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Provably Total Functions of Arithmetic with Basic Terms
A new characterization of provably recursive functions of first-order
arithmetic is described. Its main feature is using only terms consisting of 0,
the successor S and variables in the quantifier rules, namely, universal
elimination and existential introduction.Comment: In Proceedings DICE 2011, arXiv:1201.034
A proof of P!=NP
We show that it is provable in PA that there is an arithmetically definable
sequence of -sentences, such that
- PRA+ is -sound and
-complete
- the length of is bounded above by a polynomial function of
with positive leading coefficient
- PRA+ always proves 1-consistency of PRA+.
One has that the growth in logical strength is in some sense "as fast as
possible", manifested in the fact that the total general recursive functions
whose totality is asserted by the true -sentences in the sequence
are cofinal growth-rate-wise in the set of all total general recursive
functions. We then develop an argument which makes use of a sequence of
sentences constructed by an application of the diagonal lemma, which are
generalisations in a broad sense of Hugh Woodin's "Tower of Hanoi" construction
as outlined in his essay "Tower of Hanoi" in Chapter 18 of the anthology "Truth
in Mathematics". The argument establishes the result that it is provable in PA
that . We indicate how to pull the argument all the way down into
EFA
Do Goedel's incompleteness theorems set absolute limits on the ability of the brain to express and communicate mental concepts verifiably?
Classical interpretations of Goedel's formal reasoning imply that the truth
of some arithmetical propositions of any formal mathematical language, under
any interpretation, is essentially unverifiable. However, a language of
general, scientific, discourse cannot allow its mathematical propositions to be
interpreted ambiguously. Such a language must, therefore, define mathematical
truth verifiably. We consider a constructive interpretation of classical,
Tarskian, truth, and of Goedel's reasoning, under which any formal system of
Peano Arithmetic is verifiably complete. We show how some paradoxical concepts
of Quantum mechanics can be expressed, and interpreted, naturally under a
constructive definition of mathematical truth.Comment: 73 pages; this is an updated version of the NQ essay; an HTML version
is available at http://alixcomsi.com/Do_Goedel_incompleteness_theorems.ht
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