This past semester I took the first halves of Introduction to Analysis and Modern (Abstract) Algebra in which, naturally, we covered the basics of the cardinalities of infinities. I understand the concepts of what makes something countable verses uncountable. I simply have an intuitive problem with the fact that the rational numbers are countable. The other basics countable infinite sets (Integers, Evens, Odds, Natural Numbers, etc.) all sit well with me. For some reason, I just have a problem accepting that the Rationals are truly countable. I understand that a definition is not arguable, but it just does not feel right.
Part of my discontent lies in the fact that like the continueum, there is no smallest positive rational number. This is not true for the previously mentioned sets. I feel that the rationals are somehow set apart from this set. If I recall correctly, it was proven that "It cannot be proven nor disproven that there exists an infinite set whose cardinality is greater than the cardinality of the countable infinite sets and less than the cardinality of the continueum (the reals)."
If anyone knows where I may be able to find more information on this, or even has an answer to my dilemma, please let me know. Thanks!
-Will
Name:
Anonymous2004-12-27 18:13
everything is relative
the smallest rational is the one which has been agreed to be smallest
semantics really, there is no provable answer to such a question
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Anonymous2004-12-27 22:34
No, there is no smallest positive rational number. That is provable. Think of it this way: find "the smallest positive rational number". Call it t. Wouldn't t/2 also be a positive rational number? But that would be absurd since we already found the smallest positive rational number which we called t. Thus our original assumption, that t is the smallest positive rational number, must be false. Thus there is no smallest positive rational number.
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Anonymous2004-12-27 23:05
certain things are posulated as givens to allow conceptualization of higher level theories
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Anonymous2004-12-27 23:10
>>4
Understood, but sometimes definitions can be changed.
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Anonymous2004-12-28 0:41
>>5
this fact exactly is the root of the largest fallacy in bleeding-edge/theoretical science
provable != comprehensible
our best current theory is exactly that, a theory....
building a tower of theory based on incomprehensible assumptions generally leads to an inaccurate model - inherent with a lack of understanding
They're not countable by definition, they're provably countable, if you accept that integers are countable and that given a bijection for two sets, their cardinality is equal.
>>7
Right, I know. I was referring to the definition of what it means to be countable (using bijection). It just does not sit right with me. I feel that they are somehow "more infinite" than the integers.
>>6
Sometimes though what seems to be incomprehensible is in fact true. Just take Cantor for example. He is the one who came up with this whole idea of different cardinalities of infinites and they said he was a nut. In fact, he did go crazy (or at least was admitted to a sanitarium). Nevertheless, from what we understand of mathematics today, he was indeed correct.
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Anonymous2004-12-31 16:43
>>8 'I feel that they are somehow "more infinite" than the integers.'
in a naive approach i say that's because their infinitiy "is closer" ie between one and two while you have to "go far" with the integers..
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furfag2005-01-23 8:03
meditate on the diagonalization argument.
Also, a set which is both infinite and countable is said to be denumerable.
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Anonymous2005-01-23 18:10
ℵ₀
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Anonymous2005-01-24 2:18
>>10
perceptive answer, you exhibit an understanding of the misinterpretations of analytic number theory
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Anonymous2005-02-08 14:43
ℵ₀-plex.
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Anonymous2005-02-15 5:19
The above poster is correct. There are different orders of infinities, some which are, in a very strictly defined sense, larger than others.
For example, in Linear Algebra, the number of free variables, and their related homogoneous solution has different orders of infinities as solutions.
For example, the matrix [[1,1][1,1]] is of a lower order than [[1,1,1][1,1,1][1,1,1]].
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wli2005-02-19 10:28
>>15
Finite Cartesian products of infinite sets have the cardinality of the largest set in the product. Dimension is very distinct from cardinality.
The proof is that the largest set among those in the finite product has a surjection covering each of the sets in the product (the identity for itself). So a finite product of copies of the largest of those sets has a surjection onto the product. The product also has a surjection onto the largest of those sets by a canonical projection. Then finite products of an infinite set fall to Zorn's lemma on the set of bijections from products of infinite subsets of to infinite subsets of the set, and Schroder-Bernstein determines the cardinality of the mixed product.
