Q***5
发帖数: 994 | 1 It is trivial to prove that the set of irrationals is Borel.
I think what he means is the following:
Proof by contradition.
Suppose the set of irrationals (denoted by A) is a countable union of open
sets and closed sets. Since A does not contain any open segments, A must be
a countable union of closed sets, that means, its compensation set Q (the
set of all rationals) is the intersection of countable open sets. Each of
these open sets has the following property: the compensation of the open set
... 阅读全帖 |
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s****b
发帖数: 2039 | 2 集合可以分为finite和infinite，infinite又可以分为countable和uncountable。
为什么自然数，有理数的集合是countable，而实数的集合是uncountable的？看了几本
书的解释，还是不敢苟同。实数虽然是连续的，但是如果我们在实数轴上每次任选一个
实数，这样第n次选的n是自然数，所选的是实数，以至无穷，实数不就和自然数一一对
应了吗？实数不就是countable了吗？
这个想法错在哪里呢？请懂的大牛指教一下吧。多谢！ |
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w**********a
发帖数: 107 | 3 sigma algebra is closed under complement and countable union and Borel sigma
algebra is generated by open sets. So can we say that any Borel set is the
countable union of open sets and closed sets? |
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s****b
发帖数: 2039 | 4 再说，countable的本质是＂可以数的＂，每一个一个实数也是可以数的，为什么不算
countable呢？ |
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w**a
发帖数: 1024 | 5 the set of continuous functions on an interval [0,1] is countable?
how to prove? |
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n***p
发帖数: 7668 | 6 记得愚公移山么？
如果愚公只是自己这一辈子移山，那是finite
但是他想连子孙都搭进去，‘子又生孙，孙又生子，子子孙孙，无穷尽焉’，那就是
countable but not finite. |
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l******o
发帖数: 1550 | 7 countable could be infinite, and usually infinite for my first impression |
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j******w
发帖数: 690 | 9 the set of irrationals.
proof: Otherwise, there is a countable sequence open sets U_n, n\in \omega so that
\cap_n U_n=Q (Q is the set of rationals). Since each U_n is comeager, Q is comeager
which is a contradiction.
Actually there is a hierarchy of Borel sets. Each level doesn't collapse.
【 在 wakakayikaka (哇咔咔一咔咔) 的大作中提到: 】 |
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w**********a
发帖数: 107 | 10
I am sure Cantor Set is closed since its complement is the countable union
of open intervals which is open. |
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k******2
发帖数: 111 | 11 借宝地问下，borel set 到底是啥？
Union of Countable h-intervals? |
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q********e
发帖数: 1255 | 12 google 'cantor diagonal argument'
你看的几本书都没有这个证明？
countable 意思是用一种方法数这个集合你能保证每个元素都可以数到，
数学语言就是和自然数有一一对应；而不是任指一个实数你都能叫出名字。 |
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c**n
发帖数: 5275 | 13
countable是可列的，意思是你可以找到一个方法或规律一一列举出所有元素，无理数
是不行的 |
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h****e
发帖数: 2125 | 14 别老做这种白日梦了，medicaid program要查你的income和assets的：
"
Countable Assets
All other cash, stocks, bonds, investments, and retirement plan assets are
counted by Medicaid to determine eligibility for benefits. Generally, if an
applicant has access to the principal of an asset even if the principal has
never been touched or if it is subject to taxes or penalties it is a
countable asset.
According to federal law, an unmarried applicant to Medicaid may retain non-
countable assets plus no more than $ 2,000 to... 阅读全帖 |
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h****e
发帖数: 2125 | 15 别老做这种白日梦了，medicaid program要查你的income和assets的：
"
Countable Assets
All other cash, stocks, bonds, investments, and retirement plan assets are
counted by Medicaid to determine eligibility for benefits. Generally, if an
applicant has access to the principal of an asset even if the principal has
never been touched or if it is subject to taxes or penalties it is a
countable asset.
According to federal law, an unmarried applicant to Medicaid may retain non-
countable assets plus no more than $ 2,000 to... 阅读全帖 |
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O******e
发帖数: 734 | 16 "minimal" is usually a qualitative adjective,
meaning few, very little.
It can be used to describe either countable or
non-countable objects.
When used to describe a countable object, the exact
quantity of that object is not important, but the
sense that the quantity is few is important.
"He does minimal work." (non-countable work)
"He puts in minimal hours at work." (countable work hours)
Both sentences mean that he does not do a lot at work
(perhaps because he is lazy or not a good worker).
"m... 阅读全帖 |
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r********e
发帖数: 103 | 17 Note that all second-countable spaces are separable. A metric space is
separable if and only if it is second-countable. The two conditions are
equivalent here.
1) A compact metrizable space is second countable. Its subspaces are second
countable too.
