define "valid in a structure" a particular way (which is the way I prefer to deal with empty structures). So we have the following facts: I think everyone agree with the following two statements: 1. x=x is valid for nonempty structures. 2. x=x is valid for empty structures, according to one convention. 3. x=x is not valid for empty structures, according to a different convention (one that says that no formulas are valid in the empty structure). -- Daryl McCullough Ithaca, NY
From: Daryl McCullough on 30 May 2010 10:30
William Hughes says... >On May 30, 2:33=A0am, Nam Nguyen <namducngu...(a)shaw.ca> wrote: >This does not matter You claim that >there is some meaning you can give to "truth" that >makes both > > There is an x such that x is blue >and > There is no x such that x is blue > >false. Whatever, your reasoning, whomever you reference, >the final conclusion is silly. Right. To the extent that Nam's definitions agree with Shoenfeld's, so much the worse for Shoenfeld. -- Daryl McCullough Ithaca, NY |