38 research outputs found
05021 Abstracts Collection -- Mathematics, Algorithms, Proofs
From 09.01.05 to 14.01.05, the Dagstuhl Seminar 05021 ``Mathematics, Algorithms, Proofs\u27\u27 was held in the International Conference and Research Center (IBFI),
Schloss Dagstuhl.
During the seminar, several participants presented their current
research, and ongoing work and open problems were discussed. Abstracts of
the presentations given during the seminar as well as abstracts of
seminar results and ideas are put together in this paper. The first section
describes the seminar topics and goals in general.
LinkstFo extended abstracts or full papers are provided, if available
Aspects of the constructive omega rule within automated deduction
In general, cut elimination holds for arithmetical systems with the w -rule, but not for systems with ordinary induction. Hence in the latter, there is the problem of generalisation, since arbitrary formulae can be cut in. This makes automatic theorem -proving very difficult. An important technique for investigating derivability in formal systems of arithmetic has been to embed such systems into semi- formal systems with the w -rule. This thesis describes the implementation of such a system. Moreover, an important application is presented in the form of a new method of generalisation by means of "guiding proofs" in the stronger system, which sometimes succeeds in producing proofs in the original system when other methods fail
Gödel mathematics versus Hilbert mathematics. I. The Gödel incompleteness (1931) statement: axiom or theorem?
The present first part about the eventual completeness of mathematics (called âHilbert mathematicsâ) is concentrated on the Gödel incompleteness (1931) statement: if it is an axiom rather than a theorem inferable from the axioms of (Peano) arithmetic, (ZFC) set theory, and propositional logic, this would pioneer the pathway to Hilbert mathematics. One of the main arguments that it is an axiom consists in the direct contradiction of the axiom of induction in arithmetic and the axiom of infinity in set theory. Thus, the pair of arithmetic and set are to be similar to Euclidean and non-Euclidean geometries distinguishably only by the Fifth postulate now, i.e. after replacing it and its negation correspondingly by the axiom of finiteness (induction) versus that of finiteness being idempotent negations to each other. Indeed, the axiom of choice, as far as it is equivalent to the well-ordering âtheoremâ, transforms any set in a well-ordering either necessarily finite according to the axiom of induction or also optionally infinite according to the axiom of infinity. So, the Gödel incompleteness statement relies on the logical contradiction of the axiom of induction and the axiom of infinity in the final analysis. Nonetheless, both can be considered as two idempotent versions of the same axiom (analogically to the Fifth postulate) and then unified after logicism and its inherent intensionality since the opposition of finiteness and infinity can be only extensional (i.e., relevant to the elements of any set rather than to the set by itself or its characteristic property being a proposition). So, the pathway for interpreting the Gödel incompleteness statement as an axiom and the originating from that assumption for âHilbert mathematicsâ accepting its negation is pioneered. A much wider context relevant to realizing the Gödel incompleteness statement as a metamathematical axiom is consistently built step by step. The horizon of Hilbert mathematics is the proper subject in the third part of the paper, and a reinterpretation of Gödelâs papers (1930; 1931) as an apology of logicism as the only consistent foundations of mathematics is the topic of the next second part
Gödel Mathematics Versus Hilbert Mathematics. II Logicism and Hilbert Mathematics, the Identification of Logic and Set Theory, and Gödelâs 'Completeness Paper' (1930)
The previous Part I of the paper discusses the option of the Gödel incompleteness statement (1931: whether âSatz VIâ or âSatz Xâ) to be an axiom due to the pair of the axiom of induction in arithmetic and the axiom of infinity in set theory after interpreting them as logical negations to each other. The present Part II considers the previous Gödelâs paper (1930) (and more precisely, the negation of âSatz VIIâ, or âthe completeness theoremâ) as a necessary condition for granting the Gödel incompleteness statement to be a theorem just as the statement itself, to be an axiom. Then, the âcompleteness paperâ can be interpreted as relevant to Hilbert mathematics, according to which mathematics and reality as well as arithmetic and set theory are rather entangled or complementary rather than mathematics to obey reality able only to create models of the latter. According to that, both papers (1930; 1931) can be seen as advocating Russellâs logicism or the intensional propositional logic versus both extensional arithmetic and set theory. Reconstructing history of philosophy, Aristotleâs logic and doctrine can be opposed to those of Plato or the pre-Socratic schools as establishing ontology or intensionality versus extensionality. Husserlâs phenomenology can be analogically realized including and particularly as philosophy of mathematics. One can identify propositional logic and set theory by virtue of Gödelâs completeness theorem (1930: âSatz VIIâ) and even both and arithmetic in the sense of the âcompactness theoremâ (1930: âSatz Xâ) therefore opposing the latter to the âincompleteness paperâ (1931). An approach identifying homomorphically propositional logic and set theory as the same structure of Boolean algebra, and arithmetic as the âhalfâ of it in a rigorous construction involving information and its unit of a bit. Propositional logic and set theory are correspondingly identified as the shared zero-order logic of the class of all first-order logics and the class at issue correspondingly. Then, quantum mechanics does not need any quantum logics, but only the relation of propositional logic, set theory, arithmetic, and information: rather a change of the attitude into more mathematical, philosophical, and speculative than physical, empirical and experimental. Hilbertâs epsilon calculus can be situated in the same framework of the relation of propositional logic and the class of all mathematical theories. The horizon of Part III investigating Hilbert mathematics (i.e. according to the Pythagorean viewpoint about the world as mathematical) versus Gödel mathematics (i.e. the usual understanding of mathematics as all mathematical models of the world external to it) is outlined
A primordial, mathematical, logical and computable, demonstration (proof) of the family of conjectures known as GoldbachÂŽs
licencia de Creative Commons Reconocimiento-NoComercial-SinObraDerivada 4.0 Internacional.In
this
document,
by
means
of
a
novel
system
model
and
first
order
topological,
algebraic
and
geometrical
free-Ââcontext
formal
language
(NT-ÂâFS&L),
first,
we
describe
a
new
signature
for
a
set
of
the
natural
numbers
that
is
rooted
in
an
intensional
inductive
de-Ââembedding
process
of
both,
the
tensorial
identities
of
the
known
as
ânatural
numbersâ,
and
the
abstract
framework
of
theirs
locus-Ââpositional
based
symbolic
representations.
