first-order and higher-order logic, model theory, set theory, proof theory, computability theory, formal languages, definability, interplay of syntax and semantics, constructive logic, intuitionism, philosophical logic, modal logic, completeness, Gödel incompleteness, decidability, undecidability, ...

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-4
votes
0answers
24 views

profit and loss related aptitude qustion? [on hold]

A sold a watch to B at a gain of 5% and B sold it to C at a gain of 4%. If C paid Rs. 1902 for it, the price paid by A is?
2
votes
1answer
187 views

Algebras admitting quantifier elimination

I apologize if this question is meaningless or trivial: What are examples of Algebras admitting quantifier elimination? Especially are there Groups admitting quantifier elimination? I need to say ...
22
votes
5answers
2k views

Nonstandard analysis in probability theory

I am quite new at nonstandard analysis, and recently I became aware of its use in probability theory mainly through the following two books: Nelson (1987). Radically Elementary Probability Theory ...
3
votes
1answer
103 views

Number of non-isomorphic models

I had this question up on Math stackexchange: http://math.stackexchange.com/questions/1349247/number-of-non-isomorphic-models/1350763#1350763 . While it was answered partially there, I'm posting here ...
0
votes
0answers
266 views

Set as a (strict) infinite-category?

First, let me say that I have no idea if such a post has its place here. However, I believe that the ideas I'm going to present are important. The goal of this thread is three fold: 1) trying to ...
-1
votes
0answers
21 views

Freedom of speech in scientific discussions - An invitation to more tolerance in Scientific debates [migrated]

I hope this post enjoys some tolerance, and don't get closed or put on hold immediately. I believe that freedom of speech in scientific discussions is one of the key values which enriches the debates ...
7
votes
0answers
216 views

preserving saturated ideals

A reliable source made the following claim: Suppose CH there is an $\omega_2$-saturated ideal on $\omega_1$. Then this is preserved by $\mathrm{Add}(\omega_1,\omega_2)$. Question 1: How do you ...
11
votes
1answer
493 views

Can there be a global linear ordering of the universe without a global well-ordering of the universe?

This question arose in the answers to Asaf Karagila's question Does ZFC prove that the universe is linearly orderable?. The answer there was that one can have a ZFC model with no global linear ...
0
votes
0answers
36 views

Logic resolution and logic consequence [on hold]

Which of this are false? a) If some formula H results from premises D, then H could be derived from D with using resolution (reapetedly) rule. b) If some formula H results from premises D, then we ...
28
votes
3answers
2k views

What is the reverse mathematical strength of the fundamental theorem of algebra?

Reverse mathematics (RM) is that area that tries to pin down exactly which axioms are necessary to prove theorems, given some weak base theory. Harvey Friedman has pointed out several times (on the ...
16
votes
1answer
660 views

Goodstein's theorem without transfinite induction

Goodstein's theorem is an example of a theorem that is not provable from first order arithmetic. All proofs of the theorem seem to deploy transfinite induction and I've wondered if one could prove the ...
1
vote
2answers
153 views

Are monoids with zero and partial homomorphisms related?

Context: Let $\Sigma=\{U,C,A,G\}$ and $L\subset\Sigma^*$, i.e. $L$ is a language over the alphabet $\Sigma$. Let $\Sigma'=\{0,1\}$ and define a homomorphism $f:\Sigma^*\to\Sigma'^*$ by extending $U ...
3
votes
0answers
60 views

On an automatic translation of typed lambda calculus in untyped lambda calculus

I have a question regarding the "compilation" of typed lambda calculus in untyped lambda calculus. Take for example the inductive definition of lists, with introduction rules: and: We can ...
1
vote
0answers
62 views

Type theory: can multiple elimination rules be defined, in principle?

I'd like to ask a question on type theory: Consider the usual type theoretical definition of the natural numbers. We could give an elimination rule in the form: or in the form: I called the ...
2
votes
2answers
149 views

Pseudo-decision procedures for first order arithmetic

I was reading this paper http://citeseerx.ist.psu.edu/viewdoc/download?doi=10.1.1.117.2911&rep=rep1&type=pdf in which the author describes an algorithm, based on Groebner basis, ...
5
votes
0answers
75 views

Lascar strong types in fragments of arithmetic

Are Lascar strong types (definition below) in models of fragments of arithmetic always type definable? (They trivially are, in models of full induction.) Definition Given a saturated model ${\cal M}$ ...
4
votes
2answers
762 views

What lets the Square of Opposition fail in Intuitionistic Logic?

See moderator's note in the comments. I just came across the following. In intuitionistic logic and classical logic we have the following consequences: ...
9
votes
1answer
369 views

Just a little absoluteness might be cheaper?