2) A second countable metrisable space is homeomorphic
to a subspace of [0,1]^J, which leads to a metrizable compactification. |
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发帖数: 1 | 18 1. 你先给个“countable数”的定义先。希尔伯特旅馆是讨论无穷数集的概念，和可数
不可数无关。只要是无穷集合都可以有希尔伯特旅馆悖论，无论可数还是不可数。
2. 不管你那个荒唐的countable超越数是怎么定义的，一个数要么是countable,要么不
是countable,什么叫数出来的超越数会不一样?
3. 这个超越数集不可数是对的，但超越数的定义和可数不可数屁关系都没有。
总之你就是典型的在课上或者科普读物里学了几个名词，但对这些概念根本一窍不通，
还老爱拿这些名词出来唬人的民科。
关于2再教你一个：如果你说的是可计算实数，根据Church's theis,不管你用什么理论
来定义可计算性，不管你用什么程序语言，什么样的计算机，定义出来的可计算实数集
是一定的。一个数要么是可计算，要么是不可计算。 |
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A**n
发帖数: 1703 | 19 Uncountable infinity - countable infinity = uncountable infinity
Countable infinity - uncountable infinity = negative uncountable infinity
Countable infinity - countable infinity = ? Anyone? |
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l*****8
发帖数: 16949 | 20 countable一般就指Countably Infinite，已经是无穷了。
有理数是countable的。无理数是uncountable的。
细细的分，无理数还可以分成代数数（比如根号2）和超越数（比如pi)。前者是
countable,后者是uncountable. |
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c****o
发帖数: 1280 | 21 The complement of cantor set is also open and countable.
countable union of countable set is still countable. |
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t******l
发帖数: 10908 | 22 这个也是我们中小学数学教育的问题。。。你要是仔细看的话，我前面没有提 "无理数
" 这个概念。。。我前面说的很清楚的是 "根号数" 的概念，也就是有确切给定的马工
自动机生成规则的数。。。而且 "无理数" 的集合是 uncountable 的，"根号数" 的集
合是 countable。
我前面还提到了 "小数"，也就是 decimals。。。如果提到无限循环小数，那就是有对
应的分数，countable。。。而我提到无限不循环小数的时候，我后面有马上提到浮点
协处理器和 floating point number round off error 的工业界处理，所以还是可操
作 countable 的。
所以我没有使用你所提到的 perfection。。。我里面所涉及的 "无限"，都可以建模成
有限状态自动蹦蹦机的停机问题。。。所谓的 "无限" 就是可以逻辑上证明该蹦蹦机永
不停机。。。符合大众语言里的 “条条大路通罗马”，或者 “好日子望不到边”。。
。是符合时间之箭的 "无限"，也就是有使用价值的 "无限"，而不是仅仅为了数学美。
所以我对 whether the concept ... 阅读全帖 |
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s***h
发帖数: 487 | 23 这里的 decimal 泛指可以用一个符号序列表达，不一定需要十进制。
countable / decidable / computable numbers 不是同一概念。
: 不是decimal的问题。代数数很多也decimal。
: 因为超越数是不countable的，所以不可以全部在计算机/物理过程中表达
。当然
，可以
: 表达countable无穷个超越数，但对计算/物理任意给定的“一般&
rdquo;的超越数没有
意义。
: 严格一点说，物理过程能够测量的是代数数部分。举例说那个-1/12的拉
马努金
和。而
: 其它部分压根可能就不是已知的数，如R部分。 |
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f******g
发帖数: 29 | 24 For nouns with natures of countable or non-countable, like disruption, will
you use "a" or avoid using "a" in any circumstance?
According to my professor, such article usages have no underlying rule but
custom-based habits.
Now I avoid using singular in most of the countable nouns. |
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I***e
发帖数: 1136 | 25 1.
1) If A’s limit point set B is countable. We know B has to be a closed set.
Thus, B- is an open set with countable open intervals. One of these intervals
contain uncountable elements of A. Let’s assume it is (0, 1). Then you can
argue that there is a k such that (1/k, 1-1/k) contains infinite number of A’
s elements thus have an limit in (0, 1)…
2) It suffices to prove that A has one limit point in itself. Assume A and B
are disjoint. Again B is a closed set so B- is a countable union of open |
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w*********r
发帖数: 42116 | 26 美国的低保福利看个人实际拥有的财产，是否有住房，是否有存款，是否有汽车，根据
这个来估算每月给你多少钱。第一步就是确定financial resources。有人提供住房，
有车，都会从SSI中扣除掉。我在社工局当义工，帮助老墨，老黑和华人填过这种表。
第一步：financial resources
financial resources exceed the limits of $2,000 for individuals and $3,000
for couples in 2009. These resources include any money that you have in bank
accounts or brokerage accounts as well as other valuable items.
第二步：unearned income.
