How Can ZFC/PA do much of Math - it Can't Even Prove PA is Consistent(EASY PROOF)
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From: Nam Nguyen on 5 Jul 2010 13:32 Nam Nguyen wrote: > Nam Nguyen wrote: >> Aatu Koskensilta wrote: >> >>> >>> In Usenet debates, and elsewhere too, people like to make much of and >>> pontificate tediously on who has or does not have the "burden of >>> proof". In reality, such "logical" rules of formal debate have about as >>> much to do with real arguments, persuasion, conversion, as the rules of >>> cricket. Whoever wants to convince someone else of something must of >>> course present arguments, questions, reflections, examples, >>> illustrations that have some real force to the receiving person, >>> regardless of whether they make a "positive claim" according to some >>> rulebook of debating. >> >> But in real life debate or argument , say in a court, people don't >> talk about the name "Godel", inaccessible cardinalities, LST, finfinitely >> many non-logical symbols, to say a few. >> >> We do! > > Honestly, my intention here isn't to argue about the ordinary > mathematics we, > I included, do virtually all the times. > > The issue is when I tried: > > - to understand the difficulty of the arithmetical truth values of formulas > such as GC, cGC, ... In any rate, if we take the conviction we "know" what the truths about the naturals number be, we'd also have to accept that it's impossible to know if PA + cGC, PA + ~cGC is consistent - or not. There's no free lunch in mathematical reasoning, so to speak.
From: Nam Nguyen on 5 Jul 2010 14:12 MoeBlee wrote: > On Jul 3, 3:07 pm, Nam Nguyen <namducngu...(a)shaw.ca> wrote: >> MoeBlee wrote: >>> On Jul 3, 2:39 pm, Nam Nguyen <namducngu...(a)shaw.ca> wrote: >>>> MoeBlee wrote: > >>> I don't know what you find bundled with the word 'confirm', so I >>> prefer to stand by what I posted, which is clear enough, especially as >>> it is an extremely common notion discussed widely. >> Oh very simple. I can _confirm_ by sheer formal proof that the theory >> T = {(x=x) /\ ~(x=x)} is inconsistent. > > There you give an EXAMPLE of what you mean by 'confirm' but not a > definition. OK. Here it is: To confirm a T's (in)consistency is use knowledge about FOL rules of inference to assert T's (in)consistency. I've done so (confirming) the above T's inconsistency. > > But we might not need to get bogged down in that if possibly your > notion of 'confirm' is close enough to the notion of 'finitistic > proof'. Very roughly stated: a finitistic proof is one that uses > merely primitive arithmetic (which can also be viewed as purely > mechanistic pattern matching of finite strings of symbols). I would > think this would at least be subsumed by what you might mean by > "syntactic proof". FOL syntactical proof is well defined, and _does NOT require_ the concept of "primitive arithmetic". > > And, in that regard, I have said all along that there is no finitistic > proof of the consistency of PA. So you've agreed that there's no formal proof for PA's consistency and that if you go by formal proof only then you don't have knowledge of PA's consistency. If you agree on that then I rest my case.
From: Nam Nguyen on 5 Jul 2010 14:14 MoeBlee wrote: > On Jul 3, 3:12 pm, Nam Nguyen <namducngu...(a)shaw.ca> wrote: > >> Sorry, MoeBlee, you were wrong (if not hopeless) about attacking people, >> when representing a portion of somebody else's views! (Note the your >> word "these subjects"!) > > Life Too Short. Then don't complain anything about "suckhole" (your own word) because _you_ yourself created the "suckhole" here!
From: Nam Nguyen on 5 Jul 2010 14:22 MoeBlee wrote: > On Jul 3, 11:21 pm, Nam Nguyen <namducngu...(a)shaw.ca> wrote: > >> If so, why can't >> you and the person communicate with each other the results of formal >> proofs using rules of inference. > > You're mixed up about what I wrote. A crank could say anything, without elaborating his/her viewpoints. > > Moreover, you need to find out more about this basic subject matter. > If you don't know that PRA is a formal system, then indeed you need to > look into such things. A crank could use _mediocre excuses_ to hide the fact fact he/she doesn't what he/she is talking about! > >> It's a curiosity that we've been talking about formal systems such as PA >> and yet in this not-too-short post you've not mentioned inferences by FOL >> rules of inference. Would there be reasons why you've not mentioned them >> in this post? > > PRA is in FOL. You didn't seem to understand a simple English question. I asked: "why you've not mentioned them in this post?" Do you now understand what I asked?
From: Nam Nguyen on 5 Jul 2010 14:36
MoeBlee wrote: > On Jul 5, 7:05 am, Aatu Koskensilta <aatu.koskensi...(a)uta.fi> wrote: >> "Jesse F. Hughes" <je...(a)phiwumbda.org> writes: >> >>> Aatu said PA is consistent, _period_, without any formal proof? >> There seems to be some confusion over my (perfectly standard as always) >> take on these matters. > > Is there anything in my paraphrase several posts ago (in the full > context I gave it) that is inaccurate? (Note: Just for the record, the > particular quote above is not my own. However that quote is to be > understood, for the record, I did not say that you hold there is no > formal proof that PA is consistent, but rather that you hold PA is > consistent on (for lack of better term I can think of right now) even > more basic grounds than formal proof. > >> We can of course formalize this proof in any number of theories -- ACA, >> ZFC, ... -- but this is just an incidental technical observation of no >> immediate interest as far as consistency of PA is concerned. > > And that is what I have highlighted as to your view. The key question here is would the formal system containing "the formal proof that PA is consistent" be _itself_ consistent, according to Aatu? You seemed to know what his views be on this matter, can you answer this question? |