1
vote
2answers
166 views

Is there an intuitionistic generalized boolean algebra (of Stone)?

A "boolean algebra without the greatest element" was called by Stone "generalized boolean algebra" and he axiomatized it. Is there any publication about "preudo-boolean algebras without the greatest ...
9
votes
1answer
449 views

Well founded induction attributed to Noether

What I know as well founded induction, namely the rule $$ \big(\forall y.(\forall z.z\lt y\Rightarrow\phi z)\Rightarrow\phi y\big)\Longrightarrow\big(\forall x.\phi x\big), $$ whose validity is the ...
5
votes
0answers
179 views

constructive Serre classes

A Serre class (of abelian groups) is a class of abelian groups closed under subgroups, quotients, and extensions. For instance, finitely generated groups and finite groups are both Serre classes. ...
11
votes
1answer
424 views

First order decidability of rings vs Diophantine decidability

Are there known (preferably ``concrete'') examples of a ring $R$ (commutative, with 1) such that: $\bullet$ the first order theory of $R$ is undecidable, but $\bullet$ the positive existential (= ...
0
votes
1answer
175 views

Poset axioms of Boolean algebra [closed]

I found in Awodey's "Theory Category", second edition p34, a set of poset axioms to define Boolean algebras, whereas I don't see how they can be sufficient. Here are the axioms: A Boolean algebra ...
1
vote
0answers
157 views

What are the enforceable models of local artinian rings?

I was reading Hodges' "Model Theory" Chapter 8 a propos existentially closed models of $\forall_2$ theories in a countable first order language $L$. He extends the proof of the omitting type theorem ...
3
votes
2answers
318 views

What logic is modelled by generalized boolean algebra?

While it is well known that classical propositional logic is modelled by booleaan algebras, I have never heard of the logic modelled by generalized boolean algebras (GBA) defined by M.Stone (author of ...
14
votes
1answer
417 views

Nice Algebraic Statements Independent from ZF + V=L (constructibility)

Background and Motivation I've always been fascinated about algebraic statements independent from ZFC set theory. One such fascinating example comes from considering ...
2
votes
2answers
393 views

Model Theoretic Localization

This is a re-post on a previous question I asked. My first question was too vague to warrant detailed responses. Really, I have two specific questions to ask. 1) Let $\sigma = (A; \{0,1\}; +, ...
9
votes
2answers
849 views

Non-Standard Prime

Hello, My question is about the non-standard models of the integers. If we add to the Peano's axioms $P$ of arithmetic the following axioms for a fixed constant $c$: $c \neq 0$, $c \neq 1$, $c \neq ...
8
votes
2answers
1k views

Maximal ideal and Zorn's lemma

It is known that any ring A (say commutative with 1) has a maximal ideal. The proof uses Zorn's lemma. Now I heard some people saying that if we assume A to be noetherian, then we don't need to use ...
10
votes
0answers
1k views

Reverse mathematics strength of identically zero polynomials are the zero polynomial

According to wikipedia, the statement "every polynomial over a countable field that is not the zero polynomial has only finitely many roots" is equivalent to RCA0 over RCA0* (which is called ERCA-0 in ...
0
votes
1answer
946 views

Dual of Zorn's Lemma? [closed]

It seems to me that the dual of Zorn's Lemma should be true: if $S$ is a non-empty partially ordered set and every chain of $S$ has a lower bound in $S$, then $S$ has at least one minimal element. ...
4
votes
4answers
350 views

Lower bounds on the degrees of representatives of $u^n$ as $n \to \infty$

Let $k$ be an algebraically closed field and $A$ a finitely generated $k$-algebra, together with a specified surjective morphism $\phi \colon k[x_1, \dotsc, x_n] \to A$. For $f \in A$, define ...
1
vote
2answers
454 views

Group & modules of arbitrary cardinality

How do I see that there is a group of an arbitrary cardinality? Is this also true for abelian groups? Also, given a commutative ring $R\neq 0$ how do I see that there is an $R$-module of arbitrary ...
44
votes
2answers
4k views

How would you solve this tantalizing Halmos problem?

1-ab invertible => 1-ba invertible has a slick power series "proof" as below, where Halmos asks for an explanation of why this tantalizing derivation succeeds. Do you know one? Geometric series. In ...
9
votes
1answer
760 views

Does ZF prove that all PIDs are UFDs?

Main Question: Does ZF (no axiom of choice) prove that every Principal Ideal Domain is a Unique Factorization Domain? The proofs I've seen all use dependent choice. Minor Questions: Does ZF + ...
4
votes
1answer
300 views

Prime-ness checking for polynomial ideals over ACFs( algebraically closed fields).

Let $f_1,\ldots f_m \in k[X]$ have degrees bounded by $l$. and $I(\bar{f})$ be the ideal generated by $\bar{f}$. If $I(\bar{f})$ is not a prime ideal then its non-primeness is witnessed by polynomials ...
4
votes
1answer
623 views

Parametric polynomial solution of a single polynomial equation

Let $P$ be a polynomial in $n$ variables with rational coefficients, $P \in {\mathbb Q}[Z_1,Z_2, \ldots ,Z_n]$, and consider the algebraic set $Z=\lbrace (z_1,z_2,z_3, \ldots ,z_n) \in {\mathbb Q}^n ...
11
votes
6answers
901 views

When can we prove constructively that a ring with unity has a maximal ideal?

Many commutative algebra textbooks establish that every ideal of a ring is contained in a maximal ideal by appealing to Zorn's lemma, which I dislike on grounds of non-constructivity. For Noetherian ...
20
votes
5answers
1k views

Algebraic description of compact smooth manifolds?

Given a compact smooth manifold $M$, it's relatively well known that $C^\infty(M)$ determines $M$ up to diffeomorphism. That is, if $M$ and $N$ are two smooth manifolds and there is an ...