On 2/25/2019 8:55 AM, Bruno Marchal wrote:
Fictionalism does not apply to the arithmetical reality, nor to
physics, but to the naïve idea of a “physical universe” as being the
fundamental reality. The theology of the universal machine is a priori
quite non Aristotelian: there is no Creator, and there is no Creation.
Just a universal dreamer which lost itself in an infinitely surprising
structure and wake up from time to time, or from numbers to numbers.
There is according to St Anselm, who also thought that definitions bring
things into existence.
Brent
I need no more than a partial applicative algebra, and each choice of
the phi_i makes N into one, simply by defining an operation “*” in N
such that n * m = phi_n(m). There exist numbers k and s such that
((k * n) * m) = n
(((s * n) * m) * r) = (n * r) * (m * r),
for all m, n, r in N.
And, the key point, the operation “*” can be defined in the
arithmetical language, and those statements are, for each n, m, r,
provable in RA. I have shown that the converse is true. It is a very
elegant Turing complete theory. With Indexical Digital Mechanism, it
is absolutely undecidable if the Universe is bigger than the sigma_1
reality. (But here I do a blasphemy: that can only be entirely
justified by G* *only*!, It is where I have to insist that this is
presented as a consequence of YD + CT (“yes doctor” + Church-Turing
thesis).
Such theories are essentially undecidable. It means that not only they
are arithmetically incomplete, but all their effective consistent
extensions are too. They are creative, you cannot capture the semantic
in the way it could become complete, even in some imaginary domain
concevable by the machine/theory/number. The universal machine are
never entirely satisfied and a computation is always an escape
forward, but their self-reflection create a mess, and illusions.
The sigma_1 arithmetical reality, as seen by the universal numbers
which lives there, in the first person undetermined sense, is
something *very big*. It generates infinitely many surprises. There
are consistent histories.
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