An Ultrafinite Set Theory
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From: MoeBlee on 1 Mar 2010 22:02 On Mar 1, 5:24 pm, "Jesse F. Hughes" <je...(a)phiwumbda.org> wrote: > MoeBlee <jazzm...(a)hotmail.com> writes: > >> 6) Axiom of well ordering: The urelements are well ordered. > > > Assuming the ordinary definiton of 'well ordered', I guess. > > >> 7) Axiom of finiteness: There is a largest and smallest urelement. > > > WHAT 'large' and 'small'? According to WHAT relation? > > I'm with you on most of your criticisms, but I don't get this one. > > Clearly, the relation mentioned in (7) is the well-ordering mentioned > in (6). Perhaps I misunderstand, but I take it that his well ordering axiom says that the set of urelements (by the way, where's the proof that there is such a set? Maybe I skimmmed past it?) has a well ordering. That doesn't specify any PARTICULAR well ordering; it doesn't specify any particular relation that well orders the set of urelements. Rather, it just says that there exists some well ordering, perhaps many well orderings of the set of urelements. MoeBlee
From: MoeBlee on 1 Mar 2010 22:15 On Mar 1, 6:38 pm, Transfer Principle <lwal...(a)lausd.net> wrote: > On Mar 1, 1:48 pm, MoeBlee <jazzm...(a)hotmail.com> wrote: > > > On Feb 28, 6:34 pm, RussellE <reaste...(a)gmail.com> wrote: > > > 5) Axiom of complement: If A is a set there exists a set of urelements > > > not in A. > > Okay, you're [sic] own axiom. > > > 6) Axiom of well ordering: The urelements are well ordered. > > Assuming the ordinary definiton of 'well ordered', I guess. > > > 7) Axiom of finiteness: There is a largest and smallest urelement. > > WHAT 'large' and 'small'? According to WHAT relation? > > What is the purpose of your theory? Do you think it makes ordinary set > > theory otiose? If you think that, then please show how to derive > > ordinary mathematics for the sciences from your axioms. > > Again with "the sciences." It's like saying "Again with the numbers" or "again with the theorems". The question of mathematics for the sciences is a central question of this subject matter. So I mention it in certain contexts. So what? Sue me. > as > long as we believe that there are only finitely many particles > in the universe, and space and time can be quantized (using > Planck units, for example), then an ultrafinitist theory > should be sufficient for math for the sciences. Great, then let's see the derivation of it from RussellE's axioms (assuming he ever cleans up his work to present actual primitives and axioms). > If there exist a > fixed finite number (say n) urelements, then we should be able > to prove the existence of 2^n sets and 2^2^n classes yet > without need of any Separation Schema. (Certainly only finitely > many instances of such a schema should be required.) > > I'm fully aware of the possibility that MoeBlee might call me a > "liar" for saying this, but I don't care. Why should I call you a 'liar' for it? What I called you a liar for were the actual plain LIES you wrote about my postings. > This post, once again, > reveals the disdain of the standard anti-"cranks" for any > ultrafinitist theory Now you ARE lying again. I didn't express any disdain for ultrafinitism. I just asked how RussellE proposes to prove a certain amount of mathematics for the sciences IF he even proposes to do that. Stop lying about my words again. > standard theorists have never asked Y-V to "derive > ordinary mathematics for the sciences" using only the naturals > that he considers to exist (i.e., naturals n such that he will > answer "yes" to "Is n a natural number?"). Sure I might very well ask. I just haven't opined about Yessenin- Volpin. But if I were to do so, and if the context came up, I might very well ask. > Notice that RE mentions Y-V in the OP of this thread. I now > consider Y-V to be one of those figures, like Abraham Robinson, > whom both "cranks" and anti-"cranks" claim as being a member of > their side of the debate. So the standard theorists will defend > Y-V and Robinson as being non-"cranks," Since you've been talking about ME, let it be known that I've never said whether Yessenin-Volpin is a crank or not a crank or a pointillist or not a pointillist or a member of the Tea Party or not. > yet a Usenet poster who > espouses ideas similar or identical to either mathematician > regarding infinity is called a "crank." And the "cranks" will > appeal to Y-V and Robinson when arguing against Infinity with > the standard anti-"cranks." Just please stop lying about what I have and have not said. MoeBlee
From: William Elliot on 2 Mar 2010 00:57 On Mon, 1 Mar 2010, Patricia Shanahan wrote: > RussellE wrote: >> 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. > In that case, I suggest you pick a different term, to avoid confusing > yourself and others. > > I suggest "Easterly numbers" as a placeholder. Similarly, you could use > "Easterly arithmetic" for the corresponding system of arithmetic > definitions and theorems. No. It has nothing to do with any Asian cultures, not even Zen. Send the baby back home with it's father who can christen it as natural computer numbers in honor of his family name.
