19
votes
Accepted
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 ...
- 352
18
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 ...
15
votes
Accepted
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) ...
- 813
11
votes
Accepted
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 ...
- 36.2k
11
votes
Accepted
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 ...
- 22.9k
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 ...
- 2,402
9
votes
Accepted
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 ...
- 114
8
votes
Accepted
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 ...
- 24.3k
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 ...
- 42.5k
8
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 (= ...
- 24.3k
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 ...
8
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 ...
- 12.5k
6
votes
Accepted
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 ...
- 42.5k
6
votes
Accepted
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 ...
- 2,841
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 ...
- 36.2k
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 ...
- 36.2k
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 ...
- 3,525
5
votes
Accepted
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 ...
- 36.2k
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 ...
- 3,210
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 ...
- 7,580
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 ...
- 1,992
5
votes
How is the concept of a topos in mathematics relevant to philosophy?
The word 'localization' here has a specific technical meaning; it's referring to the localization of a category:
[L]ocalization of a category consists of adding to a category inverse morphisms for ...
- 420
5
votes
How is the concept of a topos in mathematics relevant to philosophy?
Could you explain what are the various ways topoi are used in philosophy?
Let's start with a quotation Robert Goldblatt's text on the matter:
The notion of topos has great unifying power. It ...
- 22.9k
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 ...
- 1,733
5
votes
What if all sets contain themselves?
The question does not define a set theory, it merely investigates the domain of non-empty sets that contain at least themselves. This domain could have a name, like NESCT.
With ordinary set theory, we ...
- 3,462
4
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 ...
- 6,466
4
votes
Is there any similarity between Kant's noumena and the empty set?
I do not see any similarity between Kant's noumenon, taken as the thing-in-itself, and the empty set from mathematical set theory.
Kant and science in accordance with him hypothesize the existence of ...
- 24.3k
Only top scored, non community-wiki answers of a minimum length are eligible