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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5
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2answers
33 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 ...
1
vote
2answers
25 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 ...
1
vote
1answer
39 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 ...
1
vote
3answers
37 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 ...
2
votes
3answers
42 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 ...
4
votes
0answers
38 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 ...
2
votes
1answer
50 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 $G:\omega^\omega\times\omega\rightarrow\omega$ ...
3
votes
4answers
111 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 ...
2
votes
4answers
65 views

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 ...
-6
votes
2answers
45 views

how is this possible please? can someon explain [on hold]

$({\sim}p \lor {\sim}q)\lor(p \lor q)=({\sim}p \lor p)\lor({\sim}q \lor q)$
2
votes
1answer
38 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: ...
2
votes
1answer
19 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 ...
1
vote
0answers
17 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 words/strings $\zeta_1, \zeta_2$ and ...
2
votes
1answer
23 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 ...
1
vote
2answers
41 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 ...
1
vote
3answers
121 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 ...
1
vote
0answers
74 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 ...
1
vote
1answer
60 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 ...
0
votes
1answer
42 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. ...
3
votes
2answers
57 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 ...
6
votes
2answers
95 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 ...
-4
votes
1answer
40 views

What does a separation in lines mean? [on hold]

does putting a problem on two different lines make it two seperate problems? 1+1 1+1 Is this the same as 1+11+1 Or does the line break indicate a seperation of problems? Is there a specific ...
4
votes
2answers
92 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 ...
1
vote
1answer
32 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 ...
0
votes
0answers
28 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' ...
0
votes
3answers
53 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 ...
5
votes
1answer
66 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?
-2
votes
0answers
33 views

Textbook with full solutions: To self-learn logic for the first time [on hold]

I never studied logic before; so I'm seeking an intelligible textbook (written in simple English) with practice problems that MUST be accompanied with full detailed solutions. I read ...
2
votes
1answer
69 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
votes
2answers
84 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 ...
2
votes
1answer
43 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 ...
0
votes
1answer
40 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 ...
0
votes
0answers
39 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 ...
2
votes
1answer
46 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. ...
1
vote
2answers
61 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? $$\neg P(2,0) \wedge P(2,1) \wedge P(2,2) \wedge P(2,3) \wedge P(2,4) \wedge P(2,5)$$
3
votes
2answers
85 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)) \\& ...
1
vote
1answer
34 views

Am I right in this discrete mathematics question?

$A = \{0, 1, 2\}$ $B = \{x \in R\mid−1 \le x \lt 3\}$ $C = \{x \in R\mid−1 \lt x \lt 3\}$ $D = \{x \in Z\mid−1 \lt x \lt 3\}$ $E = \{x \in Z+ \mid−1 \lt x \lt 3\}$ I put that $A=D$, $A=C$, and ...
2
votes
3answers
44 views

Determining if a relation is reflexive, symmetric, or transitive [closed]

Let $A = \{0,1,2,3\}$ Define a relation $T$ on $A$ as follows: $T = \{(0,1),(2,3)\}$ Is $T$ reflexive? symmetric? transitive?
0
votes
3answers
25 views

Finding the equivalence classes of a relation R

Let A = {0,1,2,3,4} and define a relation R on A as follows: R = {{0,0},{0,4},{1,1},{1,3},{2,2},{3,1},{3,3},{4,0},{4,4}}. Find the distinct equivalence classes of R. How do I solve this problem? ...
0
votes
1answer
31 views

Proving Equivalence Relations On A Set

Let X be the set of all nonempty subsets of {1,2,3}. Then X = {{1}, {2}, {3}, {1,2}, {1,3}, {2,3}, {1,2,3}} Define a relation R on X as follows: for all S and T in X, SRT if, and only if, the least ...
1
vote
1answer
40 views

Are the implicitly definable sets of a second-order theory the sets the second-order quantifiers range over?

I know that in a second-order setting, due to the failure of the Beth definability theorem, implicit and explicit definition come apart (i.e., there are predicates which can be implicitly, but not ...
0
votes
2answers
28 views

when tossing a coin ten times, what is the probability of an outcome which has a string of 3 or more heads as well as a string of 3 or more tails?

here is an experiment from my Stat textbook "Try this experiment: Write down a sequence of heads and tails that you think imitates 10 tosses of a balanced coin. How long was the longest string ...
1
vote
2answers
49 views

Discrete Math: Implication

If $\neg(P) \to \neg(Q) = Q \to P$ works as a Rule, then why doesn't $\neg(P) \to \neg(Q) = P \to Q$ work as a rule.
3
votes
3answers
66 views

How various properties of numbers, operations are found?

I know that how the term "property" is defined. Definition: An attribute, quality, or characteristic of something. Like one of the property of addition is "commutativity" which behaves like, ...
-7
votes
0answers
40 views

Logic puzzle for mathematician [closed]

Suppose in some dimension If -> 9999 -4 -> 8888-8 -> 1816-9 -> 1212-0 then what is 1919- ?
1
vote
2answers
61 views

a relation in logic

Suppose $\prec$ is a relation defined in the set of well defined formulas such that $\phi \prec \psi$ iff $\models \phi \rightarrow \psi$ and $ \nvDash \psi \rightarrow \phi$ I would like to prove ...
3
votes
0answers
40 views

What is the explicit formula (solution) to this recursively defined binary matrix?

My question concerns the following binary matrix (call it matrix $A$). Or rather the entire family of such matrices, for some number of columns $n$ and rows $2^n$. The ellipses indicate that the ...
-1
votes
0answers
66 views

Algebra and Logic [closed]

A ⇔-wff is a well-formed formula that is built out of propositional variables and the double arrow ⇔ only. (a) Characterise precisely all positive integers that arise as the number of symbols used to ...
2
votes
1answer
31 views

Enderton's logic book about completeness theorem

In page 141 of A Mathematical Introduction to Logic, Enderton simply writes, STEP 6: Restrict the structure $\mathfrak{A}/E$ to the original language. This restriction of $\mathfrak{A}/E$ ...
1
vote
1answer
32 views

Reconstructing the conditional's truth table from natural deduction

Can the conditional's truth table be reconstructed using the rules from natural deduction?