Questions about logic and 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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Use mathematical induction to prove that any integer n>=2 is either a prime or a product of primes.

Use strong mathematical induction to prove that any integer $n\ge2$ is either a prime or a product of primes. I know the steps of weak mathematical induction... basis step= $p(n)$ for $n=1$ or any ...
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1answer
38 views

Given an open statement determine if their quantification is true

The Question My Work/Question My book says for part a, iv is true. I disagree. To show an existential statement is false we have to show that for all x that statement is untrue. There are no ...
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2answers
61 views

Consider $(\mathbb N, +)$ as a model for the language with one binary function $+$ . Are the following statements true?

Consider $(\mathbb N, +)$ as a model for the language with one symbol $+$ for a binary function. Are the following statements true? $(\mathbb N, +) \vDash \forall x \exists y \forall z\ x + y\neq z$ ...
2
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1answer
34 views

How to deal with long and tedious logic problem? [closed]

I am always pretty bad at logic problems. Because most of the logics used aren't really logical (to me)So, as you might think, a long logic problem only adds to it already boring nature. The ...
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1answer
20 views

A predicate logic question about write down a sentence

Let $\mathcal{L}=\{f\}$ be a first-order language containing a unary function symbol f, and no other non-logical symbols. Write down sentences $φ$ and $ψ$ of $\mathcal{L}$ such that for any ...
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2answers
61 views

Let $\Gamma = \{\exists x \forall y (x\mathrel R y),\exists x \forall y(y\mathrel R x),\forall x\exists y(x\mathrel R y)\}$. Is $\Gamma$ consistent?

Consider the language consisting of one symbol $R$ for a binary relation. Let $\Gamma = \{\exists x \forall y (x\mathrel R y),\exists x \forall y(y\mathrel R x),\forall x\exists y(x\mathrel R y)\}$. ...
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1answer
30 views

How to negate $\forall A. \exists a,b. a \neq b \land a,b \in P(A)$?

$$ \forall A. \exists a,b. a \neq b \land a,b \in P(A) $$ My intuition tells me it is false, because given $A=\emptyset$, then $P(\emptyset) = \{\emptyset\}$, so $a=b=\emptyset$. I proceeded to ...
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1answer
41 views

If $a+b \geq x$ is known to be true does that mean $a+b\geq x-1$ contradicts it?

So I was proving something and I'm wondering if this line of argument is correct. Suppose that it is true that given conditions $M,N,O$; $a+b\geq x$. That is given those conditions the minimum value ...
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0answers
36 views

show that if A is creative then A is not computable

show that if $A$ is creative then $A$ is not computable? proof:If A is computable the $A$ andn the complement$A$ are computable enumerable. and $A$ is creative so there was a recsive function ...
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0answers
79 views

Let Γ = {p∧q,(¬p)∨q,p∨r}. Is it true that Γ ⊢ r?

I"m not sure how to solve this type of question. Here is the problem in more detail, and a similar problem: I know that given this set of formulas I'm supposed to show if its possible to deduce r ...
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0answers
24 views

Equivalent sentences using logical connectives

Using only logical connectives implication ($\to $) and negation ($\lnot $), write a sentence equivalent to the sentence: $$ (p \land q ) \lor r $$ Using logical connectives disjunction ($\lor$) ...
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1answer
35 views

what is the meaning of this predicate statement

This question appeared in the GATE exam 2011 Q.32 Which one of the following options is CORRECT given three positive integers x, y and z, and a predicate P(x) = ...
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2answers
77 views

Finding a formal deduction from an empty set of premises

I can't seem to make sense of any of this. I'm given a set of axioms schemes, modus ponens as the inference rule and I'm supposed to find a formal deduction. The question (question 1) is here. It ...
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2answers
29 views

How to specify each digit of a real number in decimal representation in set theory?

So real numbers have decimal representations. If you want to say the $n$th digit of some real number, how do you say this formally in set theory?
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1answer
30 views

a formal logic proposition about real numbers [on hold]

I have the following informal statement about real numbers: Every real number except zero has a multiplicative inverse. Can this be expressed as: $$ \forall x \exists y(x\neq 0 \implies xy=1) ...
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0answers
16 views

Find all n for which apply: If we shuffle pack 8 times we get the same order.

