29 research outputs found
Troubles with (the concept of) truth in mathematics
In the paper the problem of definability and undefinability of the concept of satisfaction and truth is considered. Connections between satisfaction and truth on the one hand and consistency of certain systems of omega-logic and transfinite induction on the other are indicated
Church's thesis and its epistemological status
The aim of this paper is to present the origin of Church's thesis and the main arguments in favour of it as well as arguments against it. Further the general problem of the epistemological status of the thesis will be considered, in particular the problem whether it can be treated as a definition and whether it is provable or has a definite truth-value
Decidability vs. undecidability. Logico-philosophico-historical remarks
The aim of the paper is to present the decidability problems from a philosophical and historical perspective as well as to indicate basic mathematical and logical results concerning (un)decidability of particular theories and problems
Truth vs. provability – philosophical and historical remarks
Since Plato, Aristotle and Euclid the axiomatic method was considered as the best method to justify and to organize mathematical knowledge. The first mature and most representative example of its usage in mathematics were Elements of Euclid. They established a pattern of a scientific theory and in particular a paradigm in mathematics
On the depth of G\"{o}del's incompleteness theorem
In this paper, we use G\"{o}del's incompleteness theorem as a case study for
investigating mathematical depth. We take for granted the widespread judgment
by mathematical logicians that G\"{o}del's incompleteness theorem is deep, and
focus on the philosophical question of what its depth consists in. We focus on
the methodological study of the depth of G\"{o}del's incompleteness theorem,
and propose three criteria to account for its depth: influence, fruitfulness,
and unity. Finally, we give some explanations for our account of the depth of
G\"{o}del's incompleteness theorem.Comment: 23 pages, revised version. arXiv admin note: text overlap with
arXiv:2009.0488
Finding the limit of incompleteness I
In this paper, we examine the limit of applicability of G\"{o}del's first
incompleteness theorem ( for short). We first define the notion " holds for the theory ". This paper is motivated by the following
question: can we find a theory with a minimal degree of interpretation for
which holds. To approach this question, we first examine the following
question: is there a theory such that Robinson's interprets
but does not interpret (i.e. is weaker than
w.r.t. interpretation) and holds for ? In this paper,
we show that there are many such theories based on Je\v{r}\'{a}bek's work using
some model theory. We prove that for each recursively inseparable pair , we can construct a r.e. theory such that
is weaker than w.r.t. interpretation and
holds for . As a corollary, we answer a
question from Albert Visser. Moreover, we prove that for any Turing degree
, there is a theory with Turing
degree such that holds for and is weaker than
w.r.t. Turing reducibility. As a corollary, based on Shoenfield's
work using some recursion theory, we show that there is no theory with a
minimal degree of Turing reducibility for which holds.Comment: 18 pages. Accepted and to appear in Bulletin of Symbolic Logi
On the relationship between meta-mathematical properties of theories
In this work, we understand incompleteness in an abstract way via
metamathematical properties of formal theories. We systematically examine the
relationship between the following twelve important metamathematical properties
of theories: Rosser, EI (Effectively inseparable), RI (Recursively
inseparable), TP (Turing persistent), EHU (essentially hereditarily
undecidable), EU (essentially undecidable), Creative,
(theories with Turing degree ), REW (all RE sets are
weakly representable), RFD (all recursive functions are definable), RSS (all
recursive sets are strongly representable), RSW (all recursive sets are weakly
representable). Given any two properties and of these properties, we
examine whether the property implies the property , and whether the
property implies the property .Comment: 23 page
Current research on G\"odel's incompleteness theorems
We give a survey of current research on G\"{o}del's incompleteness theorems
from the following three aspects: classifications of different proofs of
G\"{o}del's incompleteness theorems, the limit of the applicability of
G\"{o}del's first incompleteness theorem, and the limit of the applicability of
G\"{o}del's second incompleteness theorem.Comment: 54 pages, final accepted version, to appear in The Bulletin of
Symbolic Logi