10
votes
3answers
383 views

What is category theory?

I have read the page about category theory in wikipedia carefully, but i don't really get what this theory is. Is category theory a content in ZFC-set theory? (Just like measure theory, group theory ...
2
votes
3answers
128 views

An example of an ordered, uncountable set in $\Bbb R$?

Is there an example of an ordered, uncountable set in $\Bbb R$? My Calculus professor, who likes to keep things simple, defined a sequence in $\Bbb R$ as an "ordered and infinite list of real ...
1
vote
1answer
96 views

Definition of “Universe of Discourse” and Definition of “Set” [duplicate]

I want to axiomatize "the concept of set" in my head, but every time I face some circular definition or intuition. In predicate logic, we quantify over some "Universe of Discourse". Intuitively ...
5
votes
2answers
96 views

Definition of $H_\lambda$ (hereditary cardinality)

It seems to me that the definition of $H_\lambda$ (the set of sets of hereditary cardinality less than $\lambda$) on the web page at Cantor's Attic is not quite correct. From the page: ...
2
votes
1answer
51 views

Distinguishing characteristic and spurious properties

When working in set theory with definitions of somehow primitive notions like ordered pairs (Kuratowski) or ordinal numbers (von Neumann) one is supposed to use only the characteristic properties of ...
1
vote
1answer
89 views

What is a finite partial function from $\mathfrak c$ to $D=\{0,1\}$

While I reading a paper, there is a notation is called finite partial function. I searched by google, I cannot find its definition. So I post it here as a question: What is a finite partial ...
2
votes
1answer
126 views

Help with definition: partition mod ultrafilter.

I do not see how the partition is well-defined. By definition $A\neq\varnothing\mbox{ mod }D\iff A\notin I_{D}$. Since D is a maximal filter $A\notin I_{D}\iff A\in D$ . So ...
8
votes
3answers
279 views

How to write $\pi$ as a set in ZF?

I know that from ZF we can construct some sets in a beautiful form obtaining the desired properties that we expect to have these sets. In ZF all is a set (including numbers, elements, functions, ...
1
vote
1answer
110 views

Ordered pairs in sets

So I am asked to explain why $(x,y,z)=(x,(y,z))$ is a reasonable definition of an ordered triple. However, I am in a limbo at the moment let alone with trying to explain an ordered pair,(x,y), ...
3
votes
1answer
114 views

(Non) equivalence of regular cardinal definitions

The usual definition of a regular cardinal is "$\kappa$ is regular if $cf(\kappa) = \kappa$", which, assuming the axiom of choice, is equivalent to this definition: "$\kappa$ is regular iff it cannot ...
1
vote
1answer
82 views

Defining Test-Objects

In various categories one encounters what are referred to as 'test-objects'. For example, singleton set serves as test-object (in testing the equality of functions) in the category of sets. In the ...
6
votes
1answer
104 views

Defining $\Bbb{Q}$ without the axiom of infinity

(TL;DR version: I want a meaningful definition of $\Bbb{Q}$ without $\sf{Inf}$.) In the "conventional" construction of the rationals, we define $\Bbb{Q}$ as follows: $\omega$ is the first limit ...
1
vote
1answer
34 views

A follow up question on completeness of filter (generalised to p.o.s)

Now I'm looking at the following generalisation of $\kappa$-closedness of a filter on a set to a filter in a partial order: Let $\kappa$ be a regular uncountable cardinal. A partial order $P$ is ...
4
votes
1answer
64 views

Question about definition of $\kappa$-completeness of filter

I am looking at the following definition: Let $\kappa$ be a regular uncountable cardinal and let $\mathcal F$ be a filter on a non-empty set $X$. We say that $\mathcal F$ is $\kappa$-complete if ...
2
votes
2answers
105 views

The problem of bound variables in mathematical definitions

I was reading Paul Bernays’ Axiomatic Set Theory recently; in the book, Bernays gives the following definition of ‘ordinal number’. \begin{align} \text{On}(\alpha) \stackrel{\text{def}}{\iff} ...
4
votes
1answer
118 views

What is a “set-like class”?

Just / Weese contains the following theorem (p 126): Theorem 5: Let $\mathbf Z \neq \varnothing$ and let $\mathbf W \subseteq \mathbf Z \times \mathbf Z$ be a wellfounded set-like class. Let $\mathbf ...
0
votes
2answers
495 views

Definition of denumerable (countable) set

When we say that a set $S$ is denumerable, that is, there is a bijection $S \to \omega$, do we mean that there exists such a bijection or do we mean that we have one and are talking about a pair ...
2
votes
4answers
266 views

What's the definition of $\omega$?

This is a follow up on a comment to one of my previous questions. What's the definition of $\omega$? Are the following equivalent definition of $\omega$: $\omega$ is the initial ordinal of ...
4
votes
1answer
106 views

What is the definition of the well-founded part of a model of set theory?

I've been trying to understand John Steel's various notes on inner model theory, but the one thing that trips me up is what he calls the well-founded part of a model of set theory. What exactly is the ...
2
votes
1answer
150 views

Discussion: Differing definitions for the rank of a set

I've just identified that the definition we used for the rank of a set in my set-theory class (1.) is different than the one I commonly find on the web (2.). $\text{rank}(A)=\min\{\alpha\mid ...
3
votes
3answers
237 views

What does $H(\kappa)$ mean?

As the typical references (Wikipedia, Mathworld, etc.) don't seem to address this satisfactorily, I figured this would be a good place to put a nice formal definition. Hence: I've heard that if ...
2
votes
2answers
134 views

Am I allowed to realize one object twice within one set-theory?

Say I consider a set theory with the Axioms of Extensionality and the Axiom of Pairing. As I understand it, stating the axiom allows me to make a definition like $$(a,b):=\{\{a\},\{a,b\}\}$$ and ...
19
votes
3answers
579 views

Conflicting definitions of “continuity” of ordinal-valued functions on the ordinals

I've encountered the following definition in Kunen, Levy, and other places: A function $\mathbf{F}:\mathbf{ON}\to\mathbf{ON}$ is continuous iff for every limit ordinal $\lambda$, we have ...
2
votes
1answer
235 views

Is Gödel's completeness theorem a representation theorem?

In general a representation theorem is — according to Wikipedia — a "theorem that states that every abstract structure with certain properties is isomorphic to a concrete structure". ...
0
votes
0answers
173 views

Equinumerosity without ordered pairs

After some discouraging comments to this question let me ask straight ahead: Can the concept of equinumerosity be defined basically without the concept of ordered pairs (in any of its ...
1
vote
3answers
187 views

Equivalence relations and bijections without ordered pairs

Is it correct that — opposed to general relations and functions — equivalence relations and bijective functions can be defined without reference to ordered pairs? Especially, do the ...
4
votes
1answer
102 views

Help on a definition

In many books I find these two definitions of Ramsey ultrafilters, extremely similar but different: 1)For every partition $\mathbb{N}=\bigsqcup A_k$ with $A_k\not\in\mathcal{U}$ there exists ...
7
votes
2answers
448 views

Definition of a set

What is a set? I know that results such as Russell's paradox mean that the definition isn't as straight forward as one might expect.