An uncountable countable set
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From: cbrown on 15 Oct 2006 15:18 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. > If I remember correctly, you find this absurd because you assert that, if x and y are sets, there is no difference between saying "x is a member of y" and "x is a subset of y". From this it follows that x = {x}. But doesn't your web page say that, with weak pairing, your assertion results in there being only one set? So perhaps you will find any construction of the natural numbers to be absurd, because you don't believe that the number "2" is different from the number "1"... Cheers - Chas
From: David Marcus on 15 Oct 2006 15:31 Tony Orlow wrote: > David Marcus wrote: > > Tony Orlow wrote: > >> David Marcus wrote: > >>> Tony Orlow wrote: > >>>> David Marcus wrote: > >>>>> How about this problem: Start with an empty vase. Add a ball to a vase > >>>>> at time 5. Remove it at time 6. How many balls are in the vase at time > >>>>> 10? > >>>>> > >>>>> Is this a nonsensical question? > >>>> Not if that's all that happens. However, that doesn't relate to the ruse > >>>> in the vase problem under discussion. So, what's your point? > >>> Is this a reasonable translation into Mathematics of the above problem? > >> I gave you the translation, to the last iteration of which you did not > >> respond. > > > >>> "Let 1 signify that the ball is in the vase. Let 0 signify that it is > >>> not. Let A(t) signify the location of the ball at time t. The number of > >>> 'balls in the vase' at time t is A(t). Let > >>> > >>> A(t) = 1 if 5 < t < 6; 0 otherwise. > >>> > >>> What is A(10)?" > >> Think in terms on n, rather than t, and you'll slap yourself awake. > > > > Sorry, but perhaps I wasn't clear. I stated a problem above in English > > with one ball and you agreed it was a sensible problem. Then I asked if > > the translation above is a reasonable translation of the one-ball > > problem into Mathematics. If you gave your translation of the one-ball > > problem, I missed it. Regardless, my question is whether the translation > > above is acceptable. So, is the translation above for the one-ball > > problem reasonable/acceptable? > > Yes, for that particular ball, you have described its state over time. > According to your rule, A(10)=0, since 10>6>5. Do go on. OK. Let's try one in reverse. First the Mathematics: Let B_1(t) = 1 if 5 < t < 7, 0 if t < 5 or t > 7, undefined otherwise. Let B_2(t) = 1 if 6 < t < 8, 0 if t < 6 or t > 8, undefined otherwise. Let V(t) = B_1(t) + B_2(t). What is V(9)? Now, how would you translate this into English ("balls", "vases", "time")? -- David Marcus
From: cbrown on 15 Oct 2006 15:32 Han.deBruijn(a)DTO.TUDelft.NL wrote: > MoeBlee schreef: > > > Han de Bruijn wrote: > > > Virgil wrote: > > > > 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? > > > > Wrong. > > Don't understand why that's wrong. Please explain. > Short imprecise answer: The set x' = x u {x} is not equal to or a member of the set x described in AoI. Longer more precise answer: http://en.wikipedia.org/wiki/Ordinal Cheers - Chas
From: David Marcus on 15 Oct 2006 15:36 Han.deBruijn(a)DTO.TUDelft.NL wrote: > MoeBlee schreef: > > Han de Bruijn wrote: > > > Virgil wrote: > > > > 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? > > > > Wrong. > > Don't understand why that's wrong. Please explain. Do you want an explanation in standard Mathematics or in your mathematics? -- David Marcus
From: The Ghost In The Machine on 15 Oct 2006 16:00
In sci.math, Han.deBruijn(a)DTO.TUDelft.NL <Han.deBruijn(a)DTO.TUDelft.NL> wrote on 15 Oct 2006 11:09:58 -0700 <1160935798.582319.327050(a)b28g2000cwb.googlegroups.com>: > MoeBlee schreef: > >> Han de Bruijn wrote: >> > Virgil wrote: >> > > 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? >> >> Wrong. > > Don't understand why that's wrong. Please explain. > > Han de Bruijn > If one is going to use Peano, one needs all of his axioms. 1. (Zero) 0 is in N. 2. (Successor) There is a function S(n) such that, for every n in N, S(n) is in N. 3. (Zero is first.) For every n in N, S(n) != 0. 4. (Successor is injective.) For every m and n in N, if S(m) = S(n) then m = n. 5. (Induction) If X is a subset of N, 0 is in X, and for every x in X, S(x) is also in X, then X = N. However, that's only the start of the ordinals. Omega (? [*]) is the cardinality of N and the first transfinite ordinal. It is not clear to me whether S(x) = ? for some x. (x cannot be in N by #2 above.) And then there's things such as ?+1, ?+2, ?+?, ? sub ?, etc. http://mathworld.wolfram.com/OrdinalNumber.html [*] U+03C9, looks a bit like a 'w'. Apologies if my newsreader munges this. -- #191, ewill3(a)earthlink.net Useless C++ Programming Idea #104392: for(int i = 0; i < 1000000; i++) sleep(0); |