40,696 research outputs found
Open questions about Ramsey-type statements in reverse mathematics
Ramsey's theorem states that for any coloring of the n-element subsets of N
with finitely many colors, there is an infinite set H such that all n-element
subsets of H have the same color. The strength of consequences of Ramsey's
theorem has been extensively studied in reverse mathematics and under various
reducibilities, namely, computable reducibility and uniform reducibility. Our
understanding of the combinatorics of Ramsey's theorem and its consequences has
been greatly improved over the past decades. In this paper, we state some
questions which naturally arose during this study. The inability to answer
those questions reveals some gaps in our understanding of the combinatorics of
Ramsey's theorem.Comment: 15 page
On what I do not understand (and have something to say): Part I
This is a non-standard paper, containing some problems in set theory I have
in various degrees been interested in. Sometimes with a discussion on what I
have to say; sometimes, of what makes them interesting to me, sometimes the
problems are presented with a discussion of how I have tried to solve them, and
sometimes with failed tries, anecdote and opinion. So the discussion is quite
personal, in other words, egocentric and somewhat accidental. As we discuss
many problems, history and side references are erratic, usually kept at a
minimum (``see ... '' means: see the references there and possibly the paper
itself).
The base were lectures in Rutgers Fall'97 and reflect my knowledge then. The
other half, concentrating on model theory, will subsequently appear
Logical Dreams
We discuss the past and future of set theory, axiom systems and independence
results. We deal in particular with cardinal arithmetic
Intuition, iteration, induction
In Mathematical Thought and Its Objects, Charles Parsons argues that our
knowledge of the iterability of functions on the natural numbers and of the
validity of complete induction is not intuitive knowledge; Brouwer disagrees on
both counts. I will compare Parsons' argument with Brouwer's and defend the
latter. I will not argue that Parsons is wrong once his own conception of
intuition is granted, as I do not think that that is the case. But I will try
to make two points: (1) Using elements from Husserl and from Brouwer, Brouwer's
claims can be justified in more detail than he has done; (2) There are certain
elements in Parsons' discussion that, when developed further, would lead to
Brouwer's notion thus analysed, or at least something relevantly similar to it.
(This version contains a postscript of May, 2015.)Comment: Elaboration of a presentation on December 5, 2013 at `Intuition and
Reason: International Conference on the Work of Charles Parsons', Van Leer
Jerusalem Institute, Jerusale
Fuzzy inequational logic
We present a logic for reasoning about graded inequalities which generalizes
the ordinary inequational logic used in universal algebra. The logic deals with
atomic predicate formulas of the form of inequalities between terms and
formalizes their semantic entailment and provability in graded setting which
allows to draw partially true conclusions from partially true assumptions. We
follow the Pavelka approach and define general degrees of semantic entailment
and provability using complete residuated lattices as structures of truth
degrees. We prove the logic is Pavelka-style complete. Furthermore, we present
a logic for reasoning about graded if-then rules which is obtained as
particular case of the general result
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