My Math Forum Set size is relative - Same topology of negative size sets

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 December 20th, 2013, 07:52 AM #1 Senior Member   Joined: Jan 2013 Posts: 209 Thanks: 3 Set size is relative - Same topology of negative size sets One way to view the topology of sets is by the Choose operator: x choose y = x!/(y!*(x-y)!) x choose y means to start with the factorial of how many ways you could order all items in the set, choose any y of them in a specific order, then divide by how many orders there are within that set of y, and divide by how many orders there are within those you did not choose (since by choosing something, you also exactly define all things which are not it). Pascals Triangle is a simple repeated calculation (a 1d cellular automata of integers) that defines each cell below 2 cells as their sum, and its shape is triangle instead of the usual square grid. Every cell in pascals triangle calculates an xChooseY for some and y, the x and y defined by this grid of 6 around 1: http://en.wikipedia.org/wiki/Star_of_David_theorem In each of the 7 cells of Star Of David Theorem, there is a relative xChooseY for variables x and y plus and minus 1 in 7 combinations, with x choose y at its center. The exact numbers are not whats important... What really surprised me is that the topology of sets is exactly the same shape for positive and negative size sets and is a type of relativity where you do not need to know the sizes of any specific sets as long as you know their sizes relative to eachother. That's what Star Of David Theorem proves. Sets are a flat 2d undirected-graph topology and are therefore 4-colorable http://simple.wikipedia.org/wiki/Four_color_theorem
December 21st, 2013, 12:36 PM   #2
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Re: Set size is relative - Same topology of negative size se

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 Originally Posted by BenFRayfield The exact numbers are not whats important... What really surprised me is that the topology of sets is exactly the same shape for positive and negative size sets ...
Ben, you were lucid there for a while. But you did not previously define negative size sets.

My understanding, which is the accepted modern view, is that if a set has a cardinality, the size is nonnegative. The empty set has cardinality zero, the set of fingers on my right hand has cardinality 5, the set of integers has cardinality aleph-null, etc. Cardinality is always greater than or equal to zero.

If you have some other measure of set size in mind that involves a sensible meaning of a negative set size, I'd be interested to know what it is.

As an aside, I was careful to write that if a set has a cardinality, then that cardinality is nonnegative. That's because in the absence of the Axiom of Choice, there exist sets that do not have any cardinality at all! That's because AC is equivalent to the (non-intuitive) proposition that every set can be well-ordered. In the absence of Choice there exists a set that can not be well-ordered. Now, the cardinality of a set is defined as the least ordinal x such that x is bijectable to the set. Since our set is not well-ordered, no ordinal bijects to it hence it does not have a cardinality.

I offer this as a counterpoint to all the people who use the Banach-Tarski paradox or the Hat paradox (http://cornellmath.wordpress.com/2007/0 ... -is-wrong/) to claim that AC is false or that math is broken or some such. It is absolutely true that AC implies some very counterintuitive and arguably false conclusions; but so does the negation of AC.

I hope someone who reads this someday is spared the trouble of getting upset about the Axiom of Choice. It's obviously true about sets. The notion you have to let go of is that set theory has anything to do with the real world.

December 21st, 2013, 05:54 PM   #3
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Re: Set size is relative - Same topology of negative size se

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 Originally Posted by Maschke I hope someone who reads this someday is spared the trouble of getting upset about the Axiom of Choice. It's obviously true about sets. The notion you have to let go of is that set theory has anything to do with the real world.
In that case, why not accept the negation of the Axiom of Choice?

December 21st, 2013, 10:40 PM   #4
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Re: Set size is relative - Same topology of negative size se

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 Originally Posted by CRGreathouse In that case, why not accept the negation of the Axiom of Choice?
I find the idea of a set without a well-defined cardinality very counterintuitive. That if you take all the cardinals and line them up in order, there's some set that's not cardinally equivalent to any of them. How can that be?

December 21st, 2013, 10:58 PM   #5
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Re: Set size is relative - Same topology of negative size se

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 Originally Posted by Maschke I find the idea of a set without a well-defined cardinality very counterintuitive. That if you take all the cardinals and line them up in order, there's some set that's not cardinally equivalent to any of them. How can that be?
"Take the set X and extract a subset C. Note that there is x in X such that there is no c in C with c ~ x."

Doesn't seem that surprising, a priori, to me.

--

I should say that, for the record, I don't have a serious problem with AC. The axiom I don't like is regularity/foundation.

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