2009/6/10 Brent Meeker <meeke...@dslextreme.com>: > > Jesse Mazer wrote: >> >> >> > Date: Tue, 9 Jun 2009 12:54:16 -0700 >> > From: meeke...@dslextreme.com >> > To: everything-list@googlegroups.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. Everybody believes > arithmetic, per Peano's axioms, is consistent, but we know that can't be > proved from Peano's axioms. So it seems we are assigning (or betting > on, as Bruno might say) more existence than is implied by the definition. > > When Quentin insists that Peano's axioms are the right ones for the > natural numbers, he is either just making a statement about language > conventions, or he has an idea of the natural numbers that is > independent of the axioms and is saying the axioms pick out the right > set of natural numbers. > > Brent

No I'm actually saying that peano's axiom define the abstract rules which permits to know if a number is a natural number or not. A number is a natural number if it satisfies peano's axiom... so by definition the set created by the numbers satisfying these rules is the set of all natural numbers. So if you change the rules, you change the set hence the new set(s) created by your new rules (axiom) is(are) not the same set(s) than the one denoted by peano's axioms hence it is not N and can't be by definition. The mathematical object you define with your new rules is not the same. And please note that modulo arithmetic is not the problem here. Torgny is not talking about that, he said BIGGEST+1 is not in the set N, but BIGGEST+1 is a natural number (Question1: What is a natural number ?, Question2: How can a natural number not be in the set of **all** natural numbers ?). With your version with modulo(BIGGEST), BIGGEST+1 is in the previously defined set, it is '0'. And in your version BIGGEST+1 doesn't satisfy that it is strictly bigger than BIGGEST, but in Torgny version it does. Regards, Quentin > > > > -- All those moments will be lost in time, like tears in rain. --~--~---------~--~----~------------~-------~--~----~ You received this message because you are subscribed to the Google Groups "Everything List" group. To post to this group, send email to everything-list@googlegroups.com 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 -~----------~----~----~----~------~----~------~--~---