Questions tagged [set-theory]

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67 views

Can Mathematics be a tool to analyse immaterial existences

Doubt: Can we say that Mathematical thoughts (arguments) can be independent of physical (Time-space continuum or material) world since it is an abstract science. In other words can Mathematics be a ...
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Justification values

The concept of truth values is sometimes expressed in terms of "truth as an object vs. truth as a property." My in-a-slogan understanding of this alternative is "sentences being ...
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Forcing and justification

In "The set-theoretic multiverse," Hamkins talks about forcing giving us "glimpses" of other set-theoretic universes. He states his position as a Platonistic one, i.e. these "...
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2answers
201 views

How do we arrive at stronger theories in mathematics/logic?

A reasonable aim of formal mathematics/logic is to build systems which can "interpret" many things. As an example, ZFC can interpret a number of things. Incompleteness Theorems provide us ...
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Justification logic and set theory

I was reading the SEP article on justification logic and the question arose for me, whether the difference between intrinsic and extrinsic justifications of set-theoretic axioms (a difference that has ...
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1answer
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Does the forcing phenomenon prove some sort of set-theoretic multiverse?

It seems that, "A can be forced to equal B," allows, "A is possibly equal to B." In possible-worlds lingo, this gets us, "There is a possible world where A = B." Since ...
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In what way, if any, can the axioms of set theory be themselves well-ordered?

In "Independence and Large Cardinals", Peter Koellner writes: ... it turns out that when one restricts [attention] to those theories that "arise in nature" the interpretability ...
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119 views

How is ~CH derived in paraconsistent set theory?

This question on MathOverflow has been left unanswered. The respondents pointed mainly towards "Transfinite Numbers in Paraconsistent Set Theory", an article to which I don't have access. ...
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1answer
34 views

Can implexion be a nonwell-founded relation?

Alexius Meinong's "doctrine" of implexion is that there are complete and incomplete objects and that the latter are "implected in" the former (see the SEP article on that). Can ...
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Is the concept of sigma algebra necessary to understand the epistemic and inter-observer boundaries of 'facts'?

The idea of fact is complex in its epistemic, metaphysical and linguistic connotations, but in the end it has compelling force, and the power to shock when the predicted practical effects are not ...
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How can we trust second-order logic? [closed]

Hello Philosophy Stack Exchange, This is a question I originally asked on Mathematics Stack Exchange, but was encouraged to ask here. Unfortunately you cannot directly migrate to this site, so the ...
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1answer
143 views

What is the difference between an object and its singleton set?

I have read in books on Mathematical Logic that we have things called "Sets" and Set Theory that correspond to classes of objects in Ontology. for example { Barack Obama, Donald Trump } is a ...
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Why is the definition of the real numbers not contradictory? [closed]

I understand that a set whose members can, in principle, be enumerated (by having a formula) can be considered as a well-defined set. Therefore, set of all even numbers, multiples of 3, and so on ...
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Does Gödel believe in the existence of his rotating universe?

I am wondering whether Gödel believe ain the existence of his rotating universe since he is a mathematical Platonist. I am also wondering in what entities believe mathematical platonists. For example: ...
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2answers
167 views

Is everything in the Naive Set Theory included in the Axiomatic Set Theory?

I am trying to understand if Naive Set Theory (NST) should be understood as a "core" for Axiomatic Set Theory (AST). Is everything (all data) included in NST included in AST? Would NST be ...
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4answers
207 views

What is a set? (Is it possible to define a set?)

I've recently been studying set theory from some introductory textbooks (like Steinhart's "More Precisely" or Open Logic Project's "Sets, Logic, Computation"). I'm interested in ...
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100 views

Is a set a concept?

Follow on from this question. Since sets have both intentional and extensional definition my thought is yes they are concepts. But maybe there is a technical reason that sets aren't concepts?
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Do the set of “Concepts” contain itself?

So I gather that a set containing itself is not allowed. Yet it seems like a set of all concepts (Concepts) should contain an element denoting the idea of "concept". Is it that there is a ...
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The empty set as an atomic unit

I was reading some perspectives on the empty set in ZFC set theory. To my understanding, every other set that we can explicitly show to exist is made up of the empty set and sets of the empty set. ...
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1answer
79 views

Do you think that the 'descent class' is paradoxical?

It is well known that something goes wrong with Russell's class of all classes which are not members of themselves. If this class is a member of itself then it is not a member of itself, and if this ...
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111 views

Are there set-theoretical problems with modal metaphysics?

I'm trying to learn about modal logic and the metaphysics of modality, and there's something that has been bugging me about what I've read so far: are there set-theoretical issues with supposing we ...
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2answers
186 views

Could the axiom of infinity be in itself inconsistent?

I've seen several threads discussing the axiom of infinity but I wasn't able to find a discussion on this particular aspect. And recent conversations with some people have led me to wonder if it is ...
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Forcing and Philosophy

The only (ontological) connection between Forcing (Cohen) and Philosophy i know is the work of Alain Badiou. Are there any other philosophers who have worked on this topic?
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3answers
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Does a Cycle Based Alternative to Set Theory Exist?

My (limited) understanding of Mathematics in general and of Set Theory (being a widely accepted foundational system of Mathematics) in particular, is that the mathematical objects described therein ...
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Analogy of Set and Subset and Contracts in abstracto and Marriage in concreto/in particular

I had a talk with a professor of family law and we are frequently told that there are general ordinances for contracts in general and particular ordinances for marriage. I am problematised by the ...
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Are there versions of set theory in which a concrete object, say an apple, can be a member of a set

Certainly, when we apply set theory, we consider collections of concrete objects as sets. For example, when I count 5 apples, I establish a bijection between the number 5 ( which is defined as the ...
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1answer
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Is a set that contains itself always logically incoherent? [closed]

This is an ontological engineering question, please treat it that way. https://en.wikipedia.org/wiki/Ontology_engineering I am examining this question from the point of view of ontological ...
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1answer
72 views

Do possible worlds partition the set of all possible states of affairs?

