On 03 Sep 2012, at 14:50, Richard Ruquist wrote:
In comp, what is the function of god.
It is responsible for the existence of numbers and their relations,
notably in distinguishing what is true and false.
My hope is that the function of a god
might be to reduce 3p tp 1p.
It does exactly that. Both:
- informally, as making true the statement "I am reconstituted in
Moscow" in the case I am reconstituted in Moscow, and perhaps else
- and formally (or meta-formally) when, following Theaetetus, we
define knowledge as the conjunction of provability (ideal machine's
believability), and truth (like in Knowable('p') = provable('p') & p).
This gives God (Truth) a mean to, well, not exactly reducing, but
"awakening" the 1p, from the 3p. It makes the first person as
unnameable as God/Truth.
Everything else seems to be capable
of running according to algorithms.
In the hierarchy of complexity, what is computable is at the Sigma_0
and sigma_1 arithmetical level, but the sigma_2 is no more computable,
nor is any Sigma_n level for n bigger than 1.
Arithmetical truth is maximally non computable as being a union of all
Just to say that the computable part of the arithmetical truth is very
And the first person indeterminacy can be used to explain why the
average universal number is confronted to the whole hierarchy, and
actually even beyond, epistemologically.
Is there anything in comp
that is non-algorithmic?
The search for the truth of arithmetical sentences which are more
complex than the sigma_1 one.
By a theorem of Kleene and Mostowski, the sigma_1 sentences can be
roughly described by the sentences having the shape ExP(x) with P(x)
decidable. Their negation are already not computable and are called
Pi_1, they have the shape AxP(x). Example: Riemann hypothesis (this is
equivalent with a P1_1 arithmetical sentence, as shown by Turing).
Then you have the Sigma_2 and Pi_2, with the shape ExAyP(x,y) P
decidable, and AxEyP(x,y) respectively. Etc.
Most truth about numbers and machines are not algorithmic (we assume
Church Thesis 'course).
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