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So, disclaimer: been a while since I did analysis, and this was a random shower thought (yes, I was taking a shower lol). So, half of this is likely badly flawed in some way lol.
So, first we begin with Cantor's diagonal argument. If you do not know what it is, go to http://en.wikipedia.org/wiki/Cantor's_diagonal_argument
We will rework the terms a bit. Namely, we will establish try to establish a 1:1 between the indices and the natural numbers as follows;
#1: 1 0 0 0 0 0 0 0 0 0 0 0 0 0..... 1 in binary
#2: 0 1 0 0 0 0 0 0 0 0 0 0 0 0..... 2 in binary
#3" 1 1 0 0 0 0 0 0 0 0 0 0 0 0..... 3 in binary
etc etc where we were left-to-right least significant to most significant digits, using binary. We generate a sequence for every natural number this way.
So now we can interpret the infinite sequences as numbers in the standard way. Now do the negation (for sequence #n, flip the nth bit), so that we get 0 0 1......
One might be tempted to think we just generated a natural number which is not a natural number - but wait, how can the natural numbers not be one-to-one with the natural numbers, that is, how can a countable set be uncountable?
Well, actually it is apparent we didn't generate a natural number even though we are flipping a countable infinite number of bits, all of which belong to natural numbers. There is clearly no successor to the sequence we generated - there's no next number. So actually, we didn't generate a natural number even though we meant to do so.
I actually stood there in the shower for like 10 minutes thinking that the generated number was a natural number loool. Hot water bad for thinking I guess o.o
P.S. Infinities are awesome, and I don't just mean the pen ones.
P.P.S. I don't suppose there is a way to do something similar with the ordinals somehow used as indices rather than powers of 2?
 
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I come up with all my ideas in the shower. not even like just joke ideas, just all of the ideas that I have, they are all brewed in the shower.
I cannot assess your binary stufz though. 2hard.
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When I'm in the shower I sing some Taylor Swift.
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On January 27 2012 09:30 Jonoman92 wrote:
When I'm in the shower I sing some Taylor Swift.
Too bad you are not as cute :/
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What you constructed is a valid binary string, but not a binary representation of a natural number. All binary representations of natural numbers e = e_1,e_2,... have some index 'n' such that for all k>n, e_k = 0, where yours was an infinite sequence of 1's after the 2nd index. ^^
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On January 27 2012 10:09 ShinyGerbil wrote:
What you constructed is a valid binary string, but not a binary representation of a natural number. All binary representations of natural numbers e = e_1,e_2,... have some index 'n' such that for all k>n, e_k = 0, where yours was an infinite sequence of 1's after the 2nd index. ^^
err the ... didn't signify continuing 1s. I just didnt bother to write the flipped bits of sequence #4, #5, etc. And also what you wrote about the indicies is what I meant by there is no successor so it's not actually a natural number.
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On January 27 2012 10:21 EtherealDeath wrote:Show nested quote +On January 27 2012 10:09 ShinyGerbil wrote:
What you constructed is a valid binary string, but not a binary representation of a natural number. All binary representations of natural numbers e = e_1,e_2,... have some index 'n' such that for all k>n, e_k = 0, where yours was an infinite sequence of 1's after the 2nd index. ^^ err the ... didn't signify continuing 1s. I just didnt bother to write the flipped bits of sequence #4, #5, etc. And also what you wrote about the indicies is what I meant by there is no successor so it's not actually a natural number.
You missed his point. The reason that almost everything on that list doesn't represent a natural number in binary is that there's going to be infinitely many 1s and infinitely many 0s interleaved in some way. A binary number only has finitely many 1s in its expansion.
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I'm curious; can you explain this in words that a grade 11 guy who's terrible at math can understand?
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Me too. Shower is such a nice gound to breed ideas.
Maybe its the water, or just the rest from sleep.
On topic. sorry math hates me, so I'll just read what people share here.
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Daydreaming impossible scenarios in the shower is real fun. I can meet any fictional characters or go skiing at any time of year. I swear I've probably had more adventures in my shower than in real life.
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But you're counting with integers.... that means 4 ascends 3, and 4 is equal to 0100 (not sure why you flipped the bits backwards) regardless of how many 0's you put - they don't change the value... or maybe I am missing your point?
