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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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
67 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
71 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
25 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
28 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
83 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 $b \land \neg a = \neg a$. ...
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3answers
101 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
49 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 $x$. $(E1)$ ...
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2answers
235 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
33 views

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 $\forall P(x)\rightarrow \lnot(y=z)$. ...
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4answers
291 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.
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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 ...
5
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1answer
58 views

Compactness Theorem / Set made of formulas of infinite size

Could someone give me an example of an infinite countable set, where formulas contained in it are under the form of a conjunction or disjunction of infinite size, for which the compactness theorem ...
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1answer
40 views

What we can infer from there exists x satisfying P

If it is known that there exists x satisfying P, can we infer that there also exists x not satisfying P? I ask this question since I have a problem as follows. Given three premises: (1)if a student ...
0
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1answer
63 views

How much conservative ZF+AC and ZF+DC are over ZF?

A logical theory $T_2$ is a (proof theoretic) conservative extension of a theory $T_1$ if the language of $T_2$ extends the language of $T_1$; every theorem of $T_1$ is a theorem of $T_2$; and any ...
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0answers
30 views

Some doubts regarding decidable sets

I've been working at one of the problems, related to the decidability. Let's denote $ f: \mathbb{N} \rightarrow \mathbb{N}$ as a computable increasing function, $A \subset \mathbb{N}$ is a ...
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2answers
29 views

Inequality with respect to transitivity

Given a relation R, R is said to be transitive if aRb ∧ bRc, then aRc. The unequal relation (≠) is not transitive, for instance a≠b ∧ b≠c, then a≠c is an invalid consequent of the antecedent (a≠b ∧ ...
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2answers
42 views

Have I properly used $\,\exists !\,$ in this statement?

I want to express the following in logical notation. For every natural number, there is a unique natural number that succeeds it. Does the following statement express that proposition? ...
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2answers
31 views

How to find the contrapositive of this statement?

$if \ \ \ \forall a \forall b \in Q, \ \ \ xy \notin Q \ then \ (a \lor b) \notin Q$ I hope I wrote that correctly. In English terms, it would be: " If a and b are real numbers and ab is irrational, ...
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1answer
29 views

Prove by contrapositive: Φ∪{β} ⊨ α & Φ∪{¬β} ⊨ α iff Φ ⊨ α

We are to prove this by contrapositive (by the way: Φ is a set of formulas of predicate logic and α a formula of predicate logic) I've managed the Right to Left proof, but I struggle with the Left to ...
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1answer
25 views

How can I translate it into Logic sentence? [closed]

Let $p$ denote "it is snowing." So how can I translate the following into symbolic logic? "It is not snowing, but snowing." Please help me.
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1answer
67 views

Structural Induction, Propostitonal formulae problem

I am kind of overwhelmed by this question. Can anyone give me some hints about where to start? Propositional formulae PF are inductively defined over the Boolean constants B := {1, 0} (true and ...
0
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1answer
22 views

Logic proof using contrapositive

If n=ab is the product of two integers $a$ and $b$, then either $a\leq n^{(1/2)}$ or $b\leq n^{(1/2)}$. Use the proof by contrapositive method. The new statement is: if $a>n^{(1/2)}$ or ...
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2answers
46 views

Prove the two logic expressions are equal

Prove $\neg(a \lor b)$ is the same as $(\neg a \land \neg b)$ It makes sense when I think about it, but how does one prove it? Also is there a relationship with the above and saying: $(a \implies ...
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0answers
26 views

Decision method for a partial mapping

Assigned this definition: A decision method for a partial mapping $F$ from $A$ to $B$ is a method which, if applied to an element $a$ of $A$, will give the value $F(a)$ if $a$ is in the domain of $F$ ...
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4answers
144 views

Two plus two equals four when earth has one moon?

As is well known, we have the least intuitive of basic operations, the 'implication' or '=>'. Consider 'A => B'. Most beginners get stumped on the vacuous truth, that implication could be true even ...
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1answer
56 views

Logic behind continuity definition.

I have a question regarding the definition of continuous functions : from wikipedia and my book : $f$ is said to be continuous at the point $c$ if the following holds: For any number ...
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0answers
41 views

Find $\varphi$ of language $L$

I have the following symbols P - functional, airity=2, N - functional, airity=0, J - functional, airity=0, V - predicate, airity=2, K = functional, airity=2. I need to find closed formula $\varphi$ ...
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151 views

What does it take to divide by $2$ (or even $3$)?

Theorem 1 [ZFC, classical logic]: If $A,B$ are sets such that $\textbf{2}\times A\cong \textbf{2}\times B$, then $A\cong B$. That's because the axiom of choice allows for the definition of ...
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1answer
44 views

What does First Godel's Incompleteness theorem mean?

I am terribly confused what really "Incomplete" mean in terms of the Godel's theorem. Does it mean there are some theorems that are definable in First order theory of natural numbers and true but ...
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1answer
39 views

Example of language implementation

I'm trying to find example of implementation $M$ of language $L$ such that $M \models \varphi_1 \land \varphi_2 \land \varphi_3 \land \varphi_4$ Where $L = \{•, \blacksquare, n\}$ is language with ...
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2answers
47 views

Help with semi-formal logic

How do I write semi-formally 'there are only 2 objects in the universe'? My hypothesis is: ∃x∃y(x≠y) Any ideas?
2
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2answers
61 views

Every element of the empty set has three toes true or false? [duplicate]

This is a bonus question that we have and I cannot figure it out. Any help would be great! Is the proposition Every element of the empty set has three toes true or false? Explain your answer
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1answer
89 views

Uncountable reals in the theory

The Question I'm looking for a possibility to somehow proof the "essence" of Cantor's diagonal argument within a recursive first-order theory which is satisfied by the reals (better: within a theory ...
2
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2answers
115 views

Recursion on a class instead of a set

According to Wikipedia, the recursion theorem states the following: Let $X$ be a set, and $f:X\to X$ a function. For any $x\in X$, there exists a unique function $g:\omega\to X$ such that $g(0)=x$ ...