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Anonymous
A proof for infinite sets that contain more elements than others? 2016-09-15 04:28:52 Post No. 8346311
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A proof for infinite sets that contain more elements than others? 2016-09-15 04:28:52 Post No. 8346311
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A proof for infinite sets that contain more elements than others?
Anonymous
2016-09-15 04:28:52
Post No. 8346311
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So, /sci/, I want you guys to consider this.
1. There exists sets X and Y
2. There exists a number x, x is an element of X{R|1<x<2}
4. There exists a number y, y is an element of Y and is an element of {R|1<=y<=2}
5. X is a subset of Y
6. ∴X implies Y AND x implies y
7. ∴ 1 and 2X,1 and 2 Y
8. 1 and 2 are NOT an element of X, and ARE an element of Y AND 1 and 2 are elements of Z
10. Set Y contains exactly two more elements than set X.
11. ∴Y>X
What do you think, /sci/? Is it proof of different levels of infinity, or that infinity plus two is not infinity? Am I retarded? Could my proof be revised? Am I doing something wrong? I talked about this with my calculus prof. Also, how do I do this using the LaTeX? If anyone could do that, that'd be nice. But, my reasoning in plain words:
>Think of two sets
>In the first set, are all real numbers greater than 1, less than 2.
>In the second, are all real numbers from 1 to two.
>The second set has every element that A has.
>The second set has two more elements, 1 and 2.
>Therefore the second set has more elements.
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>A proof for infinite sets that contain more elements than others?
[math]\mathbb{N}\subsetneq \mathbb{N}\cup\{\mbox{banana}\}[/math].
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>>8346328
Fuck off latex piece of shit. This worked fine in the preview.
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>>8346334
banana
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>>8346334
try switching browsers; it doesn't seem to work on safari
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huh
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>>8346311
This is not how levels of infinity are defined. If you think about it, it's impossible to compare two disjoint sets this way. Is [0, 1] the same size as [2, 3]? What about [0, 1] and [2, 4]? It's impossible to tell with your approach. This is why we define it using bijections: two sets are the same size iff there is a one-to-one correspondence between the two. There actually is a one-to-one correspondence between your two sets (even though it can't be written down explicitly), so we say they are the same size.
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>>8346867
thanks, for your input, anon.
Also, I think [0,1] is the same size as [2,3].
[0,1] is smaller than [2,4], but it only makes sense to compare them when talking about transfinites.
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Definition of cardinality of sets defined to serve a purpose. In this sense all whole numbers and all real numbers are "equally big". But the thing is you cannot write a bijection from Z to N which preserves "distance".
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>>8348309
>you cannot write a bijection from Z to N
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>>8348313
We can write A LOT of bijections actually.
For example: 0 goes to 0, 1 goes to -1, 2 goes to 1, 3 goes to -2, 4 goes to 2 and so on.
Check out Hilbert's Hotel and other new concepts which will come up when reading about Hilbert's Hotel.
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