6
votes
4answers
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" ...
0
votes
0answers
65 views

Algebraic constructions to add a real to a sub-group of $\langle \mathbb{R}^+,.\rangle$

Consider the group $\langle \mathbb{R}^+,.\rangle$ and let $r$ be a real number (e.g. $r=\sqrt{2}$). I would like to know about all known algebraic constructions to build a sub-group of $\langle ...
0
votes
0answers
60 views

Krull's theorem and AC

Lately I have been trying to prove axiom of choice based on krull theorem. The theorem states that for every ring with a unit $R$ that is not a field, there is a maximal ideal. I know it is equivalent ...
2
votes
1answer
68 views

A well order on $\mathbb Z$ that respects addition?

Does there exists any well-ordering on $\mathbb Z$ that respects addidtion that is if $a < b$ then $a +c < b+c$ for all $c$ in $\mathbb Z$?
8
votes
2answers
76 views

Zorn's Lemma in noetherian modules

For noetherian modules, we have in particular the equivalent definitions that the Ascending Chain Condition holds and that every nonempty subset of submodules has a maximal element. Now I can prove ...
1
vote
0answers
117 views

What is “Transitive induction”?

In the book "Artinian Modules Over Group Rings" By Leonid Kurdachenko, Javier Otal, Igor Ya Subbotin (also see http://books.google.com/) one can read (on p.117) "applying transitive induction, we ...
2
votes
1answer
98 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 ...
0
votes
1answer
109 views

Operating on a set of sequences - such as adding sequences and so on possible even when sequence is coded as number?

Suppose that there is a way to code some set of sequences into number. Then one is given one number. Suppose that we want to multiply all numbers in each sequence by that number and get the coded ...
1
vote
1answer
74 views

Coding a sequence into a natural number by map $f$ with $f(k_1, .., k_n) + f(k_{n+1}, .., k_{2n}) = f(k_1 + k_{n+1}, .., k_n + k_{2n})$

Has anyone discovered a way of coding a sequence of natural numbers into a natural number by map $f$ that has the following property $f(k_1, .., k_n) + f(k_{n+1}, .., k_{2n}) = f(k_1 + k_{n+1}, .., ...
0
votes
1answer
81 views

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 ...
7
votes
4answers
236 views

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 ...
5
votes
1answer
270 views

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 ...
3
votes
2answers
133 views

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 ...
3
votes
1answer
185 views

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 $2^{2^{\aleph_0}}$ ...
1
vote
2answers
120 views

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 ...
2
votes
1answer
128 views

Prüfer groups are countable

I have read that for any prime number $p$ the Prüfer $p$-group is countable. My question is: where can I find a proof of this fact? Thanks.
4
votes
2answers
279 views

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 ...
6
votes
2answers
123 views

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 ...
5
votes
2answers
263 views

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?
1
vote
1answer
190 views

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 ...
1
vote
2answers
280 views

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 ...
0
votes
2answers
193 views

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 ...
18
votes
4answers
959 views

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 ...
1
vote
1answer
58 views

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 ...
4
votes
2answers
149 views

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 ...
2
votes
1answer
288 views

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 ...
6
votes
2answers
442 views

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 ...
3
votes
2answers
187 views

What is the significance of “classes”?

In the introduction of Hungerford's Algebra (p. 2), he gives a rather trivial example of a class that is not a set, but what is the purpose of even having this term defined? Is it useful, other than ...
9
votes
1answer
1k views

What does it mean to say a map “factors through” a set?

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 ...