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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Decidability of determining the definition of a function

Let's say a property is an SMT formula. Let's say a function has a property iff, with addition of the function symbol to a monadic predicate calculus formula over the signature of Presburger ...
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2answers
40 views

Questions about mathematical arguments

I have a few questions about mathematical arguments (1) Suppose that I want to prove that if the statements $A, B, C$ hold true, then $Z$ holds. To prove this, I would assume $A,B,C$, which then ...
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1answer
46 views

What is the theorem that shows that second-order logic is the ceiling of model characterization?

I was reading this blog posting and the following claim was made: ...[T]here's tricks for making second-order logic encode any proposition in third-order logic and so on. If there's a collection ...
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1answer
43 views

Axiomatic set theory: definitions.

In his book axiomatic set theory, Supped writes: An equivalence P introducing a new n-place operation symbol O is a proper definition if and only if P is of the form ...
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2answers
149 views

Linear Logic, what is it used for?

I read a lot about Linear Logic recently but I failed to find any real use to the logic. I'd like to know how and where Linear Logic could be applied. Something like lambda calculus can be clearly ...
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2answers
232 views

Motive for the definition of inner product

Mathematicians pride themselves on writing proofs of propositions in an elegant way, but frequently (maybe even usually?) neglect to formally write motivations of definitions with the same elegance, ...
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23 views

A question about Goodstein's theorem

It is known that if Peano's Arithmetic (PA)-which is a first order theory-is consistent, then Goodstein's theorem is an example of a sentence of PA that can be neither proved nor disproved in PA. Is ...
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1answer
48 views

Non computational approach to this equation?

I was thinking about the following problem (not homework): Let $a,b,c,d \in {0,1,2,3,4,5,6,7,8,9}$ Find all four digit numbers $abcd$ where the two digit numbers $$ ...
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0answers
49 views

First order logic and Second order logic: a question regarding domains

(1) I understood that in the language of first order logic quantification cannot be over sets (there are no predicate variables that can be bound by quantifiers), whereas the language of second-order ...
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9answers
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What is an Empty set?

We define the term "Set" as, A set is a collection of objects. And an "Empty set" as, An empty set is a set which contains nothing. First problem I encountered: How the definition of ...
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0answers
62 views

First order logic and first order set theory

(1) I understood that in the language of first order logic quantification cannot be over sets (there are no predicate variables that can be bound by quantifiers), whereas the language of second-order ...
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1answer
47 views

Semidecidability

The set of satisfiable formulas (A formula is satisfiable if it is possible to find an interpretation (model) that makes the formula true) of FOL is a subset of the set of valid formulas (A formula is ...
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1answer
65 views

How to define multiplication in $\mathbb{Z}$ with divisibility and addition?

Q: Show that $(\mathbb{Z},|,+,0,1)$ defines multiplication in $\mathbb{Z}$. I know how to do this in $\mathbb{N}$, but I'm stuck trying to do this is $\mathbb{Z}$. The idea I have is to define lowest ...
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2answers
103 views

Use of either/or in maths

I have been using these two words for a long time, especially when representing the solutions to quadratic equations. But I am little confused. These terms are often used simultaneously, but it seems ...
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2answers
45 views

Propositional Calculus, Can someone answer the following?

Can somebody please solve the following equations: \begin{align} 1. \quad (A \rightarrow B)\land (A\rightarrow \neg B)=\lnot A \quad \quad \\ \end{align} What I have got for it so far is ...
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0answers
28 views

6 teams, 5 shifts, 5 different activities

We have to plan a sportsevent/tournament. In this event 6 teams are participating and we have 5 different activities. - Each team has to meet the other 5 teams one time - And each time must try ...
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1answer
24 views

If $Q(c) \iff \Sigma \vdash \phi[c]$, is $\lnot Q(c) \iff \Sigma \not\vdash \phi[c]$?

$Q$ is a relation as described above, $\Sigma$ is consistent, and $\phi$ is a formula with one variable. I think the relation in above holds because if $c$ does not belong in $Q$, then by the relation ...
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3answers
106 views

role of definitions in proofs

Definitions are needed to define objects and such, however I am confused as to where definitions come from. I feel that they cannot be something that we arbitrarily define because simply saying ...
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2answers
35 views

Prove that the sum of two positive real numbers is equal or greater than the square root of their product.

