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Questions tagged [logic]

Questions about mathematical logic, including model theory, proof theory, computability theory (a.k.a. recursion theory), and non-standard logics. