An uncountable countable set
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From: Han de Bruijn on 17 Oct 2006 08:08 Virgil wrote: > In article <45341a3a$1(a)news2.lightlink.com>, > Tony Orlow <tony(a)lightlink.com> wrote: > >>Well, I think that, while the empty set may easily be taken to represent >>0, 1 is not the set containing 0. That doesn't seem, even at first >>glance, like a very accurate model of what 1 is. > > If TO is not happy with the set representing 1 containing a single item > does TO want the set representing 1 to contain more or less that single > item? That single item is the EMPTY set, pasted between curly braces. Han de Bruijn
From: Han de Bruijn on 17 Oct 2006 08:14 David Marcus wrote: > Tony Orlow wrote: > >>Han.deBruijn(a)DTO.TUDelft.NL wrote: >> >>>Tony Orlow schreef: >>> >>>>Han de Bruijn wrote: >>>> >>>>>So the axiom of infinity says that you can get everything from nothing. >>>>>This is contradictory to all laws of physics, where it is said that you >>>>>pay a price for everything. E.g. mass and energy are conserved. >>>> >>>>Han, you can't really be looking for conservation of energy or momentum >>>>or mass in abstract mathematics, can you? This axiom basically defines >>>>the infinite linear inductive set. Given this method of generation, >>>>there should be things we can say about the set, no? >>> >>>So to speak, Tony. In physics and economics, you can't get something >>>for nothing. Nothing just gives nothing. You must have _something_ to >>>start with. I find the idea absurd that natural numbers can be built >>>by putting curly braces around the empty set. >> >>Well, I think that, while the empty set may easily be taken to represent >>0, 1 is not the set containing 0. That doesn't seem, even at first >>glance, like a very accurate model of what 1 is. > > This is not relevant since no one ever said that the set containing 0 is > an "accurate model of what 1 is". Before criticizing something, it helps > to know its purpose. So, I'll ask you: Do you know the purpose of the > construction of the natural numbers from sets? Sure. The purpose is to ground everything on set theory. But since you have to take into account at least ONE or TWO sets to accomplish that, you have to be able to count. That is: you have to define some natural numbers before you can do any set theory. And that's a vicious circle. Han de Bruijn
From: stephen on 17 Oct 2006 08:24 Han de Bruijn <Han.deBruijn(a)dto.tudelft.nl> wrote: > Virgil wrote: >> In article <45341a3a$1(a)news2.lightlink.com>, >> Tony Orlow <tony(a)lightlink.com> wrote: >> >>>Well, I think that, while the empty set may easily be taken to represent >>>0, 1 is not the set containing 0. That doesn't seem, even at first >>>glance, like a very accurate model of what 1 is. >> >> If TO is not happy with the set representing 1 containing a single item >> does TO want the set representing 1 to contain more or less that single >> item? > That single item is the EMPTY set, pasted between curly braces. > Han de Bruijn So? It is still a single item. An empty box is still a box. A box that contains an empty box is not empty. It contains one item. Stephen
From: MoeBlee on 17 Oct 2006 12:32 Tony Orlow wrote: > MoeBlee wrote: > > Tony Orlow wrote: > >> No, set theory confuses the issue with its concentration on omega. > > > > Oh boy, here we go again with "Set theory confuses...". Please just way > > which axioms of set theory you reject and which ones you use instead. > > > > MoeBlee > > > > Uh, what axioms of set theory are specifically involved in your "proof"? > I don't remember a deduction from those axioms. Perhaps you could > refresh my memory. My proof of what? MoeBlee
From: MoeBlee on 17 Oct 2006 12:41
Tony Orlow wrote: > MoeBlee wrote: > > P.S. > > > > I lost the context, but somewhere you (Orlow) posted: > > > > "ZF and NBG don't handle sequences or their sums, but only unordered > > sets" > > > > Z set theory defines and proves theorems about ordered tuples, finite > > and infinite sequences, and infinite summations and infinite products > > and many other things like that. > > > > MoeBlee > > > > Huh! But I thought sets were unordered. Without the axiom of choice, we prove the existence of certain orderings on certain sets. With the axiom of choice, we prove that every set has a well ordering. > If the theory of infinite series > is derived from set theory, I don't say that historically it is derived from set theory. But infinite series, such as one finds in ordinary calculus and real analysis are definable in set theory and the theorems about them can be proven from the axioms of set theory. > how come they seem to contradict each other > here? A set theoretic explication of an infinitary thought experiment (which is not an explication that occurs IN set theory) differs from your own explication of that thought experiment. That is not a contradiction between set theory and any theorem regarding infinite series. > I don't recall a derivation or proof of the empty vase from the > axioms of set theory. Well, duh. Haven't people already made the point as clear as it can possibly be made that there are no vases in set theory? But anyway, even though there are no proofs about vases in set theory, you don't recall ANY proof in set theory, since you've never studied a single one. MoeBlee |