# Complete & Consistent

How familiar are you with nostalpogy?

Not at all?  Yeah, me neither.

it does not exist (at least to my knowledge as of 10 FEB 2017).  So, whatever it is that nosalpogy represents, it is something of which I cannot conceptualize.  Moreover, I’m incapable of conceptualizing it.  If no person can elucidate what nosalpogy is , if no one can help me see ‘it’ against the setting of everything else, then nosalpogy is nothing.

Get thinking about Russell & Whitehead’s attempt to derive all of mathematics from purely logical axioms and remember how Godel’s Sentence G (just one example).

Russell & Whitehead wanted to irrefutably prove that a consistent system based on a few simple assumptions (aka axioms), whose theorems can be listed by an effective procedure (i.e., an algorithm), is capable of proving all truths about the arithmetic of the natural numbers.

Well, they failed to achieve that goal, but that failure brought its own success and furthered theoretical mathematics. Godel demonstrated, for any such formal system, such as the proposed one of Russell & Whitehead,  there will always be statements about the natural numbers that are true, but that are unprovable within the system. Godel then provided proof that the system cannot demonstrate its own consistency.

To give the gist without the jargon– I imagine a  tube with 3 tennis balls inside.  Now, imagine you have 3 box each filled with 10 of these tubes, each containing three balls.  Each tube contains a set of three balls.  Each box contains a set of 10 tubes; another way to say this is, each box contains a set of 30 balls.  So a set of 3 boxes is a set of 90 balls or a set of 30 tubes.

Imagine I am shipping out boxes of tennis balls.  On each shipping pallatte, a set of 4 boxes, each containing three boxes of tennis balls, can be packed  That means a pallatte contains a set of 360 tennis balls which is equal to a set of 90 tubes which is equal to a set of of 12 boxes.  The pallatte can also hold a set of 4 boxes each holding 3 boxes.

The point is, I can define a set of tennis balls many ways.  I can also imagine a set of sets of tennis balls (a box = 10 tubes and 10 tubes = 30 balls).  A box is a set of tubes and a set of tubes is a set of tennis balls.

So if I can imagine of box of tubes containing tennis balls; and, if I can imagine a box that contains several boxes of tubes of tennis balls, and so on…at what point do hit the top?  At what point do I reach the highest possible set?  Never.  I can always conceive of one more box around boxes just as I cannot name the highest number-I can always imagine one more.

Apologies-work in progress-researching underway.