An Ultrafinite Set Theory
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From: MoeBlee on 2 Mar 2010 19:26 On Mar 2, 5:54 pm, RussellE <reaste...(a)gmail.com> wrote: > On Mar 2, 3:33 pm, "Jesse F. Hughes" <je...(a)phiwumbda.org> wrote: > > > RussellE <reaste...(a)gmail.com> writes: > > > I have often been told there are no "consistent" ultrafinite set > > > theories (UST). > > > Really? Have you been told so here on the newsgroup? > > > Can you point me to a single post in which someone said that? > > People have said that in this newsgroup (it might have been me). You "told" yourself then? > Here is what Wikipedia says:http://en.wikipedia.org/wiki/Ultrafinitism I see no claim there that there any ultrafinite set theory must be inconsistent. > but even constructivists generally view the philosophy as unworkably > extreme So, that's not saying that any ultrafinite set theory must be inconsistent. > and > > the constructive logician A. S. Troelstra dismissed it by saying "no > satisfactory development exists at present." That's not saying that any ultrafinite set theory must be inconsistent. > Why are ultrafinite theories considered "unworkable"? Whether they are unworkable or not (work for what purpose?), it seems to me that what the article may be getting at is how difficult it is to come up with axioms for such a theory that also provide us with such results as we wish to have from a foundational theory. > I would think an UST woiuld be similar to theories with universal > sets. I don't see the connection, though I'm not claiming there isn't one. Why don't you first do some systematic, organized study on formal theories, standard set theory, alternative theories, and the philosophy of mathematics? Right now you look like some guy thrashing about in a mental cellophane bag. MoeBlee
From: Jesse F. Hughes on 2 Mar 2010 21:14 RussellE <reasterly(a)gmail.com> writes: > On Mar 2, 3:33 pm, "Jesse F. Hughes" <je...(a)phiwumbda.org> wrote: >> RussellE <reaste...(a)gmail.com> writes: >> > I have often been told there are no "consistent" ultrafinite set >> > theories (UST). >> >> Really? Have you been told so here on the newsgroup? >> >> Can you point me to a single post in which someone said that? > > People have said that in this newsgroup (it might have been me). You could have simply said "no". None of the below is any evidence in your favor. You made a perfectly clear claim, you know. > > Here is what Wikipedia says: > http://en.wikipedia.org/wiki/Ultrafinitism > > but even constructivists generally view the philosophy as unworkably > extreme > > and > > the constructive logician A. S. Troelstra dismissed it by saying "no > satisfactory development exists at present." > > Why are ultrafinite theories considered "unworkable"? > I would think an UST woiuld be similar to theories with universal > sets. -- Jesse F. Hughes "I'm not going to forget what I've seen. I understand the devastation requires more than one day's attention." -- G. W. Bush reassures Hurricane Katrina victims. Two days, minimum.
From: Transfer Principle on 2 Mar 2010 21:22 On Mar 2, 1:25 pm, Virgil <Vir...(a)home.esc> wrote: > In article > <be3d057d-1a58-4a17-89a7-e312ea28f...(a)b5g2000prd.googlegroups.com>, > RussellE <reaste...(a)gmail.com> wrote: > > Any arithmetic operation with NaN as an operand equals NaN. > > For example, NaN+1 = NaN. > If NaN - 1 = NaN (i..e., the predecessor of NaN is Nan), your arithmetic > is going to be bloody useless. In computer arithmetic (IEEE 754, which is of course where RE got the idea of NaN from), NaN-1 is indeed NaN. Here's a link which explicitly lists NaN-1 as being NaN: http://users.tkk.fi/jhi/infnan.html Therefore, by Virgil's standards, IEEE 754 arithmetic must be "bloody useless," even though Virgil probably uses software that adheres to IEEE 754 every time he turns on his computer. Ironically, in another thread when I asked about ultrafinitist theories, Fred Jeffries suggested that I consider the IEEE 754 standard as an example of ultrafinitism. RE appears to be heading in that direction with his use of "NaN."