Trying to prove this: A and B are positive real numbers. A + B ≥ √ AB  This is what I wrote: Proof by Contradiction A + B < √ AB  (A + B)2 < AB A2 + AB ...
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1answer
51 views

Find all the automorphisms of $(\mathbb{R},<)$, the real numbers with the usual ordering

Find all the automorphisms of $(\mathbb{R},<)$, the real numbers with the usual ordering Obviously the identity mapping, $\iota : \mathbb{R} \to \mathbb{R}, \iota(r) = r$ and the mapping of ...
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3answers
56 views

Proving uncountability of $\mathbb R$ only using the complete ordered field axioms

If we define the real numbers abstractly as a complete ordered field (like described in the Wikipedia page), how can we prove that they are uncountable? In other words, using just the axioms of a ...
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3answers
63 views

Solving this logical puzzle by resolution doesn't work for me

In this document there is a logical puzzle: If the unicorn is mythical, then it is immortal. If the unicorn is not mythical, then it is a mortal mammal. If the unicorn is either immortal or a ...
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1answer
84 views

Definability in $L(\omega_1)$

I'm trying to solve problem II.6.35 in Kunen's book, which asks to prove that if $V=L$, then the set $B$ of $\beta<\omega_1$ such that $L(\beta)\models ZF-P$ and every element of $L(\beta)$ is ...
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1answer
74 views

Pointclass of $\text{dom}(F)$ where $F:\omega^\omega\rightarrow\omega^\omega$ is partial recursive.

The definition I am working with: A partial function $F:\omega^\omega\rightarrow\omega^\omega$ is said to be partial recursive iff the partial function $G:\omega^\omega\times\omega\rightarrow\omega$ ...
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4answers
128 views

Proving $ \neg ( \neg \alpha \wedge \neg \neg \alpha )$

I'm training to prove this statement , but first I need to know if this statement can be proved in : 1 - both in classical and Intuitionistic logic ( in this case i need to provide demonstration in ...
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4answers
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How to prove this tautology using equivalences?

I am trying to prove that the following is a tautology: $(A \implies (B \implies C)) \implies ((A \implies (C \implies D)) \implies (A \implies (B \implies D)))$ To make progress, I thought I'd ...
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1answer
49 views

Proving that a set with a ternary logical connective is functionally incomplete (i.e. inadequate)

I am stucked at trying to prove that the set $\{\lnot ,G\}$ of logical connectives is inadequate where $G$ is a ternary connective that gives $T$ (True) if most of its arguments are $T$. For example: ...
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1answer
27 views

Recursively enumerable sets and omega consistency

I have a question about a passage in Enderton's "A Mathematical Introduction to Logic", p. 241. He writes that if some formula "∃vρ" defines a recursively enumerable set, Q, in Th R, then it cannot ...
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1answer
34 views

Bound variable in a formula

Let $S$ be an arbitrary set of symbols, $x$ variable and $\Phi$ $S$-formula. Assume that $x$ occurs as bound variable in $\Phi$. I want to show: There exist strings $\zeta_1, \zeta_2$ and ...
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1answer
27 views

how to prove that a relation is antisymmetric?

I have this question that I didn't know how to prove it and need your help. $R$ is a transitive and not reflexive relation on $A$. Prove that $R$ is antisymmetric. I tried to apply the definition of ...
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2answers
47 views

What sequent does this derivation prove?

Trying to learn sequent calculus and so I am trying to work thru some examples to get a better grip/understanding but the following question is not answered at the back of the book. I wrote my guess ...
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3answers
144 views

Isn't the axiom of determinacy inconsistent with ZF? What am I overlooking?

I'm sure there's something I'm missing here; probably a naive confusion of mathematics with metamathematics. Regardless, I've come up with what looks to me like a proof that (first-order) ZF+AD is ...
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0answers
94 views

How can nontrivial elementary embeddings of the universe to some inner model be surjective?

Consider $\kappa$ to the least measurable cardinal, or equivalently $\kappa$ is the critical point for an elementary embedding $j:V \rightarrow M$ from the universe $V$ to an inner model $M$ (critical ...
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1answer
62 views

What does “Fixed-Point Lemma” says intuitively?