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Anonymous2005-02-23 15:22
there is a good way to visualize the number of rational numbers being equal to the natural numbers. imagine a grid of numbers, with 1 in the lower left corner. to the right of the 1, it goes 1/2, 1/3, 1/4, etc. above the one, it goes 2/1, 3/1, 4/1, etc. the other numbers are filled in, so up and right of 1 is 2/2. so if you continue this grid off forever, it will contain all positive rational numbers. you can get a negative version by just mirroring the thing diagonally down, so that the grid goes out in all 4 directions. to count them, start in the middle and spiral outwards.
i hope im making sense here...
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Anonymous2005-02-24 5:48
More like to count them, use the following (or any similar) method:
1-2 6-7
/ / /
3 5 8
|/ /
4 9
/
10
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Anonymous2008-01-04 17:44
FLOOD
Name:
Anonymous2008-01-04 18:06
I wasn't convinced about the countability of the rationals until I saw this pattern:
They are both countable, just the rationals are not well order with the operator (<)
Imagine it this way, if we define every even integer n to have "size" 1/n, and then ordered to integers according to this definition we have this list:
....10, 8, 6, 4, 2, 1, 3, 5, 7, 9.......
there is no least, and no greatest number, very much resembling the rationals.
There are ways of ordering the rationals such that there is a "least member" just that the ordering isn't using the operator (<)
The fact that they have equal cardinality is obvious consider the injection f from Q to Z
f(p/q)= 2^p.3^q
Name:
Anonymous2008-01-05 12:12
butt sex
Name:
Anonymous2008-01-05 17:18
[4:25] Rational numbers as a countable infinity
1 Name: MathMajor : 2004-12-27 13:40
This past semester I took the first halves of Introduction to Analysis and Modern (Abstract) Algebra in which, naturally, we covered the basics of the cardinalities of infinities. I understand the concepts of what makes something countable verses uncountable. I simply have an intuitive problem with the fact that the rational numbers are countable. The other basics countable infinite sets (Integers, Evens, Odds, Natural Numbers, etc.) all sit well with me. For some reason, I just have a problem accepting that the Rationals are truly countable. I understand that a definition is not arguable, but it just does not feel right.
Part of my discontent lies in the fact that like the continueum, there is no smallest positive rational number. This is not true for the previously mentioned sets. I feel that the rationals are somehow set apart from this set. If I recall correctly, it was proven that "It cannot be proven nor disproven that there exists an infinite set whose cardinality is greater than the cardinality of the countable infinite sets and less than the cardinality of the continueum (the reals)."
If anyone knows where I may be able to find more information on this, or even has an answer to my dilemma, please let me know. Thanks!
-Will
The 5 newest replies are shown below.
Read this thread from the beginning
21 Name: Anonymous : 2008-01-04 18:15
FLOOD
22 Name: Anonymous : 2008-01-05 01:18
But how are they less than the infinite complex numbers.
23 Name: Anonymous : 2008-01-05 01:59
old thread is old
24 Name: Anonymous : 2008-01-05 10:56
They are both countable, just the rationals are not well order with the operator (<)
Imagine it this way, if we define every even integer n to have "size" 1/n, and then ordered to integers according to this definition we have this list:
....10, 8, 6, 4, 2, 1, 3, 5, 7, 9.......
there is no least, and no greatest number, very much resembling the rationals.
There are ways of ordering the rationals such that there is a "least member" just that the ordering isn't using the operator (<)
The fact that they have equal cardinality is obvious consider the injection f from Q to Z
(Post truncated.)
25 Name: Anonymous : 2008-01-05 12:12
butt sex
Name:
Anonymous2008-01-06 23:39
>>1
This is why you don't prove math with feelings. The rationals are countable.
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Anonymous2013-05-12 5:30
countable means ordered by natural numbers
uncountable means ordered but not by natural numbers