Let S be the set of all ( logically) possible states of affairs ( I could have said " events" or " propositions" maybe). Let R be the relation : state of affairs x is compossible/ compatible with ...
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Why should a system of set theory represent the following property of x: ((x = h) or (x = k))) using the set that Frege used?

I have been considering the possibility of an approach to set theory that represents the following property of the free variable x, in the terms of parameters h and k: ((x = h) or (x = k))) by means ...
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1answer
106 views

Modal Logic: Why are Universal frames a subset of Equivalence frames?

I'm looking through the lecture notes for my course on modal logic and am having a hard time understanding why it is that U, the class of all Universal frames, is a subset of E, the class of all ...
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2answers
77 views

Linguistic explanation of a set being an element of itself or containing itself and can you be part of yourself or contain yourself?

Set theories examines whether a set can be an element of itself or contain itself. But linguistics already offers its own explanation for whether a set can be an element of itself or contain itself. ...
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84 views

Are the foundations of mathematics “doomed” to be set-theoretic in nature?

Let's say we want to come up with a foundational theory for all of mathematics and let's say that it is embedded in first-order logic. Note that the machinery of first-order logic is described with ...
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2answers
579 views

Theology of set theory

Absolute space and time are said to emanate from Aristotle. The Church acted as custodian of these concepts from early on up to recent times. I am thinking about another issue, namely that of ...
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Do Gödel's First Incompleteness Theorem imply the inconsistency of Platonic Infinity?

According to Modern Mathematics (where the majority of mathematicians agree about the notion of actual infinite sets, as established mostly by George Cantor) an inductive set (as given by ZF(C) Axiom ...
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1answer
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What's so bad about giving up the Axiom of Choice?

The Axiom of Choice (AoC) in set theory famously gives rise to controversial and counterintuitive theorems. (Examples: Banach-Tarski paradox and existence of non-measurable sets.) I'm aware of some ...
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4answers
252 views

Believing in the axiom of Power Set

I am struggling to find a philosophical reason for believing in the axiom of power set, and I was hoping you can give me some justifications. I am not looking for answers of the form "it's convenient ...
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How much this theory fulfills of criteria for a foundational theory of mathematics?

[EDIT] The criteria for a founding theory of mathematics, especially if it uses large cardinal axioms that I want to refer to are those of Harvey Friedman's 2000 criteria given in pages 5-6 of the ...
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247 views

Number, Category and Set

Can it be said that a number is a category is a set? There is such a variety of ideas on numbers, categories and sets that probably anything one says about them will be controversial, but I was ...
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Does Reflective Set Theory “RfST” fulfill the requirements of founding Category Theory and Mathematics?

On mathoverflow I've posed the question in the title in connection to Muller's 2001 criteria for a founding theory of mathematics, which largely raised in connection to Category theory [see here]. ...
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190 views

Can we define this set?

Given an infinite set S in some universe U, construct the set of complementary pairs C: {{A,Ac}} where A,Ac∈P(U), the power set of U, and A or Ac is contained in S, not necessarily a proper subset ...
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Is a set containing itself already a paradox?

This is inspired by Russel's paradox stating there is not set of all sets. It uses the presupposition that set can contain itself. However, this already seems paradoxical. Suppose a set A = {}. Then ...
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1answer
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Can any logic system provide the impossible solution to Russell's paradox in naive set theory?

In naive set theory in classical logic, we cannot describe or find a solution to Russell's set paradox (it is impossible). But is it there any logic system or any method that can provide this ...
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Is there any logical system/method where impossible/illogical/inconsistent things can exist (like a solution to Russell's paradox that makes sense)? [duplicate]

Discussing with a philosopher about impossible things existing or being allowed within a particular logic system, he told me: "This is a funny thing about logically impossible things. You can prove ...
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2answers
892 views

Can paraconsistent or other logics make the impossible happen?

A paraconsistent logic system it is defined as "a logical system that attempts to deal with contradictions in a discriminating way. Alternatively, paraconsistent logic is the subfield of logic that ...
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Is actual infinity physical infinity? Or just the axiom of infinity?

I've always been a little confused on this point. My (second-hand) understanding of Aristotle's difference between potential and actual infinity is this: We all have an intuition of the counting ...
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Is it 'natural' to hold that big sets and proper classes exist?

Various set\class theories present different kinds of ontology, broadly speaking there is the dichotomy of classes versus Ur-elements, and the former can be further subdivided into sets and proper ...
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List of topics in philosophy relevant to mathematics, and open problems in them?

I know of open problems in model theory, but would like to know about philosophical problems (philosophy of language, Husserl's phenomenology ) that have relevance in set theory or type theory.
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How does Badiou analyze natural situations?

I'm having trouble applying Badiou's method of looking at situations as sets (EDIT: specifically sets in a model of ZFC). The following example was in the introduction to one of his books, Infinite ...
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1answer
167 views

Is the following conditional true?

∀x: P(x) → Q(x) ⇒ {x | P(x)} ⊆ {x | Q(x)} I really do think this is a stupid question, but I'm stuck, so pardon my logical/set-theoretic ignorance!
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What problems get mis-conceptualised when we impose set theoretical assumptions on them?

While classical mechanics has a logic that is based on a Boolean algebra of subsets of the state space, Quantum logic is based on the subspaces of a complex Hilbert space. https://en.wikipedia.org/...