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# 086

With Great Power... - November 17, 2009
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Thanks to my buddy Joe for coming up with the phrase! Although his was:

"With great power sets, comes great redundancy."

We tried to think of other words besides redundancy that sounded like 'responsibility' and fit the theme of power sets... reachability? There's gotta be a better one than my made up subset-ability word lolz.

I don't much like the flow, but "great cardinality" wou;d be a conceptual fit.

Cardinality.
(In-)Constructibility.
http://en.wikipedia.org/wiki/Axiom_of_constructibility
Ineffability.
http://en.wikipedia.org/wiki/Ineffable_cardinal
Inaccessibility.
http://en.wikipedia.org/wiki/Inaccessible_cardinal

In fact, the area of large cardinals has plenty of "-ility"-ies.

...That's the catchphrase of old uncle Ben.
If you missed it don't worry, they'll say the line
Again and again and again!

I just can't hear that line without thinking of the song!xD

Gotta go, gotta go song...

Otherwise known as the P song...

Great Reflexibility?

If the number of elements in a power set is defined as:
|P(S)| = 2^n where S is a finite set of n elements

Then since the number of reflexive relations in a finite set of n elements is 2^(n^2 - n)
then the number of reflexive relations in a power set is:
2^( |P(S)|^2 - |P(S)| )
= 2^( (2^n)^2 - 2^n )

... which is pretty great.

I like Evan's idea, so much better than any of mine were...

Expanding 2^((2^n)^2 - 2^n):
=2^((2^n)(2^n-1))
=2^(2*(2^(n-1))(2^n-1))
but recall (2^(n-1))*(2^n-1) is perfect if 2^n-1 is prime.

Cardinality. Of course large cardinal theory has additional suitable terms: ineffability, inaccessibility, et c.

I always hate it when I am ineffable...

reducibility

"Uncountability"?

If only the original were "...accountability", that would work _so well_. Unfortunately, it isn't, but still...

Great comic, by the way! I just found it for the first time, via Abstruse Goose.

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