Absoluteness is a wonderful thing, but expensive consistency-strength wise. My question is, when can we get large amounts of absoluteness in specific situations for much cheaper? Specifically, fix a ...
10
votes
14answers
6k views

undergraduate logic textbook

I am going to teach the standard undergraduate Logic course for math and engineering majors. What are good (bad) text-books and why. I have not taught that course for a while and wonder if there are ...
10
votes
0answers
163 views

Generic $\mathbf{\Sigma}_3^1$-absoluteness for class forcings

In the paper "Generic Absoluteness" by Bagaria and Friedman (http://www.logic.univie.ac.at/~sdf/papers/bagfried.pdf) it is shown that in ZFC generic $\mathbf{\Sigma_3^1}$-absoluteness is false for ...
15
votes
1answer
334 views

Computability of Brauer groups

A friend of mine and I were talking about computable algebra, and this question came up. The answer may already be known, but I couldn't find it with Google: Suppose I have a countable field, $k$. ...
7
votes
1answer
224 views

Countable group with uncountable number of subgroups $< 2^{\aleph_0}$

Is it consistent that there is a countable group $G$ such that the cardinality of the set of subgroups of $G$ is uncountable, but strictly less than $2^{\aleph_0}$?
7
votes
2answers
274 views

Topological tameness beyond the Gandy-Harrington topology

The Gandy-Harrington topology on $\omega^\omega$ is the topology generated by all lightface $\Sigma^1_1$ sets; that is, all sets which are continuous-in-the-usual-sense images of $\omega^\omega$. ...
25
votes
5answers
1k views

Forcing as a replacement of induction and diagonal arguments

Let me give some examples motivating the question. The use of forcing instead of induction: For this consider Cantor's theorem: Theorem 1. Any two countable dense linear orders $I, J$ without end ...
-1
votes
1answer
97 views

Non-standard naturals and goodstein sequences [closed]

By the Kirby–Paris theorem, Goodstein's theorem is independent of Peano arithmetic (PA). Therefore there are non-standard models in which every Goodstein sequence terminates. However, Tennenbaum's ...
5
votes
1answer
295 views

Details for Woodin's forcing argument for a saturated ideal from the Levy collapse

Theorem 2.65 in Woodin's book shows that a saturated ideal on $\omega_1$ exists after Levy-collapsing a Woodin cardinal $\delta$ to $\omega_2$. I am confused about the part of the argument where he ...
8
votes
1answer
183 views

Is the Jordan decomposition of a self-adjoint functional constructive?

Let $A$ be an abstract C*-algebra, and let $\varphi\colon A \rightarrow \mathbb C$ be a bounded linear function. Assuming the axiom of choice, there exist unique positive bounded linear functions ...
-3
votes
0answers
28 views

Calculating minimum number of comparisons to ensure 1 and only 1 value set [migrated]

This is a math problem with software limitations, but I felt it more appropriate here than in a software SE. I am forced to work with a very inflexible tool and must check 5 fields, each of which can ...
7
votes
3answers
263 views

Do non-normal states exist in the Solovay model?

Let H be an infinite dimensional Hilbert space. Then there exist non-normal states on B(H) in ZFC (i.e. states that are not represented by a density operator). Is this also true in the Solovay model ...
9
votes
1answer
439 views

Coding a model of $0^\sharp$ from a $\Pi^1_1$ Gale-Stewart game

As a preface to this question, this is my first time asking on Math overflow, and this seemed like the sort of question that would be acceptable here. However, I apologize if it is not. A method for ...
9
votes
0answers
177 views

The Chang model after collapsing an inaccessible limit of Woodins

If $\kappa$ is an inaccessible cardinal and $G \subset \operatorname{Col}(\omega,\mathord{<}\kappa)$ is a $V$-generic filter, then in $V[G]$ the Chang model $L(\text{Ord}^\omega)$ satisfies "every ...
18
votes
3answers
2k views

Who needs Replacement anyway?