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On January 27 2012 09:30 Jonoman92 wrote:
When I'm in the shower I sing some Taylor Swift.
You understand me <3
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On January 27 2012 12:57 Ninja_Bread wrote:
But you're counting with integers.... that means 4 ascends 3, and 4 is equal to 0100 (not sure why you flipped the bits backwards) regardless of how many 0's you put - they don't change the value... or maybe I am missing your point?
This seems to sum it up for me.
By his logic, I could write 9, but because it's not 09 it leaves no room for ascention?
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On January 27 2012 12:57 Ninja_Bread wrote:
But you're counting with integers.... that means 4 ascends 3, and 4 is equal to 0100 (not sure why you flipped the bits backwards) regardless of how many 0's you put - they don't change the value... or maybe I am missing your point?
The bits were flipped backwards because otherwise for any specific number, you'd need an infinite number of 0's to the left, which is a bit weird to write and is rather nonstandard notation.
i.e.
#1 = ....00000000001 instead of #1 = 100000000.....
On January 27 2012 12:26 Iranon wrote:Show nested quote +On January 27 2012 10:21 EtherealDeath wrote:On January 27 2012 10:09 ShinyGerbil wrote:
What you constructed is a valid binary string, but not a binary representation of a natural number. All binary representations of natural numbers e = e_1,e_2,... have some index 'n' such that for all k>n, e_k = 0, where yours was an infinite sequence of 1's after the 2nd index. ^^ err the ... didn't signify continuing 1s. I just didnt bother to write the flipped bits of sequence #4, #5, etc. And also what you wrote about the indicies is what I meant by there is no successor so it's not actually a natural number. You missed his point. The reason that almost everything on that list doesn't represent a natural number in binary is that there's going to be infinitely many 1s and infinitely many 0s interleaved in some way. A binary number only has finitely many 1s in its expansion.
Actually our statements are equivalent. The statement that there can be no successor implies there is no final 1, and vice versa.
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United States4053 Posts
i think what you have stumbled upon is something called a p-adic number (in this case, a 2-adic)
i don't know a whole lot about them, but the number you generate ends up being 001011111111.... which is equal to -12 or something like that.
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On January 27 2012 14:40 Cyber_Cheese wrote:Show nested quote +On January 27 2012 12:57 Ninja_Bread wrote:
But you're counting with integers.... that means 4 ascends 3, and 4 is equal to 0100 (not sure why you flipped the bits backwards) regardless of how many 0's you put - they don't change the value... or maybe I am missing your point? This seems to sum it up for me. By his logic, I could write 9, but because it's not 09 it leaves no room for ascention?
I implied an infinite succession, namely
#1 = 1000000..... (Binary 1)
#2 = 0100000..... (Binary 2)
#3 = 1100000..... (Binary 3)
#4 = 0010000..... (Binary 4)
#5 = 1010000..... (Binary 5)
...
#n = (Binary n)
So one would be inclined to think that we have enumerated every possible infinite binary string - thus the total number of possible binary strings of undetermined length is infinite, but countably infinite. By countably infinite I mean we can put them into a 1-1 correspondence with the natural numbers - pair them up exactly one for one by some clever scheme.
However, this isn't the case. Above, we have clearly assigned every natural number an infinite binary string. However, we will now create a binary string which clearly isn't any of the strings above. We do this by flipping bits. For sequence #x, we flip the xth bit. Thus sequence #x is not the same as sequence #1, or #2, #3, etc no matter what sequence #n we use since whatever sequence we compare, one of the bits has been flipped.
So there is no one-to-one correspondence between the infinite binary strings and the natural numbers.
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On January 27 2012 15:14 infinitestory wrote:
i think what you have stumbled upon is something called a p-adic number (in this case, a 2-adic)
i don't know a whole lot about them, but the number you generate ends up being 001011111111.... which is equal to -12 or something like that.
Huh? I don't see the connection o.O
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I couldn't find you day 2 so it sat in my trunk t.t It actually sat there for 2 months...
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It's unclear what you're trying to show. Could you tell us?
It feels like you're trying to show that the set of infinitely long binary strings is uncountable? If so, you can just use Cantor's diagonalization argument instead of the awkward procedure you used in your first post.
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EPT
