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24 votes

Are identical sets distinct objects?

In math we've decided that if two sets have exactly the same elements then they are the same set. This is taken to be one of the fundamental properties of the mathematical conception of sets, and it ...
JonathanZ's user avatar
  • 452
20 votes
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Why is a set with one element distinct from the element itself?

In computing, there are data models (such as the XPath data model used for XML) in which an item and a singleton collection containing that item are treated as indistinguishable. You can build a ...
Michael Kay's user avatar
19 votes

Why is a set with one element distinct from the element itself?

One reason why this is true is because there is such a thing as the empty set - the set with no elements at all. Consider a set X that contains only the empty set, and nothing else. How many elements ...
Dawood ibn Kareem's user avatar
15 votes
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What did Russell mean when he wrote that the null-class, the class having no members, did not exist?

Let me start by slightly rephrasing what Russell wrote, since Russell is using the word "exists" in an unusual and confusing way. With my changes in bold, here is what Russell wrote: (b) ...
Tanner Swett's user avatar
14 votes

Are identical sets distinct objects?

If two things are identical, shouldn't they point to the same thing, not separate identical things? This sounds like a computer science question, not a philosophy question. In math and philosophy, it ...
Kevin's user avatar
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13 votes
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Set theory vs. type theory vs. category theory

Short Answer It sounds you're struggling to understand the relationship between three fundamental theories. Naive set theory is the theory used historically by Gottlob Frege to show that all ...
J D's user avatar
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11 votes
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Is a set containing itself already a paradox?

Russell's paradox arises within naïve set theory by considering the set of all sets that are not members of themselves. Such a set appears to be a member of itself if and only if it is not a member of ...
Mauro ALLEGRANZA's user avatar
11 votes

Is infinity a concept or a word empty of meaning?

Your comment seems to be the nub of your problem: I cannot think about infinity because my finite meaning cannot grasp even conceptually non finite objects, despite that there are perhaps infinite ...
Rushi's user avatar
  • 3,237
10 votes
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Is infinity a concept or a word empty of meaning?

Mathematics shows that we can make a one to one correspondence [of the] natural numbers with [the] even numbers. This is not right If there is a last number. There are infinite sets that look exactly ...
ac15's user avatar
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9 votes
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Is the axiom of infinity truly an axiom?

Is the axiom of infinity truly an axiom? Yes, it is an axiom of set theory. But in mathematics an axiom of a theory does not have to be plausible according to our everyday intuition. The only ...
Jo Wehler's user avatar
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9 votes

Why is a set with one element distinct from the element itself?

You may consider a collection as a container: Apparently a thing included in a container is different from the thing without container. Aside: Set theory provides operations to handle sets (= ...
Jo Wehler's user avatar
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9 votes

What if all sets contain themselves?

I think that disallowing the empty set will cause the resulting theory to lose too much. For example, the intersection of two disjoint sets, which would be the empty set, is now undefined. You will ...
Frank's user avatar
  • 2,454
9 votes

What if all sets contain themselves?

The axiom of foundation is not (usually) taken for a logical truth (although see about predicativism for an attempt to situate well-foundedness on the level of predication theory). Accordingly, its ...
Kristian Berry's user avatar
9 votes
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What if all sets contain themselves?

I think we can find a contradiction with ZFC immediately: Let S = {A, B} with A =/= B. WLOG, by your hypothesis, assume that A = S. (One of these elements needs to equal the whole set!) Then, also ...
waf9000's user avatar
  • 114
8 votes

Can paraconsistent or other logics make the impossible happen?

Logic, paraconsistent or not, does not exactly make something happen, it is applied to reshuffle information already contained in a system. Paraconsistent logic does not even have to be applied to ...
Conifold's user avatar
  • 43.4k
8 votes

Why is a set with one element distinct from the element itself?

Why do we need a zero when it's conceptually the same as nothing? Because zero, as a number, has very different properties from being nothing at all. The reasoning is similar about the empty set ...
Niklas Rosencrantz's user avatar
7 votes
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Quine on higher set theory

Quine did not specifically study "higher set theory", and his positions on the issue are mostly generalities following from his empirical holism (mathematics is the "entrenched" part of the "web of ...
Conifold's user avatar
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6 votes
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What are functions in the Peano axioms?

An Intuitive Walkthrough of PA as a formal system *Peano Arithmetic are a set of axioms in first order logic that describe how arithmetic of the natural numbers works. A first order formal language is ...
Not_Here's user avatar
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6 votes

What is the difference between an object and its singleton set?

A set is a mathematical object; a singleton set is a set with only one element. The set N of natural numbers has infinite many elements. The singleton set { N } has only one element. In general, the ...
Mauro ALLEGRANZA's user avatar
6 votes

What did Russell mean when he wrote that the null-class, the class having no members, did not exist?

In the passage above Russell discusses two uses of existence: (a) is the "common sense" use: "which occurs in philosophy and in daily life is the meaning which can be predicated of an ...
Mauro ALLEGRANZA's user avatar
6 votes

What does a set of pencils contain when we know that pencils are not physically present in the set?

You're being confused by overloaded terms. Pencil refers to the complex physical process on your desk that you can write with; to the radically simplified symbolic representation of that class of ...
g s's user avatar
  • 6,209
6 votes

Is infinity a concept or a word empty of meaning?

Literally, infinity is both a concept and a word- that should be clear to you from the fact that you have been thinking about the concept and have typed the word more than once in your post. Infinity ...
Marco Ocram's user avatar
6 votes

Why are pure powers of the empty set insufficient as a definition for ordinals?

The following sequence is called the “pure” or “irreducible” power sets of the empty set: {}, {{}}, {{{}}}, {{{{}}}}, … Conceptually, this sequence ‘represents’ order, to me. Why is it insufficient, ...
ac15's user avatar
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5 votes

Quine on higher set theory

This is just part of Quine's naturalism, a sort of science-first approach to everything. That is mainly what underlies his suspicion of higher set theory. Here is Quine discussing the matter, from ...
E...'s user avatar
  • 6,556
5 votes
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Why does the Second-Order Axiom schema of Comprehension not lead to Russell's Paradox for ZFC2?

In Second-order Logic, the comprehension schema (considering for simplicity only unary predicate variables) is: ∃X∀x [ ϕ(x) ↔ X(x) ], where x is an individual variable, X is a 1-ary predicate ...
Mauro ALLEGRANZA's user avatar
5 votes
Accepted

What's so bad about giving up the Axiom of Choice?

There is a lot of writing both in favor and against AC from a philosophical standpoint - e.g. in favor see Penelope Maddy's Believing the axioms. However, there are also more mundane issues. I think ...
Noah Schweber's user avatar
5 votes
Accepted

Can set theory be non-extensional?

For sets, extensionality is defined as follows. ∀S∀T(S=T ↔ ∀x(x ∈ S ↔ x ∈ T)) All modern set theories have this as an axiom or theorem, so they are all extensional. Russel did not reject ...
David Gudeman's user avatar
5 votes

Why is a set with one element distinct from the element itself?

They are distinct because a set is something different than most elements you can put into it. Sets and elements of sets usually are distinct categories or types of things (an element might be an ...
AnoE's user avatar
  • 2,887
5 votes

What did Russell mean when he wrote that the null-class, the class having no members, did not exist?

The definition that Russell provides for (b): To say that A exists means that A is a class which has at least one member. We translate this into modern mathematical notation as follows: "A ...
Kevin's user avatar
  • 2,038

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