# Questions tagged [set-theory]

The tag has no usage guidance.

43 questions
Filter by
Sorted by
Tagged with
37 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 ...
91 views

### 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 ...
57 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 ...
142 views

### 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 ...
67 views

### What goes wrong if we explain Russell's paradox as resulting from an overly rigid link between property satisfaction and elementhood?

An excerpt from a question at Math SE (Bounty of 100 available for an answer): What goes wrong if we explain Russell's paradox as resulting from an overly rigid link between property satisfaction and ...
82 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 ...
64 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. ...
74 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 ...
261 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 ...
394 views

### 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 ...
349 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 ...
180 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 ...
185 views

### 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 ...
232 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 ...
204 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]. ...
181 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 ...
2k views

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 ...
154 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

### 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 ...
852 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 ...
351 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 ...
97 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 ...
60 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.
72 views

### 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 ...
164 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!
104 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/...
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 ...
368 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 ...
707 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 ...
338 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 ...
89 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?
391 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 ...
3k 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 ...
397 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 ...
269 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 ...
130 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 ...
397 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)...
181 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: ...
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 ...
5k views

### 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 ...
155 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 ...