Additionally,
we
describe
that
NT-ÂâFS&L
is
able
to:
i.-Ââ
Embed
the
De
MorganÂŽs
Laws
and
the
FOL-ÂâPeanoÂŽs
Arithmetic
Axiomatic.
ii.-Ââ
Provide
new
points
of
view
and
perspectives
about
the
succession,
precede
and
addition
operations
and
of
their
abstract,
topological,
algebraic,
analytic
geometrical,
computational
and
cognitive,
formal
representations.
Second,
by
means
of
the
inductive
apparatus
of
NT-ÂâFS&L,
we
proof
that
the
family
of
conjectures
known
as
Glodbachâs
holds
entailment
and
truth
when
the
reasoning
starts
from
the
consistent
and
finitary
axiomatic
system
herein
describedWe
wish
to
thank
the
Organic
Chemistry
Institute
of
the
Spanish
National
Research
Council
(IQOG/CSIC)
for
its
operative
and
technical
support
to
the
Pedro
Noheda
Research
Group
(PNRG).
We
also
thank
the
Institute
for
Physical
and
Information
Technologies
(ITETI/CSIC)
of
the
Spanish
National
Research
Council
for
their
hospitality.
We
also
thank
for
their
long
years
of
dedicated
and
kind
support
Dr.
Juan
MartĂnez
Armesto
(VATC/CSIC),
Belén
Cabrero
SuĂĄrez
(IQOG/CSIC,
Administration),
Mar
Caso
Neira
(IQOG/CENQUIOR/CSIC,
Library)
and
David
Herrero
RuĂz
(PNRG/IQOG/CSIC).
We
wish
to
thank
to
BernabĂ©-ÂâPajaresÂŽs
brothers
(Dr.
Manuel
BernabĂ©-ÂâPajares,
IQOG/CSIC
Structural
Chemistry
&
Biochemistry;
Magnetic
Nuclear
Resonance
and
Dr.
Alberto
Bernabé
Pajares
(Greek
Philology
and
Indo-ÂâEuropean
Linguistics/UCM),
for
their
kind
attention
during
numerous
and
kind
discussions
about
space,
time,
imaging
and
representation
of
knowledge,
language,
transcription
mistakes,
myths
and
humans
always
holding
us
familiar
illusion
and
passion
for
knowledge
and
intellectual
progress.
We
wish
to
thank
Dr.
Carlos
Cativiela
MarĂn
(ISQCH/UNIZAR)
for
his
encouragement
and
for
kind
listening
and
attention.
We
wish
to
thank
Miguel
Lorca
Melton
for
his
encouragement
and
professional
point
of
view
as
Patent
Attorney.
Last
but
not
least,
our
gratitude
to
Nati,
MarĂa
and
Jaime
for
the
time
borrowed
from
a
loving
husband
and
father.
Finally,
we
apologize
to
many
who
have
not
been
mentioned
today,
but
to
whom
we
are
grateful.
Finally,
let
us
point
out
that
we
specially
apologize
to
many
who
have
been
mentioned
herein
for
any
possible
misunderstanding
regarding
the
sense
and
intension
of
their
philosophic,
scientific
and/or
technical
hard
work
and
milestone
ideas;
we
hope
that
at
least
Goldbach,
Euler
and
Feymann
do
not
belong
to
this
last
humanÂŽs
collectivity.Peer reviewe
Antirealism and the Roles of Truth
Geschiedenis van Antieke en Middeleeuwse Semantie
Property Theories
Revised and reprinted; originally in Dov Gabbay & Franz Guenthner (eds.), Handbook of Philosophical Logic, Volume IV. Kluwer 133-251. -- Two sorts of property theory are distinguished, those dealing with intensional contexts property abstracts (infinitive and gerundive phrases) and proposition abstracts (âthatâ-clauses) and those dealing with predication (or instantiation) relations. The first is deemed to be epistemologically more primary, for âthe argument from intensional logicâ is perhaps the best argument for the existence of properties. This argument is presented in the course of discussing generality, quantifying-in, learnability, referential semantics, nominalism, conceptualism, realism, type-freedom, the first-order/higher-order controversy, names, indexicals, descriptions, Matesâ puzzle, and the paradox of analysis. Two first-order intensional logics are then formulated. Finally, fixed-point type-free theories of predication are discussed, especially their relation to the question whether properties may be identified with propositional functions