Questions about mathematical logic, including model theory, proof theory, computability theory (a.k.a. recursion theory), and non-standard logics. Consider using one of the following tags: (model-theory), (set-theory), (computability), (proof-theory) if they fit the question.

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73 views

Formula that's only satisfiable in infinite structures

What formula in first order logic can I write that's only satisfiable over infinite structures, over a dictionary without the = sign?
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1answer
18 views

Prove that if $\forall A \in \mathcal F (B\subseteq A)$ then $B \subseteq \bigcap \mathcal F $

This is Velleman's exercise 3.3.10. Suppose that $\mathcal F$ is a nonempty family of sets, B is a set, and $\forall A \in \mathcal F (B\subseteq A)$. Prove that $B \subseteq \bigcap \mathcal F $. My ...
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0answers
49 views

Algorithm to answer existential questions - Reduction

Lemma 1. For any $x$ in the ring $F[t,t^{-1}]$ ($F[t,t^{-1}]$: the polynomials in $t$ and $t^{-1}$ with coefficients in the field $F$), $x$ is a power of $t$ if and only if $x$ divides $1$ and ...
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2answers
33 views

Splitting condition of forcing posets

I was looking at Wikipedia for brief reminders of what I learned in my elementary set theory class, and discovered the forcing page (which I did not learn): A forcing poset is an ordered triple, ...
2
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2answers
64 views

Is there much of difference between set models and class models?

When we talk of class models and set models, is there a need to talk about them separately? What would be an example? What I can think of is that a class cannot technically be talked inside a model ...
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0answers
13 views

Clarification of conditional propositions [duplicate]

I am studying first order logic and we have been introduced to conditional propositions.$(p \Rightarrow q)\;$ The truth table for $p \Rightarrow q$ is this: ...
9
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1answer
141 views

Is there any connection between the symbol $\supset$ when it means implication and its meaning as superset? [duplicate]

A rather old-fashioned symbol for logical implication is $\supset$ (see list of logic symbols). For example $p \supset q$ means $p \implies q$ or $p \rightarrow q$ in more recent notations. Is there ...
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3answers
66 views

Please help me understand the proof

Doubt In third line of the proof, why is Q $\rightarrow$ R ? Thanks
4
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1answer
109 views

Quantum and Paraconsistent Logics

Is there any relation between these two logics? I know both are completely different "from construction", but recently I've read that paraconsistent logics tries to explain some things in quantum ...
2
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1answer
59 views

What is the definition of a definable set of statements, and what is a constructive way to think of this regarding Tarski's Undefinability Theorem?

Logic and model theory are not my area so my thinking is probably off, but I am curious about this so please go ahead and set me straight. A definable set is one for which there is a formula that is ...
2
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2answers
45 views

Order of statements in implication

The question is from Exercise 13 of part 1.4 in Rosen's "Discrete Mathematics and Its Applications" (5th edition): "let $M(x,y)$ be "$x$ has sent y an e-mail message", where the universe of discourse ...
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1answer
61 views

Meaning of “such that”

The use of the term "such that" confuses me I've seen this like $A=\{(x,y) :x,y\in\Bbb R\ \text{and } P(x,y) \}$ and $B=\{(x,y)\in \Bbb R^2:P(x,y)\}$ for some predicate $P$. Is there any difference ...
2
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2answers
37 views

Testing whether Argument is valid or not

I am to determine if argument is valid by making truth table ATTEMPT Let W= Warning lights will come on P= Pressure is high R=Relief valve is clogged Then i have premises as W ...
3
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2answers
63 views

A simple expression to map $\mathbb N^*$ bijectively to $\mathbb N$

Let $\mathbb N = \{ 1,2,3,\ldots \}$, then by the well-known "Cantor"-Scheme we have $\mathbb N \times \mathbb N \cong \mathbb N$. But even nicer is that we can write this scheme $\varphi : \mathbb N ...
1
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1answer
38 views

Proof of a classical Theorem of Martin-Löf on complexity dips for Kolmogorov complexity,

I have a question on the first Theorem from the article Complexity of Oscillations in Infinite Binary Sequences by P. Martin-Löf, which could be downloaded from the publisher or from here. Theorem ...
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2answers
32 views

Boolean Algebra Problem ABCC'

Hi I just want to ask the answer of this Boolean Algebra problem.. $$ABCC' + B + A'B $$ How to simplify that one?
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1answer
41 views

Proof outline of a certain sentence (Introductory course on logic, proof writing et al.)

