The End of an Era - That of The Natural Numbers
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From: Jesse F. Hughes on 22 May 2010 11:38 Nam Nguyen <namducnguyen(a)shaw.ca> writes: > William Hughes wrote: >> On May 22, 2:02 am, Nam Nguyen <namducngu...(a)shaw.ca> wrote: >>> William Hughes wrote: >>>> On May 22, 1:47 am, Nam Nguyen <namducngu...(a)shaw.ca> wrote: >>>>> William Hughes wrote: >>>>>> Next question. Let L be the trivial model. >>>>>> Is >>>>>> "for all x, x=/=x" >>>>>> true in L. >>>> <snip evasion> >>>> Try again. Begin your answer with Yes or No. >> <snip evasion> >> >> Try again > > 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? -- Jesse F. Hughes "I thought it relevant to inform that I notified the FBI a couple of months ago about some of the math issues I've brought up here." -- James S. Harris gives Special Agent Fox a new assignment.
From: Nam Nguyen on 22 May 2010 12:37 Nam Nguyen wrote: > Aatu Koskensilta wrote: >> Nam Nguyen <namducnguyen(a)shaw.ca> writes: >> >>> I'm sure you've heard of "equivalent formulas" in a theory. >> >> What of them? >> > > Aren't 2 equivalent formulas either both true or both false? It's not that I don't see where your side is coming from. I do. But given the debate about whether or not x=x is true in all levels of FOL edifice then the belief in your side that the formulas is _always_ ("absolutely") true is incorrect, as demonstrated with M3 as the false model for T3. As mentioned earlier, a formula's being true/false is nothing more than a meta mapping between the formula into a binary value. As such you're still _free_ to choose any mapping available to your heart content, and there are many available even with constrains, such as for consistent theories the mapping be compatible with a _true_ model (one in which there's a non-empty n-ary predicate). In saying x=x is always true in FOL=, what your school is saying is that you have chosen a mapping base on, say, "pure logic" as exemplified by truth tables. In such mapping all FOL logical (tautological) theorems are mapped to "true", and their negations to "false". But within FOL edifice, model theoretically truths for a T are definable as well and in such context, there are cases where x=x would be false, by properties of permitted by the definition. That mean there isn't one-size-fit-all mapping. So, it's NOT true x=x is always true, as poster (Marshall) initially claimed. *** There's another matter related to your school's belief that we'd know the naturals as the standard model of arithmetic, and to the question of validity of GIT, using the knowledge of the naturals. I had: >> In his definition of structure (i.e. model where truth and >> falsehood would be defined), pg. 18, Shoenfield stipulated >> 3 conditions something to be called a structure M must consist >> of, the last one of which says: >> >> iii) For each n-ary predicate symbol p of L other than =, an n-ary >> predicate pM in |M|. >> >> (|M| means the universe of M) The implication of condition iii is that despite all theories T's must extend _the_ logical theory T0 = {x=x}, a [true] model of T0 will NOT exist until we have in our mind a [true] model of a consistent T. Iow, there's NO such thing as the standard model of T0: simply because you can NOT construct a non-empty U without having a _non-logical axiom_ stipulating _how many_ individual elements U would have (even just 1 element)! What this means is there's some sort of a commandment in FOL reasoning, something like: "Thou Shall NOT mixed up what's Logical and what's NON-Logical together and still expect Consistency in Reasoning, in FOL". But by virtually all accounts, the naturals collectively is just one out of "uncountably" many NON-logical concepts that could possibly exist in the realm of concepts. So to peg logical reasoning (such as about syntactical consistency of PA) against the non-logical concept of the naturals, is to invite weirdness if not outright inconsistency at least in meta reasoning. For example, if we re-intuit the concept of the naturals as that in which G(T) (per GIT) be false, then we'd arrive at the meta theorem GIT': For any consistent T as strong as arithmetic of the naturals [i.e. the new "naturals"], there's a formula G(T) which is false and not provable! Can we re-intuit the concept of the naturals? Sure we can, with some conditions of course: C1 - The naturals isn't a tautological or contradictory concept. (The Symmetry Principle). C2 - All theorems of Q are interpreted as true in the concept of the naturals. (Of course, if Q isn't syntactically consistent, C1 wouldn't be satisfied). Now by the Principle of Symmetry, we could name a concept or its negation as "the naturals" as long as both C1 and C2 are met.
From: Nam Nguyen on 22 May 2010 12:41 Jesse F. Hughes wrote: > Nam Nguyen <namducnguyen(a)shaw.ca> writes: > >> William Hughes wrote: >>> On May 22, 2:02 am, Nam Nguyen <namducngu...(a)shaw.ca> wrote: >>>> William Hughes wrote: >>>>> On May 22, 1:47 am, Nam Nguyen <namducngu...(a)shaw.ca> wrote: >>>>>> William Hughes wrote: >>>>>>> Next question. Let L be the trivial model. >>>>>>> Is >>>>>>> "for all x, x=/=x" >>>>>>> true in L. >>>>> <snip evasion> >>>>> Try again. Begin your answer with Yes or No. >>> <snip evasion> >>> >>> Try again >> 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!
From: Nam Nguyen on 22 May 2010 13:25 Nam Nguyen wrote: > Jesse F. Hughes wrote: >> Nam Nguyen <namducnguyen(a)shaw.ca> writes: >> >>> William Hughes wrote: >>>> On May 22, 2:02 am, Nam Nguyen <namducngu...(a)shaw.ca> wrote: >>>>> William Hughes wrote: >>>>>> On May 22, 1:47 am, Nam Nguyen <namducngu...(a)shaw.ca> wrote: >>>>>>> William Hughes wrote: >>>>>>>> Next question. Let L be the trivial model. >>>>>>>> Is >>>>>>>> "for all x, x=/=x" >>>>>>>> true in L. >>>>>> <snip evasion> >>>>>> Try again. Begin your answer with Yes or No. >>>> <snip evasion> >>>> >>>> Try again >>> 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! Fyi, the following is an example of a genuine yes/no question: Is {Axy[x=y]} |- Axy[x=y] true? WH's question: Is "for all x, x=/=x" true in L would be a yes/no question, but only if he acknowledged L be just my M3 for an inconsistent T (and in which case my answer was _already_ a no. (Iow, he didn't even notice my answer!) He never confirmed L = M3 for an inconsistent T. So it's simply pointless to answer a question that only appears as a yes/not question because crucial details are missing!
From: Jesse F. Hughes on 22 May 2010 15:08
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. If you mean is there a theory in which that formula is provable the answer is yes. Perfectly clear response, depending just on what your question means. 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? -- "Eventually the truth will come out, and you know what I'll do then? Probably go to the beach. I'll also hang out in some bars. Yup, I'll definitely hang out in some bars, preferably near a beach." -- JSH on the rewards of winning a mathematical revolution |