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The
seminar is held in hybrid
format, in person (Múzeum
krt. 4/i Room 224) and
online.
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| May
5 (Friday) 4:15 PM Room
224 + ONLINE |
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Katarina
Maksimović
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Department of
Philosophy, Faculty of
Philosophy, University of
Belgrade
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We
assert, we deduce and we
calculate… But what does it
all mean?
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This
talk will focus on the problem of
meaning in logic and mathematics,
exploring the meaning of certain
concepts and the concept of
meaning in general. The
presentation will begin by
introducing two semantic
approaches: the extensional
approach and the intensional
approach. The first one has been
the cornerstone of set theory,
model theory and classical logic,
while the other has been largely
overlooked in modern times,
despite having very successful
applications in proof-theoretic
semantics, categorial proof
theory, and lambda calculus. After
briefly explaining these
approaches, I will argue that the
intensional approach offers a more
suitable means of defining both
proof identity criteria and
algorithm identity criteria.
Toward the end of the talk, the
relationship between the concept
of deduction and the concept of
calculation will be discussed in
connection with the Church-Turing
thesis.
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| May 19 (Friday) 4:15 PM Room
224 + ONLINE |
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Amitayu Banerjee
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Rényi Institute of Mathematics, Budapest
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On a variant of Erdős--Dushnik--Miller theorem
without the Axiom of Choice (AC). |
Theories: ZFC (Zermelo–Fraenkel set theory with the Axiom of Choice (AC)), ZF (Zermelo–Fraenkel set theory without AC), ZFA (ZF with the Axiom of Extensionality weakened to allow the existence of atoms).
Known informations:
- In 1941, Ben Dushnik and Miller established the proposition "Every infinite graph contains either a countably infinite independent set or a clique with the same cardinality as the whole graph" in ZFC, and gave credit to Paul Erdős for
the proof of the result for the case in which the cardinality of the
graph is a singular cardinal. The above result is uniformly known as Erdős--Dushnik--Miller theorem.
- Consider the following variant (abbreviated as EDM): "Every uncountable graph contains either a countably infinite independent set or an uncountable clique". It is well-known that in ZFC, EDM implies
the proposition "Any partially ordered set such that all of its
antichains are finite and all of its chains are countable is countable"
(we abbreviate by K) as well as the infinite Ramsey's theorem ("Every infinite graph has either an infinite independent set or an infinite clique").
- In 1977, Andreas Blass studied the exact placement of the infinite Ramsey's theorem in the hierarchy of weak forms of AC. In particular, he proved that the Boolean Prime Ideal Theorem (a weak form of AC) is independent of the infinite Ramsey's theorem in ZF (i.e., there exists a ZF model where the Boolean Prime Ideal Theorem holds, but the infinite Ramsey's theorem fails, and there exists a ZF model where the infinite Ramsey's theorem holds, but the Boolean Prime Ideal Theorem fails) (see: https://doi.org/10.2307/2272866).
- In 2021, I studied some relations of K with weak forms of AC. (see: arXiv:2009.05368v2; to appear).
In 2022, Eleftherios Tachtsis investigated the deductive strength of K without AC in more detail. Among several results, Tachtsis proved that DC_{\aleph_{1}} (Dependent Choices for \aleph_{1}, a weak form of AC stronger than Dependent Choices (DC)) implies K in ZF (see: https://link.springer.com/article/10.1007/s00605-022-01751-9).
New Results: We study the exact placement of EDM in the hierarchy of weak forms of AC. In particular, we prove the following results (see arXiv:2211.05665v3):
1. The strength of EDM is strictly between DC_{\aleph_{1}} and K in ZFA.
2. EDM is strictly stronger than the infinite Ramsey's theorem in ZF (i.e., the infinite Ramsey's theorem does not imply EDM in ZF).
3. The Boolean Prime Ideal Theorem is independent of EDM in ZFA (specifically, neither the Boolean Prime Ideal Theorem implies EDM in ZF, nor EDM implies the Boolean Prime Ideal Theorem in ZFA).
Open Questions: Finally, I will state some open questions in this track.
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| May 26 (Friday) 4:15 PM Room
224 + ONLINE |
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Dana Slaila
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MA in Logic and Philosophy of Science, Eötvös University Budapest
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Mathematical Thinking Process
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This presentation will discuss how we think in mathematics and the
potential for introspection to communicate the thinking processes. It
will explore three key cognitive processes: abstracting, correlating,
and conjecturing. The results demonstrate that while
strong arguments can be made about how we think during math practice,
no absolute truth can be claimed. However, these arguments can be useful
for revealing properties of mathematics and aiding in teaching and
research. A unique, definitive framework of independent
categories for mathematical thinking processes, on the other hand, is
unlikely to be offered in the foreseeable future due to the difficulty
of measuring cognition and the complexity of isolating various factors.
Nonetheless, existing frameworks share similarities
and can be derived from each other.
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