I think that resorting to calling the biggest natural number BIGGEST, rather than specifying exactly what that number is, is a tell-tale sign that the ultrafinitist knows that any specification for BIGGEST will immediately reveal that it is not the biggest because one could always add one more.

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Quentin Anciaux wrote: > You have to explain why the exception is needed in the first place... > > The rule is true until the rule is not true anymore, ok but you have > to explain for what sufficiently large N the successor function would > yield next 0 and why or to add that N and that exception to the > successor function as axiom, if not you can't avoid N+1. But torgny > doesn't evacuate N+1, merely it allows his set to grows undefinitelly > as when he has defined BIGGEST, he still argues BIGGEST+1 makes sense > , is a natural number but not part of the set of natural number, this > is non-sense, assuming your special successor rule BIGGEST+1 simply > does not exists at all. > > I can understand this overflow successor function for a finite data > type or a real machine registe but not for N. The successor function > is simple, if you want it to have an exception at biggest you should > justify it. > > Regards, > Quentin > > 2009/6/9 Brent Meeker <meeke...@dslextreme.com>: > >> Quentin Anciaux wrote: >> >>> 2009/6/9 Torgny Tholerus <tor...@dsv.su.se>: >>> >>> >>>> Jesse Mazer skrev: >>>> >>>> >>>>>> Date: Sat, 6 Jun 2009 21:17:03 +0200 >>>>>> From: tor...@dsv.su.se >>>>>> To: everything-list@googlegroups.com >>>>>> Subject: Re: The seven step-Mathematical preliminaries >>>>>> >>>>>> My philosophical argument is about the mening of the word "all". To be >>>>>> able to use that word, you must associate it with a value set. >>>>>> >>>>>> >>>>> What's a "value set"? And why do you say we "must" associate it in >>>>> this way? Do you have a philosophical argument for this "must", or is >>>>> it just an edict that reflects your personal aesthetic preferences? >>>>> >>>>> >>>>> >>>>>> Mostly that set is "all objects in the universe", and if you stay >>>>>> >>>>>> >>>>> inside the >>>>> >>>>> >>>>>> universe, there is no problems. >>>>>> >>>>>> >>>>> *I* certainly don't define numbers in terms of any specific mapping >>>>> between numbers and objects in the universe, it seems like a rather >>>>> strange notion--shall we have arguments over whether the number 113485 >>>>> should be associated with this specific shoelace or this specific >>>>> kangaroo? >>>>> >>>>> >>>> When I talk about "universe" here, I do not mean our physical universe. >>>> What I mean is something that can be called "everything". It includes >>>> all objects in our physical universe, as well as all symbols and all >>>> words and all numbers and all sets and all other universes. It includes >>>> everything you can use the word "all" about. >>>> >>>> >>> It includes all set, but no all set as it N includes all natural >>> number but not all natural number... excuse-me but this is non-sense. >>> Either N exists and has an infinite number of member and is >>> incompatible with an ultrafinitist view or N does not exists because >>> obviously N cannot be defined in an ultra-finitist way, >>> >> That's not obvious to me. You're assuming that N exists apart from >> whatever definition of it is given and that it is the infinite set >> described by the Peano axioms or equivalent. But that's begging the >> question of whether a finite set of numbers that we would call "natural >> numbers" can be defined. To avoid begging the question we need some >> definition of "natural" that doesn't a priori assume the set is finite >> or infinite; something like, "A set of numbers adequate to do all >> arithmetic we'll ever need" (unfortunately not very definite). The >> problem is the successor axiom, if we modify it to S{n}=n+1 for n e N >> except for the case n=N where S{N}=0 and choose sufficiently large N it >> might satisfy the "natural" criteria. >> >> Brent >> >> >> >>> any set that >>> contains a finite number of natural number (and still you haven't >>> defined what it is in an ultrafinitist way) are not the set N. >>> >>> Also any operation involving two number (addition/multiplication) can >>> yield as result a number which has the same property as the departing >>> number (being a natural number) but is not natural number... Also >>> induction and inference cannot work in such a context. >>> >>> >>> >>>> For you to be able to use the word "all", you must define the "domain" >>>> of that word. If you do not define the domain, then it will be >>>> impossible for me and all other humans to understand what you are >>>> talking about. >>>> >>>> >>> Well you are the first and only human I know who don't understand >>> "all" as everybody else does. >>> >>> Quentin Anciaux >>> >>> >>> >>>> -- >>>> Torgny Tholerus >>>> >>>> >>>> >>> >>> >>> >> > > > > --~--~---------~--~----~------------~-------~--~----~ 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 -~----------~----~----~----~------~----~------~--~---