The End of an Era - That of The Natural Numbers
in [Math]
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From: Nam Nguyen on 22 May 2010 18:37 Jesse F. Hughes wrote: > Nam Nguyen <namducnguyen(a)shaw.ca> writes: > >> Jesse F. Hughes wrote: >>> Nam Nguyen <namducnguyen(a)shaw.ca> writes: >>> >>>> I had a posted with technical examples of T1, T2, T3 >>>> and M1, M2, M3; and M3 would have the answer already. >>>> >>>> Can't you know how to read _technical arguments_ ? >>>> >>>> (You really sound like a kid wanting to "win" at all cost!) >>>> >>> Is it not a yes/no question? Why not just answer yes or no? >> Why don't you answer this "yes/no" question: is Axy[x=y] >> provable? >> >> That is, just answer yes or no - and NOTHING ELSE! > > > If you mean is Axy[x = y] provable in classical FOL, then the answer > is no. But no has asked me the yes/no question "Is Axy[x = y] provable in classical FOL?". > > If you mean is there a theory in which that formula is provable the > answer is yes. And no one asked me the yes/no question "Is there a theory in which that formula is provable?". > > Perfectly clear response, depending just on what your question means. So, what _exactly_ did WH mean by L? If by L he meant M3 for the inconsistent T3 then, for the nth time, my answer is no: there's no formula is true in M3 as a (false) model of T3. If he meant something other than M3 then of course his question wasn't, as you said, "clear". > > Now you have a go at it. Let L be the trivial model, i.e., the model > with empty support. Is (Ax)~(x = x) true in L? Since, as above, "no formula is true" then (Ax)~(x = x) is NOT true in L = M3. And that's by the definition of being true, being false in a model: if a predicate is an empty set then no predicate-formula is true and if it's not true then by definition of model it's false. In such L = M3, all predicates, explicitly stipulated in the language, or by FOL=, or implicitly definable such as '~=', are empty. So, no formula is true in it. It's now your turn (as well as WH's) to answer my question: if you think my answer is incorrect, then where and why (in some technical details)?
From: William Hughes on 22 May 2010 19:18 On May 22, 7:37 pm, Nam Nguyen <namducngu...(a)shaw.ca> wrote: >there's no formula is true in M3 Nonsense The formula (not(exits x) or blue(x)) is true in M3 and provable in T3. - William Hughes as a (false) model of T3. If he meant something > other than M3 then of course his question wasn't, as you said, "clear". > > > > > Now you have a go at it. Let L be the trivial model, i.e., the model > > with empty support. Is (Ax)~(x = x) true in L? > > Since, as above, "no formula is true" then (Ax)~(x = x) is NOT true in > L = M3. And that's by the definition of being true, being false in a > model: if a predicate is an empty set then no predicate-formula is true > and if it's not true then by definition of model it's false. In such > L = M3, all predicates, explicitly stipulated in the language, or by > FOL=, or implicitly definable such as '~=', are empty. > > So, no formula is true in it. > > It's now your turn (as well as WH's) to answer my question: if you think > my answer is incorrect, then where and why (in some technical details)?
From: Nam Nguyen on 22 May 2010 20:41 William Hughes wrote: > On May 22, 7:37 pm, Nam Nguyen <namducngu...(a)shaw.ca> wrote: > >> there's no formula is true in M3 > Nonsense The formula > > (not(exits x) or blue(x)) > > is true in M3 and provable in T3. Did I already say "by definition of model it's false" or similar more than one time? How could any formula be true in a model in which its universe U and any n-ary predicate are empty? _By definition of model_ do you not agree the formula P(x1, x2, ..., xn) would be false in an empty predicate P? Btw, how would you express "not(exits x)" in L(T3)? >> >> It's now your turn (as well as WH's) to answer my question: if you think >> my answer is incorrect, then where and why (in some technical details)? >
From: William Hughes on 22 May 2010 21:32 On May 22, 9:41 pm, Nam Nguyen <namducngu...(a)shaw.ca> wrote: > William Hughes wrote: > > On May 22, 7:37 pm, Nam Nguyen <namducngu...(a)shaw.ca> wrote: > > >> there's no formula is true in M3 > > Nonsense The formula > > > (not(exits x) or blue(x)) > > > is true in M3 and provable in T3. > > Did I already say "by definition of model it's false" Many times. You are under the delusion that by saying something many times you can change it from incorrect to correct. > or similar more than one time? How could any formula > be true in a model in which its universe U and any > n-ary predicate are empty? Note that by FOL any statement about the property of the elements of an empty set is true. So use the quantifier For All. Eg. For All x:(blue(x)) - William Hughes
From: Jesse F. Hughes on 22 May 2010 21:43
Nam Nguyen <namducnguyen(a)shaw.ca> writes: > It's now your turn (as well as WH's) to answer my question: if you > think my answer is incorrect, then where and why (in some technical > details)? Not my turn. I don't bother with substantive discussions with you. It is obvious that you don't really understand logic at all and these discussions never seem to have a positive effect on your ignorance. I just wanted to see if you'd be prodded to actually answer William's question. I'll leave the replies to him. -- "Is that possible? Could it be that easy? No way. [...] There must be a mistake. Right? "But I am the top mathematician in the world." -- James S. Harris |