The exercise asks me to outline a direct proof that if $\mathbf A$ is a diagonal matrix, then $\mathbf A$ is invertible whenever all its diagonal entries are nonzero. To me this sounds like ...
6
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1answer
313 views

Elaboration of calculus in finitistic math

I was just curious if there were some approaches to prove major theorems of calculus in finitistic systems like PRA? I'm aware of some questions which were already considered, e.g. ...
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1answer
28 views
0
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1answer
57 views

Existential quantifier axioms in Halmos' system, equivalence proof needed

I have to refer to page 21 of the book cited in the link below. There is a list of axioms Q1-Q5, and an assertion that they are equivalent to a shorter set, namely Q1, Q2 and Q6. I am trying to derive ...
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2answers
65 views

What is the formal negation of the statement “There is much X in Y”. [on hold]

What is the formal negation of the statement "There is much X in Y"? The answer to me is that "It is not the case that there is much X in Y" But I want a more useful negation. Can I say that its ...
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3answers
62 views

Distribution of universal quantifiers over implication

I want to prove that $∀x(φ(x)⟹ψ(x))$ implies $∀x(φ(x))⟹∀x(ψ(x))$. I read they are not equivalent, but I am not sure why. I tried the following: $∀x(φ(x)⟹ψ(x))$ $⟹[φ(a)⟹ψ(a)]$ is true. $⟹φ(a)$ is ...
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2answers
56 views

Are the quantifiers interchangeable?

In other words, is it true that $\forall x \; \exists y\;\phi(x, y) \iff \exists y\;\forall x \; \phi(x, y) $?
2
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1answer
36 views

what is essentially universal or existential?

In Lambda-Prolog , I see essentially universal quantifier or essentially existential quantifier such terms, I am confused. It seems the universal quantification of a variable in program or goal is ...
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1answer
35 views

Solve the following proof : M |- M ∨ {[(Z∨S) ∧ (¬] → (C↔D)}

Solve the following proof : M |- M ∨ {[(Z∨SC↔D)} I try to proof above question with the following (F⋀Z)⋀ → (C↔D) 1 (F⋀Z)→C 2 F⋀Z 1⋀E 3 F 2⋀E really confused :( this ...
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0answers
62 views

Satisfiability/compactness theorem

I am trying to solve the following problem: Let $\mathcal{F}$ be a set of propositional formulas. Assume that for any valuation map $v$ there is some $F$ $\in$ $\mathcal{F}$ such that $v^*(F) = ...
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2answers
1k views

What is the purpose of implication in discrete mathematics?

I would be obliged if you can show me an example of a truth table for implication where there is a also a real life aspect to it. (i.e., where would someone use the scenario to make F->F = T and also ...
2
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2answers
41 views

Inference in Predicate Logic

I have stumbled upon the following reasoning, but I'm not sure if it's correct. Here it goes: Domain X $\forall x :\phi(x)⟹\gamma(x)$ Let $E\subseteq X⟹[\forall x\in E :\phi(x)⟹\gamma(x)]$ Suppose I ...
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4answers
117 views

Proving that if $a+b$ and $ab$ are of the same parity, then $a$ and $b$ are even.

Here is the proof: Let $a,b \in \mathbb{Z}$. Prove that if $a+b$ and $ab$ are of the same parity, then $a$ and $b$ are even. When working these problems, I do try to set them up logically. My ...
2
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1answer
128 views

Is there a rule for uniform substitution of predicate symbols in FOL?

In a Hilbert-style axiomatization of first-order logic (FOL), there is a rule for variable substitution but I don't see any rule for substituting predicate symbols. Consider a theorem like: $\forall ...
2
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1answer
67 views

(totally) (M,P)-generic forcing condition

We say a cardinal $\theta$ is sufficiently large for a forcing $Q$ if $\mathcal{P}(\mathcal{P}(Q)) \in H(\theta)$. And a set $M$ is a suitable model for $Q$ if $Q \in M$ and $M \prec H(\theta)$, $M$ ...
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0answers
59 views
+50

Reference request: fixed point and first-order logics

I'm looking for materials on the relationship between first-order and fixed-point logics, specifically on the condition for a formula in a fixed-point logic to have an equivalent first-order formula. ...
4
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1answer
90 views

If $\phi$ holds for all standard models of ZF and ZF proves this, then does ZF prove $\phi$?