This includes unearned income like interest and stock dividends. 减掉20美元。
第三步：earned income
Subtract $65 from you... 阅读全帖 |
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F*********s
发帖数: 223 | 27 其实很多貌似简单的概念都不是那么简单的。比如有多少人能深刻理解R?为什么那么多
数学定理公式都定义在R上？为什么有countable和不countable 之分。在R上到底发生
着什么?别说 正负无穷了，就是在你眼皮底下[0,1]闭区间就有无穷的数置身其间。其
中无穷多的不可数无理数是无穷多的可数有理数的无穷倍多。如果果真有无穷个数在[0
,1]之间，那你从0出发应该永远都不可能到达1.这些都是非常矛盾的现象和概念。 |
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t******l
发帖数: 10908 | 28 我没有咋看连续的 random walk，因为我对真正连续空间实数集上面的概率排列组合不
是很理解。。。或者说，我觉得都是一大堆 patch，不如 countable set 简洁。
实际上，我现在把整个高中连续代数都理解为基于 extendable countable set，也就
是始于 algebraic numbers，extend on-demand。。。这样实用上够了，比一大堆不可
计算数的实数集对我个人好理解一些。
看来一下你 link 里的 quantum mechanics on cantor set。。。不太理解。。。但是
提出新理论得有用，不能仅仅是数学等价。
: no you did not get it. You completely missed the point of continuous-
time
: and discrete-time random walking.
: The continuous-time does not need "traversal" or the step can just
take no
阅读全帖 |
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发帖数: 1 | 29 不是decimal的问题。代数数很多也decimal。
因为超越数是不countable的，所以不可以全部在计算机/物理过程中表达。当然，可以
表达countable无穷个超越数，但对计算/物理任意给定的“一般”的超越数没有意义。
严格一点说，物理过程能够测量的是代数数部分。举例说那个-1/12的拉马努金和。而
其它部分压根可能就不是已知的数，如R部分。 |
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s***h
发帖数: 487 | 30 数学里面确实有很多问题，从 CS 角度看，本质是 sum of undecidable numbers。
这类问题，CS 不能说 Math 有错，因为是 out of scope of CS 。
: 不是decimal的问题。代数数很多也decimal。
: 因为超越数是不countable的，所以不可以全部在计算机/物理过程中表达
。当然
，可以
: 表达countable无穷个超越数，但对计算/物理任意给定的“一般&
rdquo;的超越数没有
意义。
: 严格一点说，物理过程能够测量的是代数数部分。举例说那个-1/12的拉
马努金
和。而
: 其它部分压根可能就不是已知的数，如R部分。 |
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b*****c
发帖数: 1103 | 31 我觉得可以绕圈圈数数。
不过要用到有理数是countable set，如果不是countable set就不行 |
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t*******r
发帖数: 22634 | 32 mapping 是对小学生而言的，正确说法是 indexing，所以 countable。。。集合的
countable 的概念，本身对小学生并不重要。。。理论数学的坑爹 indexing 的方式，
对小学生更是过犹不及。。。对小学生重要的是，分数的 visual modelling，任何时
候都是应该可以数数的，当然也是要完备的。。。 |
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t******l
发帖数: 10908 | 33 发现关于 real number 的这段还是有点意思。。。其实数学到一定的虚无缥缈的抽象
程度后，我觉得就跟神学确实差不多，或者说，艺术。
http://en.wikipedia.org/wiki/Real_number#In_computation
A real number is called computable if there exists an algorithm that yields
its digits. Because there are only countably many algorithms,[12] but an
uncountable number of reals, almost all real numbers fail to be computable.
Moreover, the equality of two computable numbers is an undecidable problem.
Some constructivists accept the existence of only those reals that are
co... 阅读全帖 |
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p****x
发帖数: 980 | 34 Still, I have more hair than you.
Coz your hair is countable.
BTW, your brain cells are countable too.
There is nothing right in your left mind.
And not much left in your right mind. |
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J***o
发帖数: 7166 | 35 I was wondering if in another language the below Chinese could have an
easier way to be understood.
"Right, rational numbers are algebraic . I say quite accurate. Strictly
speaking, algebraic number is divided irrational and rational algebraic
number
Number of algebra. But both are countable (added together is countable)."
