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Questions tagged [set-theory]

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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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2answers
175 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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2answers
2k views

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
132 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 ...
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1answer
88 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 ...
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2answers
802 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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4answers
281 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 ...
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0answers
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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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0answers
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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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0answers
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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 ...
3
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1answer
162 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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0answers
112 views

How did Cantor understand set theory as a weapon?

In a letter to J. Hontheim Cantor wrote on 21 December 1893: "The time is not far, however, that my teaching will turn out to be a really exterminating weapon against all pantheism, positivism and ...
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4answers
97 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/...
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5answers
885 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 ...
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2answers
315 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 ...
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1answer
569 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 ...
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5answers
323 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 ...
0
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1answer
82 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?
7
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4answers
368 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 ...
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2answers
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What does “self-contradictory” mean?

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 ...
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1answer
366 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 ...
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3answers
249 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 ...
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1answer
123 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 ...
4
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2answers
272 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)...
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1answer
176 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: ...
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7answers
833 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 ...
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7answers
4k 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 ...
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2answers
147 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 ...
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5answers
1k 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? ...