From: RussellE on 2 Mar 2010 01:48 On Mar 1, 1:48 pm, MoeBlee <jazzm...(a)hotmail.com> wrote: > On Feb 28, 6:34 pm, RussellE <reaste...(a)gmail.com> wrote: > > > Simpler is better. Here is a simple ultrafinite set theory (UST). > > > Primitives: > > > Urelement - an element of a set. A set or proper class can not be an > > urlelement. > > Set - a collection of urelements. > > Proper Class - a collection of sets. > > If they're primitives, then what is the part following the dash > symbol? > Are those definitions or axioms or combination above? Are the > primitives 'collection' and 'element'? Or what? OK. The primitives are element and collection. urelement - Only objects defined to be urelements can be elements of a set set - A collection of elements. proper class - a collection of sets. > PLEASE look up how primitives, defintitions, and axioms work! > > > 1) Axiom of extensionality: Two sets are equal (are the same set) if > > they have the same elements. > > > 2) Axiom of singletons: If x is an urelement there exists a set, {x}, > > with x as its only element. > > > 3) Axiom of union: If A and B are sets there exists a set with the > > elements of both A and B. > > > 4) Axiom of intersection: If A and B are sets there exists a set with > > the elements common to both A and B. > > Okay, all Z set theory so far. OK > > 5) Axiom of complement: If A is a set there exists a set of urelements > > not in A. > > Okay, you're own axiom. I am not sure I need this axiom. > > 6) Axiom of well ordering: The urelements are well ordered. > > Assuming the ordinary definiton of 'well ordered', I guess. You got me. I don't define well ordering. I can't define well ordering the way ZFC does. My theory doesn't have sets of ordered pairs. I could define proper classes as ordered pairs. Any suggestions for a well ordering axiom would be welcome. > > 7) Axiom of finiteness: There is a largest and smallest urelement. > > WHAT 'large' and 'small'? According to WHAT relation? Again, this is not a great axiom. A better finitenes axiom would be: 7) Axiom of finitenes: The set U = {u_0, u_1, ..., u_k} exists. u is an element of U implies u is an urelement. > What is the purpose of your theory? I want to show it is possible to have a consistent, finite set theory. > Do you think it makes ordinary set > theory otiose? No. Why would you think that? Are anti-foundational set theories otiose? > If you think that, then please show how to derive > ordinary mathematics for the sciences from your axioms. What do you mean by "ordinary mathematics for the sciences"? Can you derive E=MC^2 from ZFC? If you mean Peano arithematic, my theory can't do that. I can derive part of PA. I can show there is a set of "small" natural numbers for which addition is completely defined. > > I probably don't need the axiom of complement. > > It can be derived from the other axioms. > > I included the axiom of intersection because I don't really > > understand > > how set theories like ZFC define intersection. > > Why don't you just READ how it's done? > > > Maybe intersection > > can also be derived from the other axioms. > > Yes. You can read about it in virtually any textbook on set theory. ZFC doesn't have an axiom of intersection. I assume intersection can be derived from the axiom schema of specification. My theory doesn't have an axiom schema of specification. > > I don't need an axiom schema of specification. > > The singleton axiom and union axiom are enough to create any set. > > Depends on what you think are "enough" sets. Do we really need a continuum number of sets to do "science"? Russell - Zeno was right. Motion is impossible.
From: Virgil on 2 Mar 2010 02:29
In article <bbc61265-cb81-487d-95f9-21d2187ada9f(a)k36g2000prb.googlegroups.com>, RussellE <reasterly(a)gmail.com> wrote: > > Assuming the ordinary definiton of 'well ordered', I guess. > > You got me. I don't define well ordering. With your "axioms" you can't define any sort of ordering, so we see that your set theory is disorderly. |