From: Jesse F. Hughes on 2 Mar 2010 21:31 Transfer Principle <lwalke3(a)lausd.net> writes: > In computer arithmetic (IEEE 754, which is of course where RE > got the idea of NaN from), NaN-1 is indeed NaN. Here's a link > which explicitly lists NaN-1 as being NaN: > > http://users.tkk.fi/jhi/infnan.html > > Therefore, by Virgil's standards, IEEE 754 arithmetic must be > "bloody useless," even though Virgil probably uses software > that adheres to IEEE 754 every time he turns on his computer. > > Ironically, in another thread when I asked about ultrafinitist > theories, Fred Jeffries suggested that I consider the IEEE 754 > standard as an example of ultrafinitism. RE appears to be > heading in that direction with his use of "NaN." Yes, it's very ironic when two wholly unrelated persons have a difference of opinions. -- "It is my opinion that since neither Spight nor Hughes can see or understand their moral trespass [namely, quoting AP in .sigs], that their degrees from whatever university they earned their degree should be annulled." -- Archimedes Plutonium (12/1/09)
From: Transfer Principle on 2 Mar 2010 22:46
On Mar 2, 10:03 am, MoeBlee <jazzm...(a)hotmail.com> wrote: > On Mar 2, 12:48 am, RussellE <reaste...(a)gmail.com> wrote: > > 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. > Primitives: > 1-place predicate - 'x is a ret' > 1-place predicate - 'x is an urment' > 2-place predicate - 'xey' ('x is an element of y') This is the second time that MoeBlee has played around with rhyming words like "ret" and "urment" in trying to describe RE's theory. (The first time was back in the second post of this thread.) Also the standard theorists Patricia Shanahan William Eliot have also criticized RE for trying to steal terminology from the standard theories (such as ZFC and PA) and use them in his own theory. I don't agree with this notion that standard theories have a monopoly on these terms. If we accept ZFC as the standard theory, then what about the theory ZF (or to be explicit, ZF+~AC)? Now ZF+~AC proves the existence of nonempty sets without choice functions. But according to the standard theory ZFC, every nonempty set has a choice function. So what if I were to claim that therefore, these nonempty objects in ZF+~AC that lack choice functions aren't really sets, so we should call them "rets" or "tets" instead? Similarly, ZFA proves the existence of illfounded sets. This is in contrast with the standard theory ZFC, which proves that every set is wellfounded. So what if I were to claim that therefore, these illfounded objects in ZFA aren't really sets, so we should call them "prets" or "vlets" instead? Just as NFU proves the existence of non-Cantorian sets. This is in contrast with the standard theory ZFC, which proves that every set is Cantorian. So what if I were to claim that therefore, these non-Cantorian sets in NFU aren't really sets, so we should call them "nfets" or "wrets" instead? Finally, bringing this back to ultrafinitism, we know that there exist standard naturals n such that Y-V can't answer yes to the question "Is n a natural number?" The sum or product of two standard naturals is also a standard natural, whereas the set of all naturals n such that Y-V can answer "Is n a natural number?" isn't closed under either addition or multiplication. So what if I were to claim that therefore, Y-V isn't really talking about natural numbers, but something called "yvatural numbers" instead? Of course, this is silly. Adherents of ZF+~AC, ZFA, and NFU aren't going to call their objects "rets" just because they aren't sets in ZFC. The "yvatural numbers" example is even worse, since the set of all "yvatural numbers" is a proper subset of the set of all natural numbers, and so every "yvatural number" literally _is_ a natural number, whether the adherents of PA like it or not. Thus, one shouldn't say that the objects that RE describes in his theory aren't sets or natural numbers, unless one is prepared to do the same with NFU's sets or Y-V's naturals. Of course, at this point the standard anti-"cranks" are likely thinking about a "slippery slope" argument -- if one can call RE's objects "natural numbers," what's to stop another so-called "crank" from calling Q the set of naturals, or R the set of naturals, or {e, i, pi, 42} the set of naturals, or some other crazy set? Where do we draw the line? Here's where we draw the line: we can call an object defined in a nonstandard theory by the same name as an object defined in a standard theory, if the nonstandard object is an _analog_ of the standard object in the new theory, satisfying some basic property of the standard object. An example: RE wishes to define "urelement" in his theory. To me, a basic property of "urelements" is that they contain no elements (and aren't the empty set). Since RE's objects don't contain elements, I believe that RE has the right to keep on calling them "urelements." On the other hand, if RE were to define "urelements" so that they have elements, then I'd agree that RE would be disingenuous in calling them "urelements," so that MoeBlee and the others would be justified in making him change their name to "urments" or "burblements." RE's "sets" can contain urelements. Sets in ZFCU and NFU may contain urelements. So I see no reason for RE to change the name "sets," unless we're going to make adherents of ZFCU and NFU stop calling their objects "sets" too. RE's "classes" can contain sets as elements. Classes in NBG may contain sets. So I see no reason for RE to change the name "classes," unless we're going to make adherents of NBG stop calling their objects "classes" too. Finally, ultrafinitists wish to work with a finite subset of the set of standard naturals. So I see no reason for them to stop calling their objects "natural numbers" simply because there are only finitely many of them in their theories. To repeat, ZFC/PA don't have a monopoly on the names "sets," "natural numbers," etc., no matter how much the standard theorists may desire this. |