An uncountable countable set
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From: David Marcus on 9 Oct 2006 01:21 Ross A. Finlayson wrote: > Have you heard of Burali-Forti, a "paradox"? There's no set of > ordinals nor cardinals in ZF. > > There is a universe, and there's not in ZF. There are smaller sets > than the universe that are sufficient for many statements, in terms of > quantifying over their elements. There is quite regular use of the > word universe in the practice of naive set theory. > > Basically it's an anti-foundation mindset, and then some. Quantify > over sets: the result is not a set. There's nothing else over which > to quantify. So, foundation is an exercise in vacuity. > > In set theory, where basically "pure" set theory means every item is a > set, there are only sets, every thing is a set, and there's reason to > consider why the transfer principle holds true, that everything is a > set. Basically I see a contradiction in the existence of the universal > quantifier without a universe. > > That gets into things along the lines of that infinite sets are > irregular, and then some. I thought you said there was a contradiction in ZF. In the context of ZF, the Burali-Forti argument shows that there is no set of all ordinals, but does not lead to a contradiction. So, do you still say there is a contradiction in ZF? If so, what is it? -- David Marcus
From: David Marcus on 9 Oct 2006 01:30 Han.deBruijn(a)DTO.TUDelft.NL wrote: > David Marcus wrote: > > > Consider this situation: At time 5 one ball is added to a vase. At time > > 6, the ball is removed. > > > > Is the following a valid translation into mathematics? > > > > Let the value 1 denote that that the ball is in the vase and the value 0 > > that the ball is not in the vase. Let A(t) be the location of the ball > > at time t. Let > > > > A(t) = { 1 if 5 < t < 6; 0 if t < 5 or t > 6 }. > > No. Then what is the translation into Mathematics? -- David Marcus
From: cbrown on 9 Oct 2006 01:58 Tony Orlow wrote: > Sure, that sounds okay to me. In the naturals, the first is 1, the > second 2, etc. What is the aleph_0th? I don't know. What is the "3/2"th natural number? Cheers - Chas
From: Han de Bruijn on 9 Oct 2006 03:52 Virgil wrote: > In article <1160332251.241188.301420(a)i3g2000cwc.googlegroups.com>, > Han.deBruijn(a)DTO.TUDelft.NL wrote: >> >>But ah, a picture says more than a thousand words: >> >>http://hdebruijn.soo.dto.tudelft.nl/jaar2006/ballen.jpg > > But the function one should be graphing is not continuous at any of the > time of entry or exit of any ball, nor at noon (except it is > right-continuous at noon). Correct. I have "continuized" the function. The true (discrete) function should be like a staircase with a stair at each red ball. But a discrete version likewise explodes (i.e. not implodes) at noon. Han de Bruijn
From: Virgil on 9 Oct 2006 04:05
In article <e3ab0$4529ffd9$82a1e228$22026(a)news2.tudelft.nl>, Han de Bruijn <Han.deBruijn(a)DTO.TUDelft.NL> wrote: > Virgil wrote: > > > In article <1160332251.241188.301420(a)i3g2000cwc.googlegroups.com>, > > Han.deBruijn(a)DTO.TUDelft.NL wrote: > >> > >>But ah, a picture says more than a thousand words: > >> > >>http://hdebruijn.soo.dto.tudelft.nl/jaar2006/ballen.jpg > > > > But the function one should be graphing is not continuous at any of the > > time of entry or exit of any ball, nor at noon (except it is > > right-continuous at noon). > > Correct. I have "continuized" the function. The true (discrete) function > should be like a staircase with a stair at each red ball. But a discrete > version likewise explodes (i.e. not implodes) at noon. Actually not precisely "at" noon. Let A_n(t) be equal to 0 at all times, t, when the nth ball is out of the vase, 1 at all times, t, when the nth ball is in the vase, and undefined at all times, t, when the nth ball is in transition. Note that noon is not a time of transition for any ball, though it is a cluster point of such times. let B(t) = Sum_{n in N} A_n(t) represent the number of balls in the vase at any non-transition time t. B(t) is clearly defined and finite at every non-transition point, as being, essentially, a finite sum at every such non-transition point. Further, A_n(noon) = 0 for every n, so B(noon) = 0. Similarly when t > noon, every A_n(t) = 0, so B(t) = 0 |