# Tagged Questions

361 views

### Are there counter intuitive interpretations of ZF or ZFC?

Are there interpretations of ZF or ZFC that are non standard in the sense that $\epsilon$ is interpreted in a counter intuitive way that intuitively has nothing to do with "belongs to" or "is part of" ...
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### Sections, Transversals and Quotient Maps

Here I read: Given a quotient space $\bar X$ with quotient map $\pi\colon X \to \bar X$, a section of $\pi$ is called a transversal. I asked myself how such sections are possible. It must be a ...
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### Categorial definition of subsets

I cannot see why this definition http://en.wikipedia.org/wiki/Subobject is equivalent to subsets in the category of sets. I am confused by the following facts, 1) the partial order is defined ...
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### Bourbaki Proof of Zorn's Lemma in Lang's Algebra

Serge Lang in his book Algebra has a nice appendix on set theory at the end of the book. In particular, in paragraph 2, pp. 881-884 he provides a proof of Zorn's Lemma from "other properties of sets ...
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### Axiom of Choice and Ascending Chain Condition

Matsumura, in his "Commutative Ring Theory" p. 14 proves that "A partially ordered set $\Gamma$ satisfies the ascending chain condition $\Leftrightarrow$ every nonempty subset of $\Gamma$ has a ...
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### Number of non-isomorphic subgroups of $p$-adic integers.

What is the cardinality of the set of non-isomorphic subgroups of $p$-adic integers $\mathbb Z_p$ for a given $p$? The obvious upper bound is $2^{2^{\aleph_0}}$. But are there $2^{2^{\aleph_0}}$ ...
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### Give an example of a field of order $\beth_2$

I'd like an example of a field of order $\beth_2$ (that is, the cardinality of the power set of the continuum). I'd prefer more explicit constructions, if possible. This is just out of curiosity, as I ...
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### Prüfer groups are countable

I have read that for any prime number $p$ the PrÃ¼fer $p$-group is countable. My question is: where can I find a proof of this fact? Thanks.
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### pseudo numbers and surreal numbers

A surreal number $\{x_L\|x_R\} \in No$ is a number when for all $\xi\in x_L$ and all $\eta \in x_R$ we have $\eta > \xi$. All the things $\{x_L\|x_R\}$ which are not of that form are called ...
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### What's the name of this operator?

Let $f,g$ be functions in $C^A$ and $C^B$ respectively. Let $\boxtimes:C^A \times C^B \to (C\times C)^{A \times B}$ s.t. $f\boxtimes g(a,b)=(f(a),g(b))$ It seems not the tensor product, nor ...
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### Constructing the reals from fractions of ordinals

We can construct the positive rationals from ratios of positive integers (and thus from pairs of finite ordinals). Can we analogously construct the reals from pairs of countable ordinals?
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### Formal Definition/counter part in mathematics for “Objects” of Object Oriented Models [closed]

I'm a newbie in both formal mathematics and theoretical computer science, so please bear with me if you find my question is not properly framed. Object Oriented Modeling seems very useful in defining ...
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### Infinite associativity condition

How associativity condition may be formulated for a function taking an arbitrary ordinal number of arguments? For a binary operation $\ast$ it is $(a\ast b)\ast c = a\ast (b\ast c)$, but I want an ...
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### Infinite rooted binary tree

Let $T$ be a rooted infinite binary tree and let $\text{Sym}(T)$ be the group of all symmetries of $T$. Show that any $\alpha \in \text{Sym}(T)$ sends the root to the root, even if you just view ...
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### Algebraic closure for $\mathbb{Q}$ or $\mathbb{F}_p$ without Choice?

I know the usual proof of the existence of an algebraic closure for any field using Zorn's Lemma. The answer to this previous question makes it clear that in general, some nonconstructive axiom (not ...
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### How many sieves are there on a given rational number $q$?

Consider the poset category $\mathbb{Q}^{op}$, i.e. where $p \rightarrow q$ iff $p \geq q$. Take any $q \in \mathbb{Q}$. Then how many sieves are there for $q$ in $\mathbb{Q}^{op}$? Supposedly, the ...
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### Is $\operatorname{Hom}_A(M,N)$ a set without axiom of choice?

Let $M$ and $N$ be $A$-modules, $\operatorname{Hom}_A(M,N)$ the set of all $A$-module homomorphisms $M\rightarrow N$. $\operatorname{Hom}_A(M,N)$ can be viewed as a subset of the cartesian product ...
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### Does a 'universal' group/ring/field/topology/etc. exist?

My question is inspired by the fact that there is no universal set (at least in ZF). There are many abstract objects such as group, ring, field, vector space, topology, etc. such that we can say about ...
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### Zorn's lemma in abstract algebra?

It is well konwn that Zorn's lemma implies: Prop.1 Every commutative unital ring has a maximal ideal. Prop.2 Every proper ideal is contained in a maximal ideal in a unital ring. Question: Can we ...
Consider the following diagram: commutative diagram (Sorry, It wouldn't let me directly post the image.) What does it mean precisely to say "$f$ factors through $G/\text{ker}(f)$"? Does it mean \$f ...