The lemma as stated in Enderton's logic says: Fixed-Point Lemma.   Given any formula $\beta$ in which only $v_1$ occurs free, we can find a sentence $\sigma$ such that $$ A_E \vdash ...
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1answer
55 views

Assumptions, Axioms and Premises

The following attempt of mine at defining these terms, reflects my current understanding of them: Assumption: $\quad$ A statement accepted as true without proof being required. ...
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58 views

Help in proving a tautology

I am having real trouble deriving this tautology: $\forall(x) ((x=a) \lor (x\neq a))$ It is easy to solve this by assuming the negation, unpack the negation with DeMorgan's Law, and derive from ...
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2answers
109 views

Propositional Logic: Proof involving only the symbols $\{\rightarrow,F \}$

I feel like I literally tried everything. I'm exhausted, and could really use some help. I was instructed to prove some logic proposition using only the symbols $\{\rightarrow,F \}$. Let me first ...
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2answers
106 views

How should the relativized Kleene pointclass $\Sigma^1_1(A)$ be defined?

How should the relativized Kleene pointclass $\Sigma^1_1(A)$ be defined, when $A$ is a set of reals ($A \subset \omega^\omega$)? I assume that there is a standard definition, but I can't seem to find ...
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1answer
62 views

Positive and negative logical connectives

By inspecting the rules of inference for (intuitionistic) predicate calculus (or, alternatively, thinking about double negation translation), one sees that there is a certain dichotomy between two ...
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30 views

Conservative extensions and elementary equivalence- anything in common?

What is the difference between a Conservative extension, T', of a theory T, and a theory that is elementarily equivalent to T (but non-isomorphic, having, say, more elements). As far as I gathered, T' ...
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3answers
63 views

proving logic equation in logic algebra

Im trying to prove the following logic equations are equal and am having trouble. $ab'e'f + a'b'ef + acd'e' + a'cd'e + b'c'f + b'df = acd'e' + a'cd'e + b'c'f + b'df$ $a' = \neg a$ I am pretty new ...
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1answer
78 views

Is there any formula of monadic second-order logic that is only satisfied by an infinite set?

Is there any formula, of monadic second-order logic, that is only satisfied by an infinite set?
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1answer
80 views

Hilbert–Bernays provability conditions

Let "provability formula" ${\rm Prf}(x, y)$ written in the manner that provability operator $\square A$ defined as $\exists x\ {\rm Prf}(x, \overline A)$ satisfying Hilbert–Bernays axioms: If ZF ...
4
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2answers
88 views

How to intuit 'only if'?

I already know, and so ask NOT about, the proof of:   $A$ only if $B$   =   $A \Longrightarrow B$. Because I ask only for intuition, please do NOT prove this or use truth tables. My problem: I try ...
3
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1answer
70 views

Satisfiability proof of formulas with pure literals

Let $\varphi$ be any propositional formula in negation normal form (NNF). A literal $\ell$ is pure in a formula $\varphi$, if the complement of $\ell$, $\ell^c$, does not occur in $\varphi$, where ...
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1answer
76 views

Which mistake(s) in my argument re: representability, definability and the halting problem?

I'd like to ask for your help in showing me the (quite likely: several) flaws in my argument below, relating weak and strong representability in a formal system and the halting problem. At least ...
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44 views

Validity of a first-order formula

How can I see (and prove) whether the given first-order formula $\varphi$ is valid or not? $\varphi = \forall x \forall y [ (r(x,y) \rightarrow (p(x) \rightarrow p(y))) \land (r(x,y) \rightarrow ...
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1answer
51 views

Show For any language L two L-structures M and N are elementarily equivalent iff they are elementarily equivalent for every finite sublanguage.

Setting For any language $\mathcal L$, two $\mathcal L$-structures $\mathcal M$ and $\mathcal N$ are elementarily equivalent iff they are elementarily equivalent for every finite sublanguage. ...
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2answers
62 views

Write the negation of the following

$P(x,y)$ is the set $\{0,1,2,3,4,5\}$ $ \forall\ y\ \neg P(2,y)$ I solved this is it correct? $$\neg P(2,0) \wedge P(2,1) \wedge P(2,2) \wedge P(2,3) \wedge P(2,4) \wedge P(2,5)$$
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2answers
95 views

Determine whether or not $\neg q \to \neg (q \land (p \to \neg q))$ is a tautology

I have been trying to solve this but I got stuck at the end. $$\begin{align} \neg q \to \neg (q \land (p \to \neg q)) &\equiv \neg \neg q\lor \neg (q \land ( \neg p\lor \neg q)) \\& ...