An Ultrafinite Set Theory
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From: RussellE on 1 Mar 2010 03:49 On Feb 28, 9:47 pm, Patricia Shanahan <p...(a)acm.org> wrote: > RussellE wrote: > > ...> The simplest way to represent natural numbers in this > > system is to assume each natural number is an urelement. > > This gives us the finite set of all natural numbers. > > ... > > How do you define the term "natural numbers"? I define a natural number to be an urelement. The set of all natural numbers is the set of all urelements. This isn't the same definition as Peano's axoims or ZFC. My natural numbers serve the same purpose as natural numbers in these other systems. Natural numbers have an order. I have a well ordering axiom. Russell - 2 many 2 count
From: Virgil on 1 Mar 2010 04:15 In article <45b1afa9-21f1-45a9-8cb8-48679accc446(a)k18g2000prf.googlegroups.com>, RussellE <reasterly(a)gmail.com> wrote: > On Feb 28, 9:47�pm, Patricia Shanahan <p...(a)acm.org> wrote: > > RussellE wrote: > > > > ...> The simplest way to represent natural numbers in this > > > system is to assume each natural number is an urelement. > > > This gives us the finite set of all natural numbers. > > > > ... > > > > How do you define the term "natural numbers"? > > > I define a natural number to be an urelement. And are all your urelements natural numbers? If not, you so called definition is unusable. > The set of all natural numbers is the set of all urelements. > This isn't the same definition as Peano's axoims or ZFC. > My natural numbers serve the same purpose as natural numbers > in these other systems. Natural numbers have an order. > I have a well ordering axiom. But you do not have any arithmetic.
From: William Elliot on 1 Mar 2010 07:09 On Mon, 1 Mar 2010, Virgil wrote: >>> RussellE wrote: >>> >>> ...> The simplest way to represent natural numbers in this >>>> system is to assume each natural number is an urelement. >>>> This gives us the finite set of all natural numbers. >>> >>> How do you define the term "natural numbers"? >> >> I define a natural number to be an urelement. > > And are all your urelements natural numbers? If not, you so called > definition is unusable. > Even if all urelements are natural numbers, to call a finite set of urelements the natural numbers is a misnomer. He needs another term such as natural computer numbers, bounded natural numbers, initial segment of natural numbers, minimus numbers. >> The set of all natural numbers is the set of all urelements. >> This isn't the same definition as Peano's axioms or ZFC. >> My natural numbers serve the same purpose as natural numbers >> in these other systems. Natural numbers have an order. >> I have a well ordering axiom. > The natural numbers are also well ordered. > But you do not have any arithmetic. > It will be worse than the arithmetic of the extended naturals.
From: Patricia Shanahan on 1 Mar 2010 07:14 RussellE wrote: > On Feb 28, 9:47 pm, Patricia Shanahan <p...(a)acm.org> wrote: >> RussellE wrote: >> >> ...> The simplest way to represent natural numbers in this >>> system is to assume each natural number is an urelement. >>> This gives us the finite set of all natural numbers. >> ... >> >> How do you define the term "natural numbers"? > > > I define a natural number to be an urelement. > The set of all natural numbers is the set of all urelements. > This isn't the same definition as Peano's axoims or ZFC. > My natural numbers serve the same purpose as natural numbers > in these other systems. Natural numbers have an order. > I have a well ordering axiom. In that case, I suggest you pick a different term, to avoid confusing yourself and others. If you had a combination of zero element and successor operation that satisfied the Peano axioms, you could use the normal definitions of natural number arithmetic and any theorem about natural numbers that has been proved from the Peano axioms. That is how ZFC gets its arithmetic, using the empty set as zero and the set containing only x as the successor of x. Obviously, you cannot do that given the fact that your numbers do not satisfy the Peano axioms. If you go on using the term "natural numbers" you may fool yourself into assuming that something is already defined or proved because it has been defined or proved for systems satisfying the Peano axioms. If you want arithmetic in your system, you will need to go back to the drawing board to define it, and prove each theorem you want using only your definitions, axioms, and any theorems you have already proved. I suggest "Easterly numbers" as a placeholder. Similarly, you could use "Easterly arithmetic" for the corresponding system of arithmetic definitions and theorems. As you develop your definitions you may be able to prove your numbers are isomorphic to some previously defined system, and adopt the name of that system for them. Patricia
From: Brian on 1 Mar 2010 09:59
> You are virtually completely uninformed about how 'set' may be defined > in certain ordinary set theories (which does not contradict that also > we may take the basic intuitive notion of set to be undefined). > MoeBlee I have not seen this before. How is the word set defined in "certain ordinary set theories"? |