We have pack of cards 2n. Each mixing changes the order of cards of $a_1$, $a_2$, ..., $a_n$, $b_1$, $b_2$, ..., $b_n$ to $a_1$, $b_1$ ,$a_2$, $b_2$, ..., $a_n$, $b_n$. Find all n ...
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0answers
20 views

Formalize: “Every mail message larger then one megabyte will be compressed” [duplicate]

Formalize: Every mail message larger then one megabyte will be compressed
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2answers
47 views

what happens in a universal implication when the premise is false

I have just started learning Mathematical logic and couldn't figure out the answer to the above question . my question is what happens to the truth value if the premise in a universal implication is ...
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0answers
75 views

Logic puzzle of two numbers

The puzzle goes like this.. ...
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1answer
36 views

Proof for $∃xA⇔¬∀x¬A$

I want to prove, that $∃xA⇔¬∀x¬A$, using classic axioms. I think, I have to start with the following step: $∃xA⇔∃x¬¬A$ But I do not know, how to make this step, using axioms: $∃x¬¬A⇔¬∀x¬A$
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2answers
72 views

Hilbert's style proof (FO logic)

I am stuck with this question to check whether the following formulas are valid and if they are valid, then derive them using Hilbert's axiom schema and Modes Ponens for First Order Logic. ...
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0answers
30 views

Help with logical equivalence proof regarding a lemma for equivalence relations.

So the question is this: Suppose A is a set. Let ~ be an equivalence relation on A and let a,b be elements of A. Then Ta = Tb if and only if a ~ b. I need to prove this statement to be true. I know ...
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3answers
111 views

When does $\,2x=14\iff x\neq7\,$ hold?

I am a non-mathematiciam who is taking some classes in computer engineering. My question is: For which real numbers $x$ does the following hold? $$\,2x=14\iff x\neq7\,$$ I am not interested in ...
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2answers
22 views

Question regarding the arithmetic hierarchy notation used in the corollary of Post's theorem

A set $B$ is $\Delta_{n+1}$ if and only if $B \leq_T \emptyset^{(n)}$. More generally, $B$ is $\Delta^C_{n+1}$ if and only if $B \leq_T C^{(n)}$. This is from ...
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1answer
46 views

Proof of Tarski's self-reference lemma

In http://www.math.hawaii.edu/~dale/godel/godel.html, Tarksi's self reference lemma is mentioned but the proof is omitted. Tarski's Self-Reference Lemma. For any formula $p(x)$ in an adequate ...
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2answers
68 views

Is this theory complete?

I have a language $L=\{P\}$ with equation, where $P$ is binary predicate symbol. Language's formulas are: $\varphi \equiv \forall x \forall y (\neg P(x,x) \land (P(x,y) \to P(y,x)))$, $\psi \equiv ...
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2answers
92 views

Is it possible to prove that some point belongs to Mandelbrot set? Is this an example of Gödel's theorem?

Everybody knows about Mandelbrot set drawing computer programs. Program takes some point, builds sequence from it, and if found that sequence goes out of circle with 2 radius, then knows that this ...
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1answer
72 views

Which if the following three propositions are logically equivalent? [closed]

Which if the following three propositions are logically equivalent? $(p \wedge q) \Rightarrow (p \wedge r)$ $p \wedge (q \Rightarrow (p \wedge r)) $ $(\lnot p) \vee (\neg q) \vee (r \wedge p)$ ...
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0answers
27 views

What is principle of duality?

What is principle of duality? What is difference between principle of duality and De Morgan's law?
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2answers
29 views

When there is a proposition $(P\rightarrow Q)$, which row in the truth table of $\rightarrow $ should I use?

I solved one question in a book of analysis, and although I used an informal method to check it, I'd like to know more about what should be done. The question was the following: $A\subset X$ ...
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4answers
84 views

Question about logical implication $P\to Q$ [duplicate]

Having come across mathematical logic, a question suddenly came into my mind. We commonly know that the truth value of $P\to Q$ given as: $\begin{matrix} P&Q&P \Rightarrow Q \\ ...
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0answers
29 views

$\exists G \in L'. G \iff \mathtt{True}(gn(\neg G))$ in the language $L'$ with Godel numbering $gn$ and $\mathtt{True}$ predicate?

I am reading a paper Definability of Truth in Probabilistic Logic . Given a language $L$ with the Godel numbering $gn:L \to \mathbb{N}$ the authors extend it with a predicate $\mathtt{True}$ to a ...
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0answers
48 views

Can we unify every pair of inner models of ZFC by a same hierarchy?

