An uncountable countable set
in [Math]
|
Prev: integral problem
Next: Prime numbers
From: Virgil on 13 Oct 2006 03:48 In article <7fe78$452f3cb1$82a1e228$29638(a)news2.tudelft.nl>, Han de Bruijn <Han.deBruijn(a)DTO.TUDelft.NL> wrote: > Michael Stemper wrote: > > > In article <1160613324.049353.270480(a)k70g2000cwa.googlegroups.com>, Ross A. > > Finlayson writes: > > > >>David Marcus wrote: > >> > >>>Ross A. Finlayson wrote: > >>> > >>>>David Marcus wrote: > > > >>>>>So, you aren't saying that ZF is inconsistent. You are saying that you > >>>>>prefer to use a different system of axioms. Is that correct? > > > >>>>No, David, Dave, I say ZF is inconsistent. That doesn't mean all its > >>>>results are false. It just lets me say whatever I want about a less > >>>>inconsistent set of "axioms". > > > >>>So, you are saying that ZF is inconsistent. By "inconsistent", do you > >>>mean that ZF proves both P and not P, for some statement P? If so, > >>>please be explicit: what is the statement and what is the proof in ZF of > >>>it? > > > >>Build a set: {x: true}, it's a set, > > > > No, "{x: true}" is gibberish. Just putting down a couple of braces and > > throwing random text inside them does not generate a set. At least, no > > according to the axioms of ZF that you're trying to refute. > > No more glibberish than defining the ordinals by putting ever more curly > braces around the empty set: > > 0 = {} , 1 = {{}} , 2 = {{},{{}}} , 3 = {{},{{}},{{},{{}}}} ... > > http://www.jboden.demon.co.uk/SetTheory/ordinals.html > > Han de Bruijn Except that in ZF or NBG there are axioms which allow one to construct x union {x} from any x, but no axioms to allow {x:true}.
From: Virgil on 13 Oct 2006 03:49 In article <117b5$452f3ff6$82a1e228$30232(a)news2.tudelft.nl>, Han de Bruijn <Han.deBruijn(a)DTO.TUDelft.NL> wrote: > Virgil wrote: > > > http://en.wikipedia.org/wiki/ZFC > > Axiom of infinity: There exists a set x such that the empty set is a > > member of x and whenever y is in x, so is S(y). > > Which is actually the construction of the ordinals. Right? > > Suppose we write 0 = { } > then 1 = { { } } = { 0 } > and 2 = { { } , { { } } } = { 0, 1 } > and 3 = { { }, { { } }, { { }, { { } } } } = { 0, 1, 2 } > > http://www.jboden.demon.co.uk/SetTheory/ordinals.html > > 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. Which points up again the foolishness of those who would tie all mathematics to physics.
From: Virgil on 13 Oct 2006 03:52 In article <9869d$452f40f1$82a1e228$30514(a)news2.tudelft.nl>, Han de Bruijn <Han.deBruijn(a)DTO.TUDelft.NL> wrote: > Virgil wrote: > > > In article <1160669820.603144.288450(a)e3g2000cwe.googlegroups.com>, > > mueckenh(a)rz.fh-augsburg.de wrote: > > > >>Dik T. Winter schrieb: > >> > >>> > > You question whether "all x in N" does exist, apparently. Based on > >>> > > what? > >>> > > >>> > Based on the impossibility to index the positions of our 0.111..., > >>> > >>>False. > >>> > >>> > based on the vase, based on many other contradictions arising from "all > >>> > x in N do exist". > >>> > >>>False. > >>> > >>>No proof given. > >> > >>No proof possible because every proof must be dismissed unless the game > >>of set theory should perish. > > > > The "game of set" theory, as defined by ZF or NBG or something similar, > > will survive "Mueckenh". > > We will see. As it will survive HdB, and me, as well, we shall not see. > > The future is not what happens to us, but what we make of it. HdB has not the power to condemn mathematics to what he wants to make of it, an appendix to science. > > Han de Bruijn
From: Han de Bruijn on 13 Oct 2006 03:55 Virgil wrote about the Balls in a Vase problem: > The point is that after a ball is removed from the vase it is no longer > in the vase, and every ball is removed before noon. The point is that after a ball is removed from the vase, ten others are inserted. And every ball is inserted before noon. Because different, but valid, logics lead to contradictory conclusions, the end result is undecidable (: just take a look at the length of this Balls in a Vase debate ...) Han de Bruijn
From: Virgil on 13 Oct 2006 03:55
In article <66a8$452f4298$82a1e228$30886(a)news2.tudelft.nl>, Han de Bruijn <Han.deBruijn(a)DTO.TUDelft.NL> wrote: > Randy Poe wrote about the Balls in a Vase problem: > > > Tony Orlow wrote: > > >>Specifically, that for every ball removed, 10 are inserted. > > > > All of which are eventually removed. Every single one. > > All of which are eventually inserted. Every single one. None are reinserted after being removed but,each is removed after having been inserted, so that leaves them all outside the vase at noon. > > Thus the end result is _undefined_. Not in logic. |