I did, but nothing changed. Pain always tastes bitter. |
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R********n
发帖数: 90 | 36 Florida State is scheduled to play in a record 36th consecutive bowl game,
the Independence Bowl, against Southern Miss on December 27. Their 6-6
record includes a win over Delaware State, an FCS program. For an FCS
opponent to be countable towards bowl eligibility, the FCS program must have
awarded at least 90% of the FCS scholarship limit. After our own
investigation, we have determined and confirmed that Delaware State has not
met the 90% threshold set by the NCAA. As a result, Florida State'... 阅读全帖 |
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M*****e
发帖数: 11621 | 37 你可以这样理解，但不能这么写。
对于任意y_1 > y_2..... >0, with lim_{i\to \infty} y_i = 0
(-\infty, x]是 所有(-\infty, x+y_i]的交集
但是你不能把 y_i直接换成h, 因为在mearuable space的定义里面
the intersection of countably many events is an event
but the intersection of arbitrarily many events is not necessarily so
所以你的证明里引用的那个continuity from above property也是只适用于countably
many events的。
靠，老子中英文夹杂真恶心。 |
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s**f
发帖数: 365 | 38 fewer if countable
less if uncountable
have plural form = countable |
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w*z
发帖数: 71 | 39 i'm not sure what u'r asking for. what I'm talking about is:
If A=sum(i in I) A_i, then
mu(A)=sum(i in I) mu(A_i),
where mu() is a measure, A_i's are disjoint mearurable sets and I is a
countable index set.
Therefore, mu(A_i)=0 (for all i in I) implies mu(A)=0.
However, if I is uncountable, then it's possible that mu(A)>0 but
the RHS is still 0. The reason is that a measure only allows countable
additivity.
I believe that the sum of 'uncountably many' 0's is still 0. But I don't
know what's the |
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z**e
发帖数: 10 | 40 Suppose A is countable set :
A * A *A .....stands for infinite Cartesian product of A , is the product
countable?
How to prove ? |
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H****h
发帖数: 1037 | 41 No.
Suppose A is countable set :
A * A *A .....stands for infinite Cartesian product of A , is the product
countable?
How to prove ? |
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B****n
发帖数: 11290 | 42 我對於Icare的證明非常佩服
我的證明確實比較囉唆 不過它應該可以推廣到任意有對稱軸的二維圖形都是對稱軸
為最短的等分曲線 我稍微修改了一點之前的證明 讓它可以推廣
I denote the graph as (t,f(t)), t in [0,1], with x-axis as its symmetric axis.
1. We can only consider bounded variation curve otherwise its length can not
be minimal.
2. For bounded variation curve, its root is countable. Here I define root as a
real number a such that f(a)=0 and there exists a epison such that f(a-c)<=0<
f
(a+c) or f(a-c)<0<=f(a+c) for any 0
3.Since the root is countable, we can deno |
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j******w
发帖数: 690 | 43 no
Here is just a hint:
If \Sum_i \mu(E_i)<\infty, then the index set of positive measure sets must
be countable.
Here I just simply assume ``the sum" is the least upper bound of all
countable sums.
Actually one can show that there exists no uncountably many mutually
disjoint positive measure sets. |
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p******n
发帖数: 46 | 44 2)中的 2nd countable必要吗？为什么？
metrizable space是normal space.因此必然是 completely regular space. 一个
completely regular space is homeomorphic to a subspace of [0,1]^J.
为什么还要 2nd countable呢？我的推理有问题吗？
second |
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r********e
发帖数: 103 | 45 metrizable是completely regular Hausdorff space 或者说是Tychonoff space没错，
但是这个只保证homeomorphic to a subspace of the cube [0,1]^I, rather than
the Hilbert cube [0,1]^N. Notice that the cube may not be metrizable since
the index set I may not be countable (think about the cantor cube).
Actually from Urysohn Metrization Theorem, we have the following:
For a Hausdorff space X, TFAE:
1. X can be embedded in the Hilbert cube
2. X is separable metrizable space
3. X is regular and second countable
You can see |
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Q***5
发帖数: 994 | 46 The sigma field generated by {Xt, 0<=t<=1} consists of the sets either
countable or whose complement ([0 1] \ the set) is countable.
0,
, |
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L******k
发帖数: 33825 | 47 (a) Prove or disprove: an intersection of countably many closed sets is
closed.
(b) Prove or disprove: A union of countably many closed sets is closed. |
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F**S
发帖数: 13 | 48 I am embarrassed to ask such elementary questions but they have bothered me
like a bone in my throat for quite a while. Please be so kind to enlighten
me...
The questions are centered closely about infinite dimensionality:
How exactly to see that a mapping space is infinite dimensional? To be
specific, take the simplest example of the vector space, C(R, R), of
continuous functions from R to R, where R stands for the space of real
numbers. The statement must be obvious since books I have consulte... 阅读全帖 |
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D**u
发帖数: 204 | 49 The question does not require F to be countable additive.
In the traditional measure theory, every measurable set has a non-negative
measure, so if a measurable set S_1 is a subset of S_2, then measure(S_2) >=
measure(S_1). Then for a P in the 2-d plane, we can use a countable union
of rectangles to "approximate" the measure of P.
The same approximation fails in this question because F does not guarantee a
non-negative "measure" on any polygon. |
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a**********2
发帖数: 3726 | 50 That would be awesome. There are countable communism dictatorship countries
left in the earth. |
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