Jesse Mazer wrote: > > > > Date: Tue, 9 Jun 2009 15:22:10 -0700 > > From: meeke...@dslextreme.com > > To: firstname.lastname@example.org > > Subject: Re: The seven step-Mathematical preliminaries > > > > > > Jesse Mazer wrote: > >> > >> > >>> Date: Tue, 9 Jun 2009 12:54:16 -0700 > >>> From: meeke...@dslextreme.com > >>> To: email@example.com > >>> Subject: Re: The seven step-Mathematical preliminaries > >>> > >> > >>> You don't justify definitions. How would you justify Peano's axioms > >> as being > >>> the "right" ones? You are just confirming my point that you are > >> begging the > >>> question by assuming there is a set called "the natural numbers" > >> that exists > >>> independently of it's definition and it satisfies Peano's axioms. > >> > >> What do you mean by "exists" in this context? What would it mean to > >> have a well-defined, non-contradictory definition of some mathematical > >> objects, and yet for those mathematical objects not to "exist"? > > > > A good question. But if one talks about some mathematical object, like > > the natural numbers, having properties that are unprovable from their > > defining set of axioms then it seems that one has assumed some kind of > > existence apart from the particular definition. > > Isn't this based on the idea that there should be an objective truth > about every well-formed proposition about the natural numbers even if > the Peano axioms cannot decide the truth about all propositions? I > think that the statements that cannot be proved are disproved would > all be ones of the type "for all numbers with property X, Y is true" > or "there exists a number (or some finite group of numbers) with > property X" (i.e. propositions using either the 'for all' or 'there > exists' universal quantifiers in logic, with variables representing > specific numbers or groups of numbers). So to believe these statements > are objectively true basically means there would be a unique way to > "extend" our judgment of the truth-values of propositions from the > judgments already given by the Peano axioms, in such a way that if we > could flip through all the infinite propositions judged true by the > Peano axioms, we would *not* find an example of a proposition like > "for this specific number N with property X, Y is false" (which would > disprove the 'for all' proposition above), and likewise we would not > find that for every possible number (or group of numbers) N, the Peano > axioms proved a proposition like "number N does not have property X" > (which would disprove the 'there exists' proposition above). > > We can't actual flip through an infinite number of propositions in a > finite time of course, but if we had a "hypercomputer" that could do > so (which is equivalent to the notion of a hypercomputer that can > decide in finite time if any given Turing program halts or not), then > I think we'd have a well-defined notion of how to program it to decide > the truth of every "for all" or "there exists" proposition in a way > that's compatible with the propositions already proved by the Peano > axioms. If I'm right about that, it would lead naturally to the idea > of something like a "unique consistent extension" of the Peano axioms > (not a real technical term, I just made up this phrase, but unless > there's an error in my reasoning I imagine mathematicians have some > analogous notion...maybe Bruno knows?) which assigns truth values to > all the well-formed propositions that are undecidable by the Peano > axioms themselves. So this would be a natural way of understanding the > idea of truths "about the natural numbers" that are not decidable by > the Peano axioms.
I think Godel's imcompleteness theorem already implies that there must be non-unique extensions, (e.g. maybe you can add an axiom either that there are infinitely many pairs of primes differing by two or the negative of that). That would seem to be a reductio against the existence of a hypercomputer that could decide these propositions by inspection. > > Of course even if the notion of a "unique consistent extension" of > certain types of axiomatic systems is well-defined, it would only make > sense for axiomatic systems that are consistent in the first place. I > guess in judging the question of the consistency of the Peano axioms, > we must rely on some sort of ill-defined notion of our "understanding" > of how the axioms should represent true statements about things like > counting discrete objects. For example, we understand that the order > you count a group of discrete objects doesn't affect the total number, > which is a convincing argument for believing that A + B = B + A > regardless of what numbers you choose for A and B. Likewise, we > understand that multiplying A * B can be thought of in terms of a > square array of discrete objects with the horizontal side having A > objects and the vertical side having B objects, and we can see that > just by rotating this you get a square array with B on the horizontal > side and A on the vertical side, so if we believe that just mentally > rotating an array of discrete objects won't change the number in the > array that's a good argument for believing A * B = B * A. So thinking > along these lines, as long as we don't believe that true statements > about counting collections of discrete objects could ever lead to > logical contradictions, we should believe the same for the Peano axioms. > > Jesse So we believe in the consistency of Peano's arithmetic because we have a physical model. Brent --~--~---------~--~----~------------~-------~--~----~ You received this message because you are subscribed to the Google Groups "Everything List" group. To post to this group, send email to firstname.lastname@example.org To unsubscribe from this group, send email to everything-list+unsubscr...@googlegroups.com For more options, visit this group at http://groups.google.com/group/everything-list?hl=en -~----------~----~----~----~------~----~------~--~---