Questions tagged [set-theory]

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Can a collection be a single 'thing'?

Can a collection be any single object in itself, for example Collection A is 'one collection' or set A is 'one set' or even one 'mathematical object' if viewed as a whole? For example a set containing ...
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ZFA vs. ZFC vs. ~Con(ZFA)

Among other places, I've seen it here said that ZFC and ZFA (cofoundational instead of strictly well-foundational) are equiconsistent. I feel like I should know the answer to my question, but if ZFA ...
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11 answers
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Why is a set with one element distinct from the element itself?

Why do we consider a set which is treated for all intents and purposes as a 'collection' with one element as being different from the element itself? In this 'collection' there is one element, and ...
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2 votes
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Constructivism and the effects of constructing definitions on morality

My friend states that 'morality is subjective' since one can construct definitions based on arbitrary intensions and extensions of a well-defined set viz. 'My own morally not bad actions' from the ...
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Constructing natural numbers from nothing

I found that many of us (mathematicians) try to construct natural numbers defined from the intuitive concept 'size of the set'. They take ϕ, the empty set, as the starting point, then define and ...
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If the Continuum's cardinality could physically change because of a physical manifestation of forcing

Suppose that the Continuum has a specific cardinality assigned to it when it exists "in" a physical universe. (This, or a similar-sounding supposition, occurs frequently enough in the ...
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Transconsistency and Diaconescu's theorem

Let T be some set theory. Now there is at least one constructivist T such that T(AOC) ⊨ LEM, that is, via the axiom of choice in T, the law of the excluded middle becomes a theorem (Diaconescu's). So ...
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Is there a notion analogous to “the inductive conception of set” in modal logic?

In many introductory treatments of modal logic, one defines a Kripke model with respect to some domain D. In variable domain semantics, each world in the Kripke model is assigned a different subset of ...
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Can set theory be non-extensional?

Here is Juliet Floyd stating "Wittgenstein's non-extensionalism, like Russell's in Principia, precluded development of an extensional theory of the infinite (set theory). https://youtu.be/...
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Can something be both a type and a token?

I wonder, can something be both a type and a token (in reference to the type-token distinction in philosophy)? For example, an individual dog is a token of the type of dog, but the type of dog is ...
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Is "antifoundation" the most pragmatic word for the underlying concept?

Aczel's antifoundation principle is not a statement that there are no well-founded sets. It is just the statement that every accessible pointed graph images a set,⭯ and since a graph consisting in a ...
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Intuition behind existence of "function sets"

The usual axioms ensure the existence of certain sets that serve as functions. For example (which is chosen arbitrarily) the function f which maps real values of x to x^2+2 can be represented by the ...
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References on Metaphysical Dependence on Set Theory

I'm new here. I was having a hard time reading and comprehending John Wigglesworth's thesis Metaphysical Dependence and Set Theory. I was hoping if there is anything out there talking about the matter;...
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1 answer
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Cardinality of the logical space according to David Lewis

I just read an extract of David Lewis's Counterfactuals and he claims there in a footnote on page 90 that there are at least beth_2 possible worlds. He also claims in the very same footnote that "...
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Are implementations to semiotics what proofs are to syntax, and models to semantics?

I see this term implementation and its family used here and there in writings on set theory. There are implementations of natural numbers, of ordered pairs, of functions, of "mathematics in ...
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2 answers
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Set theory with full comprehension

A few years ago I was reading an entry in the Stanford Encyclopedia of Philosophy related to set theory and I stumbled across a statement along these lines: There is a set theory where full ...
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How to write that two specific sets are the same set in first-order naive set theory?

Consider the following true statement (1) The set whose only members are the prime numbers between 6 and 12 is the same as the set whose only members are the solutions to the equation x^2-18x+77=0. ...
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Erotetic quantifiers?

I can't find it, but I have a distinct memory of there being an SEP article that says something like as if the ordinals were there before the sets in relation to a question about whether ordinals (or ...
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Can Physical and Mathematical objects exist in the same set?

I'm fairly comfortable dealing with sets of purely Mathematical objects, when I remembered something I heard a long time ago that sets can have elements of any type. Is it possible to define a set ...
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Mathematical "forms" as a relation of varying arity

This might be more a MathSE question, but on the other hand, it would involve a peculiar reimagining of the relation between set theory and type theory, so I'll try it out here. OK, so earlier I ...
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4 votes
2 answers
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Set theory vs. type theory vs. category theory

IIRC, in the univalent-foundations program (per Voevodsky), category theory is represented as a possible sort of evolution or new wave of type theory. Maybe my memory is off, but anyway, in nlab they ...
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Rehabilitating Zermelo's description of the universe of sets?

I was reading various essays about Cantor's doctrine of absolute infinity, and it came up again that Zermelo's doctrine, by contrast, was of V as an "unfinished totality." Initially, this ...
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What are sufficient grounds for establishing a theory?

This question delves into the definition of a theory, but somewhat into the grounds of Set Theory as well. I was wondering on what grounds is theory established and accepted. To what degree do the ...
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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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2 votes
3 answers
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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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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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1 answer
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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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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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1 answer
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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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1 vote
1 answer
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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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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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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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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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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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1 answer
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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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2 votes
2 answers
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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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2 votes
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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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3 answers
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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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1 vote
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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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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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1 vote
1 answer
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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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