On 8/29/2012 10:52 AM, meekerdb wrote:
On 8/29/2012 5:18 AM, Stephen P. King wrote:
On 8/29/2012 2:17 AM, meekerdb wrote:
On 8/28/2012 11:08 PM, Quentin Anciaux wrote:
Until there is a precise explanation of what this phrase
"generation by the UD" might mean, we have just a repeated
meaningless combinations of letters appearing on our computer monitors.
Seems pretty precise to me. The UD executes all possible
computations, one step at a time. If 'you' are a computation, then
it must eventually generate you.
Yes it will "eventually" generate me, but with a measure zero
chance. The UD seems to be ergodic on the Integers.
Not zero, only zero in the limit of completing the infinite
computations. So at any stage short the infinite completion the
probability of "you" is very small, but non-zero. But we already knew
I agree but the details of this are being crudely glossed over and
they are of utmost importance here! We need a precise definition of the
"at any stage short of the infinite completion" term. I suspect that we
can capture this using the uncountable infinity of non-standard models
relations between the models to give us a nice formal model.
"The existence of non-standard models of arithmetic can be
demonstrated by an application of the compactness theorem. To do this, a
set of axioms P* is defined in a language including the language of
Peano arithmetic together with a new constant symbol x. The axioms
consist of the axioms of Peano arithmetic P together with another
infinite set of axioms: for each numeral n, the axiom x > n is included.
Any finite subset of these axioms is satisfied by a model which is the
standard model of arithmetic plus the constant x interpreted as some
number larger than any numeral mentioned in the finite subset of P*.
Thus by the compactness theorem there is a model satisfying all the
axioms P*. Since any model of P* is a model of P (since a model of a set
of axioms is obviously also a model of any subset of that set of
axioms), we have that our extended model is also a model of the Peano
axioms. /The element of this model corresponding to x cannot be a
standard number, because as indicated it is larger than any standard
The x would play the role of the inverse of the epsilon of
proximity to infinite completion.
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