An Ultrafinite Set Theory
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From: Patricia Shanahan on 1 Mar 2010 19:27 RussellE wrote: > On Mar 1, 4:14 am, Patricia Shanahan <p...(a)acm.org> wrote: >> RussellE wrote: > >>>> 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. > > Nope. I am calling my numbers the "natural numbers". > Mostly to annoy Moeblee. .... I was treating clear communication as desirable, and annoying others as a bad thing. Given your priorities, it probably does not matter what you call your numbers. Patricia
From: Transfer Principle on 1 Mar 2010 19:38 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." We already know how anti-"cranks" doubt that so-called "crank" theories can provide enough math for the sciences, while the "cranks" believe that standard theory doesn't provide the right math for the sciences. (Of course, "sciences" as used by both "cranks" and anti-"cranks" usually refers to _physics_.) So the question we ask is, can one derive enough math for physics from an ultrafinitist theory like RE's? As mentioned numerous times in other threads (especially AP threads), 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. If we take the AP upper bound of 10^500, then we can have a model of RE's theory with 11 urelements. Then this should give us 2^11=2048 sets and 2^2048 classes. We note that since 2^2048 > 10^616, so we've covered. Since the largest of the RSA numbers is around 2^2048, this theory should be sufficient for math for the science of cryptography, at least. If the standard anti-"cranks" feel that science requires the existence of numbers larger than this, then let N be what they feel is the largest natural number required by science, then give a model of RE's theory in which there exist exactly ceil(lg(lg(N))) urelements. (Of course, lg here is the base-2 logarithm function.) The standard theorists are hard-pressed to argue that N exceeds 10^500, especially 10^10^500. But even if they argue that science requires numbers on the order of 10^^10 (ten tetrated to the tenth), Moser's Number, or even Graham's Number, these numbers are still _finite_, and so we could still use these limits and still have a theory that axiomatizes physics without need of an Axiom of Infinity. > > 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. Obviously, RE thinks that finitely many sets are "enough." I think that one should be able to have a theory like RE's that is finitely axiomatized without any schemata -- since the desired model of such a theory is _finite_. 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. This post, once again, reveals the disdain of the standard anti-"cranks" for any ultrafinitist theory -- Yessenin-Volpin notwithstanding. Of course, 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?"). 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," 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."
From: RussellE on 1 Mar 2010 19:40 On Mar 1, 1:15 am, Virgil <Vir...(a)home.esc> wrote: > In article > > 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. If you mean Peano arithmetic can't be derived in my UST, you are correct. This is a good thing. Since UST's can't derive Peano arithmetic, they are not subject to Godel's Incompleteness theorem. An UST might be able to prove its own consistency. Assume this is the set of all urelements: {0,1,2,3} + 0 1 2 3 0 0 1 2 3 1 1 2 3 x 2 2 3 x x 3 3 x x x There are 16 possible ordered pairs of natural numbers Addition can be defined for 10 of them, Addition is undefined for the other six pairs in this UST. The purpose of a set theory is to examine the impact of certain assumptions. Peano arithematic assumes addition can be defined for any ordered pair of natural numbers. My UST doesn't assume this. I can completely define addition for "small" natural numbers. Addition is completely defined for the numbers 0 and 1 in the UST above. I don't have to assume arithematic is defined for all natural numbers because Peano arithematic makes this assumption. Russell - 2 many 2 count
From: Aatu Koskensilta on 1 Mar 2010 20:00 Transfer Principle <lwalke3(a)lausd.net> writes: > I like RE's idea of having sets have only urelements as elements and > being elements only of the proper classes. You'll find learning about third-order arithmetic a blast, then. -- Aatu Koskensilta (aatu.koskensilta(a)uta.fi) "Wovon man nicht sprechan kann, dar�ber muss man schweigen" - Ludwig Wittgenstein, Tractatus Logico-Philosophicus
From: RussellE on 1 Mar 2010 20:29
On Mar 1, 12:14 am, William Elliot <ma...(a)rdrop.remove.com> wrote: > On Sun, 28 Feb 2010, RussellE wrote: > > On Feb 28, 9:11�pm, William Elliot <ma...(a)rdrop.remove.com> wrote: > >> On Sun, 28 Feb 2010, RussellE wrote: > >>> 7) Axiom of finiteness: There is a largest and smallest urelement. > > >> That doesn't make U finite. The ordinal number omega_0 + 1 > >> has a smallest and largest element and isn't finite. > > > Yes, I know. I am still having problems coming up with > > an axiom of finiteness. > > You could include in the language, k constant symbols u1,.. uk, > define U = { u1,.. uk } and state that if u is an urelement, > then u in U. This seems to be the simplest solution. It would be nice to have something more "elegant". > > I could use my bijection proof. The axiom says if A and B > > are sets and have a bijection there exists a bijection > > between A-B and B-A. > > > This would eliminate sets having a bijection with a proper subset. > > But, I would have to define bijection. > > >>> The singleton axiom and union axiom are enough to create any set. > >> No. Even assuming U is finite, you can't construct an empty set. > > I think I can derive that from intersection. > > You can't if there's only one urelement. Yes. I am not sure this is a problem. I could add an empty set axiom. I want to minimize the number of axioms. If I remember correctly, if an axiomatic set theory is consistent, it is still consistent when we negate an axiom. I am not sure how I can deal with an anti-empty set axiom. I use to think anti-foundational set theories were strange. Lately, I have been considering anti-union theories and anti-comprehension theories. There exists two sets with the same elements that are not equal. > >>> 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. > > >> No it doesn't. It shows that the natural > >> numbers cannot be represented in UST. > > > Which natural numbers? > > Most of them. > > > This certainly isn't the same set of natural numbers > > defined by ZFC. ZFC defines natural numbers > > as the intersection of all inductive sets with > > the empty set as a member. > > It also excludes the positive integers of Piano's axiom. Of course. It's not an UST if it doesn't exclude these. > Your natural numbers are unnatural. If it doesn't smell > like a dog nor bark or look like a dog, then it isn't a dog. Are you saying my UST is "counter-intuitive"? I find it amusing that a finite set theory is "counter-intuitive". We all have pre-conceived intuitions about "natual numbers". I don't think natural numbers can grow without limit. I want my set theory to formalized my pre-conceived notion of natural numbers. > >>> Many people have told me all known UST's are inconsistent. > >>> Obviously, no UST will be consistent with axioms from other > >>> set theories. No UST will be consistent with the axiom > >>> "if n is a natural number then n+1 is a natural number". > >>> My UST doesn't have this axiom. > > >> Of course it doesn't. You haven't even defined incrementation. > > > I have a well ordering axiom. What else do I need? > > A definition of n + 1 as the successor urelement. How does ZFC define the successor function? Is there a "successor" axiom? "n+1" is meaningless for certain n in my UST. > >>> Some natural numbers are just too big to be added together. > > >> Most natural numbers are too big for computers to comprehend. > > > Actually, this is true. At least, it is true for the natural > > numbers defined by ZFC. > > Some numbers are too small for a computer to comprehend > and others are too precise for a computer to comprehend. > > In fact it's worse than computers not being able to comprehend most > numbers. All they can ever hope to do is to comprehend almost no numbers. Some people think the universe is a computer. If so, there are numbers too big and too small for the universe to comprehend. You can almost derive the uncertainty principle from this. Position and momentum can't be computed beyond a certain precision. It could be worse. If physicists come up with a set theory it will be something like "there is a probability 1=1, a probability 1=2, ..." Russell - 2 many 2 count |