The set theory ETCS famously comes without the Replacement axiom schema (or an equivalent) that is part of ZFC. One (to me, not apparently useful) set that one cannot build in ETCS is $\coprod_{n\in ...
4
votes
0answers
173 views

Primitive Closure Arithmetic

I am researching a system that was designed by myself and what I call PCA (Primitive Closure Arithmetic), because it looks like PRA. The differences are: - PRA uses recursive definition with a ...
0
votes
1answer
152 views

Definition of “Expected/Unexpected Event”

Background of my question is Martin Gardner's "unexpected hanging" paradoxon, which has once again be the subject of an article in a popular-scientific magazin (this time because this year it has been ...
11
votes
0answers
186 views

Adding a saturated ideal

Is it consistent that there is no $\omega_2$-saturated ideal on $\omega_1$, but one is introduced by an $\omega_2$-closed forcing?
10
votes
0answers
197 views

c.c.c forcing notions and adding minimal generic reals

Is the following statement consistent: ``There is no non-trivial c.c.c forcing notion adding a minimal generic real''? The question is related to Prikry's question: Is it consistent that any ...
11
votes
3answers
418 views

The size of Lindelof space

Question. Suppose that $X$ is a Lindelof space such that every point of $X$ is a $G_{\delta}$-point. Then is it true that $|X| ≤ 2^{\omega}$?
9
votes
1answer
376 views

Is $\clubsuit_{\omega_1}$ enough to get Suslin tree?

This is problem 15.3 in Arnie Miller's problem list: (Juhasz) Suppose there exists $\langle A_{\alpha} : \alpha \in L \rangle$, where $L$ is the set of limit ordinals below $\omega_1$ and for each ...
7
votes
1answer
242 views

For a ring $k$ and a set $X$, what are the $k$-algebra homomorphisms $k^X \to k$?

Let $k$ be a commutative ring. Feel free to assume it's a field. Let $X$ be a set. This question is only interesting when $X$ is infinite. Write $k^X$ for the $k$-algebra of functions $X \to k$, ...
4
votes
1answer
161 views

Cohen algebra and $\mathcal P(\omega)/ \mathrm{fin}$

Let $C$ be the Cohen algebra, the boolean completion of the partial order of finite partial functions from $\omega$ to 2, ordered by reverse inclusion. Does there exist an ideal $I$ on $C$ such that ...
2
votes
0answers
197 views

Is there a $\Sigma^0_3$-complete ideal on $\omega$?

In Kechris's book "Classical Descriputive set theory" Chapter 23 (Exercise 23.4), it claim that there is a $\Sigma^0_\xi$-complete ideal on $\omega$ for each $\xi\geq 3$. There is a candidate ...
1
vote
0answers
67 views

BL Algebras that allow for Compactness to hold

Say we have a model $M$ of a theory $T$ of some core fuzzy logic. When dealing with compactness, we run in to a situation where the new model being built (by the use of compactness over $M$), will ...
11
votes
1answer
192 views

Does Peano's existence theorem admits a constructive proof?

$$y(t)=y_0+\int_0^t b(y(s))ds$$ $b\in C(R^d)\cap L^\infty(R^d)$ The classical proof for Peano's existence theorem in ODE need use the Ascoli's theorem, so it's not constructive. When $d=1$, in the ...
5
votes
1answer
474 views

$\Sigma_1$ elementary substructure

This is a question I was given in the exam: show that if $\lambda, \kappa$ are uncountable cardinals then $$(H_\lambda,\in) \prec_1 (H_\kappa,\in)$$ where $H_\lambda$ is the class of all sets $x$ such ...
5
votes
1answer
203 views

Conservativity of multiplicative linear logic over intuitionistic multiplicative linear logic

It is well known that multiplicative linear logic (MLL) is conservative over intuitionistic multiplicative linear logic (IMLL). In other words, if an IMLL formula is provable in MLL then it is already ...
7
votes
2answers
492 views

What does “simplification of proofs as evaluation of programs” mean?

I am currently going through Philip Walder's "Proposition as Types" and a passage of the introduction has struck me: for each way to simplify a proof there is a corresponding way to evaluate a ...
10
votes
0answers
286 views

On sentences true in all finite groups (revisited)

Let $w$ be a group word with variables $\bar x, \bar y$, where $\bar x=(x_1,\dots ,x_m)$ and $\bar y=(y_1,\dots ,y_n).$ I am interested in the following questions. (1) Is the sentence $(\forall\bar ...
8
votes
1answer
263 views

Can there be a tree of height $\omega_2$ having all levels countable, with no cofinal branch?

For many years I had the idea that if a well-founded tree is both very tall and very narrow, then it must have a cofinal branch. For example, it is a fun exercise to show that any $\omega_1$-tree ...
5
votes
0answers
90 views

O-Minimal sentences in $L_{\omega_1,\omega}$?

Is there any meaningful sense in which we can talk about o-minimal sentences of $L_{\omega_1,\omega}$? I can give a first attempt, easily; given a countable fragment $F$ and a sentence $\Phi$ in that ...
7
votes
1answer
187 views

Introducing meets while preserving directed closure

A poset $\mathbb{P}$ is called well-met iff every pair of compatible conditions in $\mathbb{P}$ has a greatest lower bound. Question: Suppose $\mathbb{P}$ is a separative partial order which is ...