On 18 Jan 2015, at 15:55, Stathis Papaioannou wrote:
On 18 January 2015 at 18:43, Jason Resch <[email protected]> wrote:
On Sun, Jan 18, 2015 at 1:12 AM, meekerdb <[email protected]>
wrote:
On 1/17/2015 9:27 PM, Jason Resch wrote:
Do you believe that one and only one of the following statements
is true?
the 10^(10^(10^100))th decimal digit of pi is 0
the 10^(10^(10^100))th decimal digit of pi is 1
the 10^(10^(10^100))th decimal digit of pi is 2
the 10^(10^(10^100))th decimal digit of pi is 3
the 10^(10^(10^100))th decimal digit of pi is 4
the 10^(10^(10^100))th decimal digit of pi is 5
the 10^(10^(10^100))th decimal digit of pi is 6
the 10^(10^(10^100))th decimal digit of pi is 7
the 10^(10^(10^100))th decimal digit of pi is 8
the 10^(10^(10^100))th decimal digit of pi is 9
Either you answer yes, or no to that question. If you answer yes,
I don't
see how you can escape mathematical realism.
By observing that "real" and "true" are different attributes.
True is enough to yield reality, under computationalism.
Isn't that circular?
It would be circular if existence was some property. But existence is
defined by belonging to the intended model I am studying through my
theory.
We cannot derive that something exists without assuming that something
else exists. This should not forbid the physical universe to emerge
from the quantum vacuum, as the quantum vacuum, whatever it is, is
already provably sigma_1 complete. It assumes already something Turing-
equivalent to a Turing universal machine.
The arithmetical truth, which can be represented by the set of all
true arithmetical sentences, or by the knower of all solutions and non
solutions of all diophantine equations (or just the universal one I
gave) does determined a the arithmetical reality, which contains all
the possible machine's relative state. The soul or first person
associated to those machines cannot associate themselves to machines,
but only to infinities of machines, with a blurred reality below their
substitution level (the comp "quantum"). That reality does not need a
priori to be arithmetical.
All this needs only what Jason said, and what I said to Brent.
Existence of a number follows from the truth of the existential
ExP(x). The truth of "ExP(x)" entails the existence (reality) of a n
such that P(n).
ExP(x) can be seen as an infinite disjunction. It means P(0) v P(1) v
P(2) v P(3) v ..., like
AxP(x) can be seen as an infinite conjunction. It means P(0) & P(1) &
P(2) & P(3) & ...
Like in modal logic, E and A are dual, you can define ExP(x) by ~Ax
~P(x), like you could define AxP(x) by
~Ex ~P(x).
Using first order logic, or second order logic (which quantify on
infinite sets) with some caution, we can effectively reason without
any implicit metaphysical assumption, or ontological commitment. All
the carts are put on the table.
Bruno
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Stathis Papaioannou
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