The singularity at the Schwarzschild radius befalls a boundary inside which
space and time do no longer exist in the standard sense. Our complex symmetric
framework reveals a black-hole-like structure, displaying precise quantum features.
One recognizes that the present configuration is uncharged due to the particleantiparticle superposition instigated by the transformation in Eq. (3.7) above, see
also Refs. [6, 7] for more details. Extensions to rotating black hole-like objects will
be discussed separately in Sect. 12, necessitating the incorporation of the Kerr
metric [8].
Although not explicitly indicated, the present operator array is equipped with an
abstract linear space, spanned by suitably defined vectors and their dual reciprocals,
see e.g. Löwdin [9] for details.
In passing we recognize a significant result regarding the likelihood of any size
limitations to our physical universe. The black-hole object derived above, imparts
that space-time have a finite lower bound given by the Schwarzschild radius. The
authentic space-time property is not defined inside this boundary, which means that
their conjugate partners, energy and momentum, must both be limited at infinity.
This conclusion follows naively from the commutation relationships between conjugate variables (operators). In the same way the energy and the momentum will
never evolve arbitrary close to zero and hence, mutatis mutandis, the spatio-temporal
“size” will also be correspondingly limited. This veracity prompts the conception of
a zero-energy universe in that the energy-momentum will be limited with most of its
energy positioned in a black-hole structure around “zero”. We will say more about it
below in connection with self-referential interactions and a reformulation of Gödel’s
theorem(s).
4 Gödel’s Theorem and the Law of Self-Reference
To begin with, it is imperative to know that we do not intend to advise a new
reading of the logical derivation of Gödel’s arguments in connection with the proof
of his first incompleteness theorem. In fact the interpretation of the mathematical
theorem(s) has been manifold as was recently stated by Feferman [10] in his
critique of the utilization of the results of Gödel [4] and Turing [11] in connection
with the positions of Artificial Intelligence, AI: it is hubris to think that by mathematics alone we can determine what the human mind can or cannot do. While we
will here argue that communication in biological systems starts already on the basic
level of molecular interactions within and between cellular aggregates, transfer of
information instigates on a built-in structure of mathematics, see e.g. Brändas [12],
we will recognize that Gödel’s paradox exhibits a consistent mathematical formulation within the logical framework of quantum mechanics. Consequently we
will belabour Penrose’s contention [13] that “physical action evoking awareness
cannot be properly simulated computationally” by simply formulating the Gödel
inconsistency theorem as a higher order singularity in linear algebra, i.e. as a Jordan
254
E.J. Brändas
space and time do no longer exist in the standard sense. Our complex symmetric
framework reveals a black-hole-like structure, displaying precise quantum features.
One recognizes that the present configuration is uncharged due to the particleantiparticle superposition instigated by the transformation in Eq. (3.7) above, see
also Refs. [6, 7] for more details. Extensions to rotating black hole-like objects will
be discussed separately in Sect. 12, necessitating the incorporation of the Kerr
metric [8].
Although not explicitly indicated, the present operator array is equipped with an
abstract linear space, spanned by suitably defined vectors and their dual reciprocals,
see e.g. Löwdin [9] for details.
In passing we recognize a significant result regarding the likelihood of any size
limitations to our physical universe. The black-hole object derived above, imparts
that space-time have a finite lower bound given by the Schwarzschild radius. The
authentic space-time property is not defined inside this boundary, which means that
their conjugate partners, energy and momentum, must both be limited at infinity.
This conclusion follows naively from the commutation relationships between conjugate variables (operators). In the same way the energy and the momentum will
never evolve arbitrary close to zero and hence, mutatis mutandis, the spatio-temporal
“size” will also be correspondingly limited. This veracity prompts the conception of
a zero-energy universe in that the energy-momentum will be limited with most of its
energy positioned in a black-hole structure around “zero”. We will say more about it
below in connection with self-referential interactions and a reformulation of Gödel’s
theorem(s).
4 Gödel’s Theorem and the Law of Self-Reference
To begin with, it is imperative to know that we do not intend to advise a new
reading of the logical derivation of Gödel’s arguments in connection with the proof
of his first incompleteness theorem. In fact the interpretation of the mathematical
theorem(s) has been manifold as was recently stated by Feferman [10] in his
critique of the utilization of the results of Gödel [4] and Turing [11] in connection
with the positions of Artificial Intelligence, AI: it is hubris to think that by mathematics alone we can determine what the human mind can or cannot do. While we
will here argue that communication in biological systems starts already on the basic
level of molecular interactions within and between cellular aggregates, transfer of
information instigates on a built-in structure of mathematics, see e.g. Brändas [12],
we will recognize that Gödel’s paradox exhibits a consistent mathematical formulation within the logical framework of quantum mechanics. Consequently we
will belabour Penrose’s contention [13] that “physical action evoking awareness
cannot be properly simulated computationally” by simply formulating the Gödel
inconsistency theorem as a higher order singularity in linear algebra, i.e. as a Jordan
254
E.J. Brändas