Monday, June 13, 2011

something from nothing.

"There were others who had forced their way to the top from the lowest rung by the aid of their bootstraps."

Von Neumann proposed that all numbers could be bootstrapped out of the empty set by the operations of the mind:

The mind observes the empty set. The mind's act of observation causes the appearance another set - the set of empty sets. The set of empty sets is not empty, because it contains one non-thing - the empty set. The mind has thus generated the number 1 by producing the set containing the empty set.

Now the mind perceives the empty set and the set containing the empty set, so there are two non-things. The mind has generated the number 2 out of emptiness. And so it goes on all the way up.

Stated stepwise the operation is:

STEP 0 {} Empty set has 0 parts.

STEP 1 { {} } Mind generates set containing empty set, which has one part.

STEP 2 { {}, { {} }} Mind generates set containing two parts, the set and the set containing the empty set.

etc, etc

So the entire number system, from zero to infinity, is generated purely and simply by the play of mind on emptiness, in the complete absence of the need to refer to any material objects being counted.

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