Questions tagged [set-theory]
Use for questions about sets, functions on sets, cardinality, set-theoretic axioms, set-theoretic paradoxes, philosophical interpretations of set theory, etc.
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Does set-theoretic pluralism, about axiom systems, inevitably become an invitation to non-axiomatic systems of set theory?
Per Hamkins[[11][12]] (see also his [22]), if no individual axiom is too sacred to be denied in some possible world,Q and so if no collection of such axioms is so sacred either, yet then:
The ...
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What papers or books should I read in order?
I have been reading literature on modal set theory and am currently reading Putnam's "Mathematics Without Foundations," which is known for being one of the earliest presentations of this ...
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Question regarding 0-ary relations
EDIT-
Definition. A denotes a unary relation iff
∀x[if x∈ A then ∃y[x= (y)]]
Using this definition, since individuals don't contain elements, any individual is a unary relation. Since the empty set ...
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Why are constants considered as nullary operations?
On page 16 of Axiomatic Set Theory by Patrick Suppes, he states:
Individual constants may in fact be treated as operation symbols of degree zero.
Then I came across the following definition of an n-...
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Are sets unary relations, and are unary relations sets?
On page 57 of Axiomatic Set Theory by Patrick Suppes, he defines a binary relation as a set of ordered pairs.
Definition 1. A is a binary relation iff
∀x[if x∈A then ∃y∃z[x=(y,z)]]
He defines an ...
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Why do we have problem of concept of set?
I am reading George Boolos's "The Iterative Concept of Set," and in the first chapter of this paper, he criticizes Cantor's definition of a set as a whole or totality of objects, pointing ...
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If a predicate doesn't determine a set, does that predicate even exist in the first place?
I thought of asking this in the Math Stack Exchange, but then I thought this stack exchange is better. Certain predicates define sets, such as "x is not equal to x". Other predicates do not, ...
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Groupings of set theoretic axioms, or even “algebras of axioms”
I want to understand how to group the axioms of a set theory to study the effect that each axiom has in relation to the others. Here’s what I mean:
First of all, assume “a set theory” is not a well-...
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Can set elements be predicated of objects?
If we define a set as a cartesian product of properties, do you think that the cartesian product can be predicated of an object?
For example if we define:
Set of names of Captials N = { The name ...
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Is Russell's Paradox a semantic paradox or a syntactic paradox?
Is Russell's Paradox a semantic paradox or a syntactic paradox? I ask because of the following:
Let P be a predicate
Let SEP be the property of being a set of things that satisfies P
Let SP be the ...
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Need help understanding how certain mathemetical statements across the landscape can seemingly contradict (e.g. Cantor-Hume vs Euclid)
Here are the main components to my understanding on this issue:
Almost all of math can be given in a foundation of set theory
Different math can seemingly contradict, e.g. in Euclidean geometry ...
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Can a part of the Lowenheim-Skolem Theorem be proven using only first order logic?
Can a part of the Lowenheim-Skolem Theorem be proven using only first order logic? I ask because of the following:
Let SM be the predicate set of mathematical structures
Let CA be the predicate ...
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Can a set be uncountable in one sense and countable in another sense?
Can a set be uncountable in one sense and countable in another sense? Or in other words are there senses in which the set of all real numbers satisfies the Peano Axioms? I ask because of the following,...
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numerable diagonalization [closed]
If I understand Cantor demonstrated by using a strategy using the diagonal of a matrix, that there are decimal numbers which cannot be a part of an enumeration.
We can build a Table, the rows ...
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Is it true that every set has less elements than its power set? [closed]
Define Σ to be the set of all sets and only sets. Consider the following proof that A < ℘(A), for any set A, found on page 97 of Axiomatic Set Theory by Patrick Suppes.
The function f on A such ...
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