Tagged Questions

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Meaning of “There exists a proper class of…”

How a statement of the form "There exists a proper class of..." can be formalized in $\sf ZFC$? It sounds a bit like an oxymoron to me, because it's the very essense of a proper class that it does not ...
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Ordered tuples of proper classes

From time to time I encounter notation like this: A triple $\langle \mathbf{No}, \mathrm{<}, b \rangle$ is a surreal number system if and only if ... The confusing part is that a proper class ...
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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 ...
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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 ...
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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 ...
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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: ...
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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 ...
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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 ...
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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 ...
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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, ...
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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), ...
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(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 ...
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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 ...
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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 ...
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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 ...
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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 ...
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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} ...
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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 ...
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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 ...
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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 ...
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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". ...
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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 ...
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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 ...
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 ...