Questions tagged [set-theory]

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Selection of logical connectives {¬,∨,∧,⇒,⇔} in set theory?

Nearly every treatment of set theory, whether Paul Halmos' Naive Set Theory, Herbert Enderton's Elements of Set Theory, Patrick Suppes' Axiomatic Set Theory, etc. introduce a common set of logical ...
402 views

Quine on higher set theory

In The Oxford Handbook of Philosophy of Mathematics and Logic, Stewart Shapiro states in his introductory section: Quine himself accepts mathematics (as true) only to the extent that it is applied ...
1k views

Is the axiom of infinity truly an axiom?

I hope I can communicate my concerns effectively, so I can reach an understanding about a topic that I've been reflecting and researching intensely on for a few days. I am thinking about actually ...
2k views

Proof Universe Came From Nothing?

Consider the following proof: (1) Let the Universe be defined as the set of all things. (2) It is impossible for a thing to come from itself. (You can't be your own parent) (3) 2 implies a set of ...
439 views

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 ...
2k views

How many empty sets are there?

Thre are many sets with a single object, for example the set which only contains the statue of liberty or the set which contains my copy of Catch-22. But how many sets are there that contain nothing? ...
1k views

What is the distinction between being and having?

A human has hair but is not hair. Braided hair has hair as a quality but is also just hair. A man is a male but a man also has the quality of having masculine quality and body parts. How might I ...
759 views

What are functions in the Peano axioms?

I'm posting this here because it's more of a philosophical question than a mathematical one. In set theory, we define a function as a particular type of set; and since the natural numbers are defined ...
224 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 ...
279 views

Is there any similarity between Kant's noumena and the empty set?

Kant's ding-an-sich or noumena were roundly criticized by Fichte, Hegel, and other near contemporaries as incomprehensible, meaningless, or at least very unsatisfactory. How can we "know" or talk ...
393 views

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 ...
412 views

In what sense is set theory the “meta theory” of analysis?

Here, Terence Tao said: I deliberately chose not to be excessively formal with regards to the set-theoretic foundations of mathematics here, which I am regarding as part of the metatheory of ...
468 views

Subformulas of the WFF (∀x) ((∀y) ((x ∈ y) ∨ (y ∈ x )))

Consider the well-formed formula in set theory (∀x) ((∀y) ((x ∈ y) ∨ (y ∈ x ))). I believe there are 5 subformulas: (x ∈ y) (y ∈ x) ((x ∈ y)∨(y ∈ x)) (∀y) ((x ∈ y)∨(y ∈ x)) (∀x) ((∀y) ((x ∈ y)∨(y ∈ x)...
194 views

Philosophers who use formal systems to make arguments about the world, and their detractors

I was given a photocopy of an article: "In Defense of Alain Badiou" by Robert Michael Ruehl, published in Philosophy Now. The article is behind a paywall, but here's the idea that caught my attention: ...
133 views

Prove ∀w(∀v((v=w∧φ(v))⇔φ(w)))

In this math question of mine, an answer pointed me to this theorem: ∀w(∀v((v=w∧φ(v))⇔φ(w))) which in turn, the answerer stated, implies another theorem: ∃v(v=t∧φ(v))⇔φ(t) which was the fact I ...
872 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 ...
410 views

Why does the Second-Order Axiom schema of Comprehension not lead to Russell's Paradox for ZFC2?

Let ZFC2 be the Second-Order formalisation of ZFC. The Second-Order Axiom schema of Comprehension (part of the deductive system for SOL) says that for every formula (of SOL) there is a relation with ...
88 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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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 ...
105 views

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/...
160 views

Prove a logical formula is equivalent to the contradiction if and only if the set it describes is empty

Let ψ be a well-formed-formula (wff). Prove that (ψ ≡ ⊥) ⇔ {x:ψ(x)}=Ø that is, the formula ψ is a contradiction if and only if the set it describes has no members. Note This question is not about ...
479 views

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 ...
161 views

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 ...
118 views

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 ...
97 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 ...
72 views

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?
207 views

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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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 ...
234 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 ...
179 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 ...
165 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!
64 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 ...
148 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 ...
70 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. ...
185 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 ...
86 views

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. ...
42 views

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 ...
82 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 ...
99 views

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 ...
62 views

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.
502 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 ...
347 views

If God existed, would he know that he knows everything?

If we define God's knowledge as a proper class (not a set, because that would create contradictions), that would mean he does not know he knows everything. Since a class cannot contain itself, he ...
109 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 ...
4k views

In this video, the mathematician Gregory Chaitin states that "the notion of the set of all sets is self-contradictory". What does "self-contradictory" mean? Is it different from "contradictory"? There ...
72 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 ...
95 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 ...
150 views

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 ...
91 views

Given that Γ is a maximal set, show that P → ¬P ∈ Γ only if P → Q ∈ Γ

To show that P → ¬P ∈ Γ only if P → Q ∈ Γ, would I have to use soundness, completeness, or could I prove it using derivation rules?