An Ultrafinite Set Theory
in [Math]
|
Prev: integration limit notation
Next: Arcs And Marks
From: Jesse F. Hughes on 2 Mar 2010 18:33 RussellE <reasterly(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? > I suspect people don't mean we can always derive a contradiction from > the axioms of a UST. I think they mean UST's aren't consistent with > their idea of arithematic. I suspect that you're making things up. Or, perhaps more charitably, misremembering. -- "Am I am [sic] misanthrope? I would say no, for honestly I never heard of this word until about 1994 or thereabouts on the Internet reading a post from someone who called someone a misanthrope." -- Archimedes Plutonium
From: RussellE on 2 Mar 2010 18:54 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). 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. Russell - Integers are an illusion
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." |