I apologize if this is a nonsensical question. Suppose $\phi$ holds in all standard models of ZF. Suppose further that ZF proves this. Then does ZF prove $\phi$?
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2answers
40 views

Max/Min to logical operator transformation and viceversa

I have some doubts in transforming conditions that involve max/min in logical operator condition and viceversa. In particular, should be (I put some examples, I would know if I'm right and the ...
4
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1answer
111 views

The existential theory is undecidable

Lemma 1. For any $x$ in the ring $F[t,t^{-1}]$ ($F[t,t^{-1}]$: the polynomials in $t$ and $t^{-1}$ with coefficients in the field $F$), $x$ is a power of $t$ if and only if $x$ divides $1$ and ...
1
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3answers
171 views

Is there any commonality between Math induction and Logic induction?

Logic induction is reasoning by probability. Math induction seems to be related to just Natural numbers and used to prove a statement for every natural number. From these definitions there is no ...
1
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3answers
72 views

Suppose $A\subseteq \mathscr P (A)$. Prove that $ \mathscr P (A)\subseteq \mathscr P ( \mathscr P (A))$

This is Velleman's exercise 3.3.4. Suppose $A\subseteq \mathscr P (A)$. Prove that $ \mathscr P (A)\subseteq \mathscr P ( \mathscr P (A))$. I started reexpressing the terms in their equivalent forms ...
2
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2answers
77 views

Why are some conditionals regarded false even if the antecedent is false?

In the Mendelson's logic book, there are 2 conditionals which Mendelson says they are regarded false even if their antecedent is false. One of them is the following: If this piece of iron is ...
2
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1answer
83 views

What does it mean to axiomatize a logic?

I'm sorry if this question is not clearly formulated: An axiomatization, or an axiomatic system, usually means a set of axioms (i.e. a theory). A formal theory is such a set of formulas in some formal ...
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1answer
25 views

Logic question and proportions [closed]

there's something that have been bugging me. If we have quantities A, C, E And if we have quantities B, D, F And if we take the equimultiples G, H, K from A, C, E And if we take the equimultiples L, ...
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2answers
106 views

Explicit construction of a nonmeasurable set, where only the proof of correctness uses choice?

By Solovay's theorem, assuming the existence of an inaccessible cardinal, the axiom of choice is necessary to prove the existence of nonmeasurable sets. In the past, I've thought that one consequence ...
2
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0answers
59 views

Is this a typo in Jech's Set Theory?

In Jech's Set Theory, p. 603 in the chapter about Proper Forcing, the proof of Theorem 31.7. In the second but last paragraph, the proof says By Theorem 8.27 (Menas), $\lbrace M \cap \lambda ...
2
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1answer
37 views

Prenex normal form of $\neg \big(\forall x \ P(x) \vee \forall x \ Q(x) \big)$

I have the statement $\neg \big(\forall x \ P(x) \vee \forall x \ Q(x) \big)$ and I have to write it in prenex normal form. First I use the second De Morgan law $\neg \big(\forall x \ P(x) \vee ...
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0answers
43 views

Which one of the following well formed formula is a tautology?

Which one of the following well formed formula is a tautology? (a) $\forall x \, \exists y \, R(x, y) \equiv \exists y \, \forall x \,R(x, y)$ (b) $\left[\forall x \, \exists y \,(R(x, y) \implies ...
2
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3answers
50 views

Prove that if $x \neq 0$, then if $ y = \frac{3x^2+2y}{x^2+2}$ then $y=3$

The problem is the following (Velleman's exercise 3.2.10): Suppose that $x$ and $y$ are real numbers. Prove that if $x \neq 0$, then if $ y = \frac{3x^2+2y}{x^2+2}$ then $y=3$. My approach so ...
1
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1answer
27 views

Why $C(n\mid l(n)) \ge C(n) - C(l(n))$ for Kolmogorov complexity

Denote by $C(n)$ the plain Kolmogorov complexity of $n$ and the length of a binary encoding of $n$ by $l(n)$, why do we have $$ C(n\mid l(n)) \ge C(n) - C(l(n))? $$ If I have a shortest program $p$ ...
2
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1answer
32 views

Real Closed Fields with Predicate for a Dense Subfield

Consider $M = (\mathbb{R};+,<, \times, 0, 1, K)$ where $K$ is a unary predicate which holds on $\mathbb{Q}$ (or any dense subfield of $\mathbb{R}$). Question: Is it true that the parametrically ...
6
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2answers
88 views

Law of Clavius explained

Law of Clavius states $ \sim P \Rightarrow P \vdash P$ And the only explanation I sort of understand is ...
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0answers
12 views

Each recursive approximating sequence for Kolmogorov complexity is not uniform

Denote the plain Kolmogorov complexity by $C(x)$. Let $\phi(t,x)$ be a recursive function and $\lim_{t\to\infty} \phi(t,x) = C(x)$ for all $x$. For each $t$ define $\psi_t(x) := \phi(t,x)$ for all ...