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Ahmet Çevik

Publications and source records attributed to Ahmet Çevik.

7 recordsLinked to original sources
arXiv2023

Hierarchical Multiverse of Sets

In this paper, I develop a novel version of the multiverse theory of sets called hierarchical pluralism by introducing the notion of `degrees of intentionality' of theories. The presented view is articulated for the purpose of reconciling epistemological realism and the multiverse theory of sets so as to preserve a considerable amount of epistemic objectivity when working with the multiverse theory. I give some arguments in favour of a hierarchical picture of the multiverse in which theories or models are thought to be ordered with respect to their plausibility, as a manifestation of endorsing the idea that some set theories are more plausible than others. The proposed multiverse account settles the pluralist's dilemma, the dichotomy that there is a trade-off between the richness of mathematical ontology and the objectivity of mathematical truth. The view also extends and serves as an alternative position to Balaguer's intention-based Platonism from which he claims that a certain version of mathematical pluralism follows.

math.LO↗
arXiv2023

On the Cardinality of Future Worldlines in Discrete Spacetime Structures

We give an analysis over a variation of causal sets where the light cone of an event is represented by finitely branching trees with respect to any given arbitrary dynamics. We argue through basic topological properties of Cantor space that under certain assumptions about the universe, spacetime structure and causation, given any event $x$, the number of all possible future worldlines of $x$ within the many-worlds interpretation is uncountable. However, if all worldlines extending the event $x$ are `eventually deterministic', then the cardinality of the set of future worldlines with respect to $x$ is exactly $\aleph_0$, i.e., countably infinite. We also observe that if there are countably many future worldlines with respect to $x$, then at least one of them must be necessarily `decidable' in the sense that there is an algorithm which determines whether or not any given event belongs to that worldline. We then show that if there are only finitely many worldlines in the future of an event $x$, then they are all decidable. We finally point out the fact that there can be only countably many terminating worldlines.

gr-qc↗
arXiv2021

Most-Intersection of Countable Sets

We introduce a novel set-intersection operator called `most-intersection' based on the logical quantifier `most', via natural density of countable sets, to be used in determining the majority characteristic of a given countable (possibly infinite) collection of systems. The new operator determines, based on the natural density, the elements which are in `most' sets in a given collection. This notion allows one to define a majority set-membership characteristic of an infinite/finite collection with minimal information loss, compared to the standard intersection operator, when used in statistical ensembles. We also give some applications of the most-intersection operator in formal language theory and hypergraphs. The introduction of the most-intersection operator leads to a large number of applications in pure and applied mathematics some of which we leave open for further study.

math.GM↗
arXiv2019

Axiom of Neutrosophic Choice

We introduce neutrosophic choice functions, the neutrosophic counterpart of the Axiom of Choice, prove some results, and discuss how it effects the foundations of mathematics in a neutrosophic setting.

math.GM↗
arXiv2019

Natural Density and The Quantifier 'Most'

This paper proposes a formalization of the class of sentences quantified by \textit{most}, which is also interpreted as {\em proportion of} or {\em majority of} depending on the domain of discourse. We consider sentences of the form "\textit{Most A are B}", where \textit{A} and \textit{B} are plural nouns and the interpretations of $ A $ and $ B $ are infinite subsets of $ \mathbb{N} $. There are two widely used semantics for \textit{Most A are B}: (i) $C(A \cap B) > C(A\setminus B) $ and (ii) $ C(A\cap B) > \dfrac{C(A)}{2} $, where $ C(X) $ denotes the cardinality of a given finite set $ X $. Although (i) is more descriptive than (ii), it also produces a considerable amount of insensitivity for certain sets. Since the quantifier {\em most} has a solid cardinal behaviour under the interpretation {\em majority} and has a slightly more statistical behaviour under the interpretation {\em proportional of}, we consider an alternative approach in deciding quantity-related statements regarding infinite sets. For this we introduce a new semantics using {\em natural density} for sentences in which interpretations of their nouns are infinite subsets of $ \mathbb{N} $, along with a list of the axiomatization of the concept of natural density. In other words, we take the standard definition of the semantics of \textit{most} but define it as applying to finite approximations of infinite sets computed to the limit.

math.LO↗
arXiv2018

Countable chains and infinite joins in effectively closed sets of Cantor space

We prove that there exists a countable infinite sequence of non-empty special $Π^0_1$ classes $\{\mathcal{P}_i\}_{i\inω}$ such that no infinite union of elements of any $\mathcal{P}_i$ computes the halting set. We then give a generalized form of lower and upper cone avoidance for infinite unions. That is, we show that for any special $Π^0_1$ class $\mathcal{P}$ and any countable sequence of sets in $\mathcal{P}$, $\mathcal{P}$ has a member that is not computable by the infinite union of elements of the sequence. We also prove the upper cone counterpart, that for any non-recursive set $X$, every non-empty $Π^0_1$ class contains a countable sequence of members whose join does not compute $X$. We finally show that there exists a $Π^0_1$ class whose degree specrum is a countably infinite strict chain.

math.LO↗