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Completeness: From Henkin's Proposition to Quantum Computer
The paper addresses Leon Henkin's proposition as a "lighthouse",
which can elucidate a vast territory of knowledge uniformly: logic, set theory,
information theory, and quantum mechanics: Two strategies to infinity are
equally relevant for it is as universal and thus complete as open and thus incomplete.
Henkin's, Godel's, Robert Jeroslow's, and Hartley Rogers'
proposition are reformulated so that both completeness and incompleteness to
be unified and thus reduced as a joint property of infinity and of all infinite sets.
However, only Henkin's proposition equivalent to an internal position to
infinity is consistent . This can be retraced back to set theory and its axioms,
where tha t of choice is a key. Quantum mechanics is forced to introduce infinity implicitly by Hilbert space, on which is founded its formalism. One can
demonstrate that some essential properties of quantum information,
entanglement, and quantum computer originate directly from infinity once it is
involved in quantum mechanics. Thus, these phenomena can be elucidated as
both complete and incomplete, after which choice is the border between them.
A special kind of invariance to the axiom of choice shared by quantum
mechanics is discussed to be involved that border between the completeness
and incompleteness of infinity in a consistent way. The so-called paradox of
Albert Einstein, Boris Podolsky, and Nathan Rosen is interpreted entirely in
the same terms only of set theory. Quantum computer can demonstrate
especially clearly the privilege of the internal position, or "observer'' , or "user" to infinity implied by Henkin's proposition as the only consist ent ones as to infinity. An essential area of contemporary knowledge may be synthesized from a single viewpoint.
which can elucidate a vast territory of knowledge uniformly: logic, set theory,
information theory, and quantum mechanics: Two strategies to infinity are
equally relevant for it is as universal and thus complete as open and thus incomplete.
Henkin's, Godel's, Robert Jeroslow's, and Hartley Rogers'
proposition are reformulated so that both completeness and incompleteness to
be unified and thus reduced as a joint property of infinity and of all infinite sets.
However, only Henkin's proposition equivalent to an internal position to
infinity is consistent . This can be retraced back to set theory and its axioms,
where tha t of choice is a key. Quantum mechanics is forced to introduce infinity implicitly by Hilbert space, on which is founded its formalism. One can
demonstrate that some essential properties of quantum information,
entanglement, and quantum computer originate directly from infinity once it is
involved in quantum mechanics. Thus, these phenomena can be elucidated as
both complete and incomplete, after which choice is the border between them.
A special kind of invariance to the axiom of choice shared by quantum
mechanics is discussed to be involved that border between the completeness
and incompleteness of infinity in a consistent way. The so-called paradox of
Albert Einstein, Boris Podolsky, and Nathan Rosen is interpreted entirely in
the same terms only of set theory. Quantum computer can demonstrate
especially clearly the privilege of the internal position, or "observer'' , or "user" to infinity implied by Henkin's proposition as the only consist ent ones as to infinity. An essential area of contemporary knowledge may be synthesized from a single viewpoint.