Definition: Fix a ground model $V$ of ZFC. Let $F:V\rightarrow V$ be a definable class function (we call it an operator). The hierarchy $W^F$ corresponding to $F$ is defined as follows: ...
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3answers
40 views

At Most Two Distinct Members of A

The quantified predicate logic statement that describes at most two distinct members of A, where A, is some arbitrary set is: $\forall$xyz( (Px $\land$ Py $\land$ Pz) $\Rightarrow$ (x=y $\lor$ x=z ...
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3answers
41 views

How to show that if $\neg b = a \land d$ then $a \land \neg b = \neg b$ and $b \land \neg a = \neg a$

I'm new to boolean algebra and am having trouble proving the following simple theorem. Many thanks for any help. If $\neg b = a \land d$ then $a \land \neg b = \neg b$ and $b \land \neg a = \neg a$. ...
3
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3answers
102 views

Self-studying Russell's Paradox

I'm self-studying and having trouble wrapping my head around Russell's paradox, even after looking here. I'd really appreciate a more intuitive explanation of the concept before I move on to ...
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0answers
50 views

how to show and prove the above axioms are valid

at this theorem: the logical axioms are valid. i want to check the equlity axioms and quantifier axioms. i consider $x=x$ for each variable $x$. $(E1)$ ...
12
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2answers
239 views

What does it mean to say that a particular mathematical theory is a foundation for mathematics?

We usually hear that set theory is a foundation for contemporary mathematics. Category theory is also another foundation of maths. There are other theories which deemed to be a foundation for maths. ...
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0answers
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Is $ \phi$ a propositional consequence of $\boldsymbol{\Gamma}$?

Let $\boldsymbol{\Gamma}$ be the set $$\{\forall P(x)\rightarrow \exists yQ(y) ,\exists yQ(y)\rightarrow P(x), \lnot P(x) \leftrightarrow(y=z)\}$$ $ \phi$ is $\forall P(x)\rightarrow \lnot(y=z)$. ...
6
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4answers
296 views

Motivation for natural deduction

I've been learning natural deduction recently. I've seen many problems and am starting to be able to solve problems more easily. For some reason I feel the need to ask what high school math students ...
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1answer
40 views

Are these two notions of “computable function” the same or related?

From http://en.wikipedia.org/wiki/Semicomputable_function, we have: "If a partial function is both upper and lower semicomputable it is called computable." Is this the same kind of "computable ...
6
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1answer
75 views

Why are models in logic called models?

A model is an interpretation of a given formal language under which any wff in a given set of wffs of this formal language is true. Why are models called models? What's the reasoning behind the name? ...
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0answers
57 views

Divisibility lattice and duality with topological spaces

Consider the integers $\mathbb{N}$ seen as a poset with divisibility as an order relation. See it as a distributive bounded lattice with gcd and lcm, with gcd being the meet and lcm the join. Clearly, ...
2
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1answer
78 views

Problem with proving formally tautology using given rules

Using the rules below prove that the following assumeptions leads to the following conclusion by tautology. $A\vee B \vee C, A\to C, B\to C \Rightarrow C$ What I did: $A\vee B \vee ...
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3answers
109 views

Is there such a thing as the number of axioms?

This question was inspired by this question. Does it really make sense to say that a formal system has some number of axioms, say three, or ten, etc? E.g., take a formal system that admits ...
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0answers
23 views

Logic negate and simplify

Negate and Simplify: [(pvq)->~r]v~q Can someone show me step by step how to go about this. I am a little confused about negating over an implication.
2
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0answers
25 views

Confusion on particular step in this weak induction proof

I am studying for a midterm, and I came across this proof. Use mathematical induction(weak) to show that for all integers n $\geq$ 2 , if x$_{1}$, x$_{2}$, ... x$_{n}$ are strictly between 0 and 1 ...
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1answer
26 views

Logical formula with natural numbers

How to write a formula using only quantifiers, variables, brackets, logical operators and $\in$, $\mathbb{N}$, $+$, $\cdot$, $=$, $\leq$ : Among any three natural numbers exist pair of them such ...
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1answer
26 views

(∀x ∈ X, P(x)) or (∀x ∈ X, Q(x)) ⇒ ∀x ∈ X,(P(x) or Q(x)) where X is nonempty and P(x) and Q(x) are statements.

I know this is an obvious statement but how would one go about in showing that this is true ? My answer is : Consider the 2 cases; Case 1) ∀x ∈ X, P(x) holds. Then clearly ∀x ∈ X, P(x) or Q(x) ...
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1answer
55 views

Propositional calculus logic question

In my assignment I have the following question: For every proposition $\theta$ let $E(\theta)$ be the set of basic propositions. Prove the following: For every two propositions, $\alpha$ and ...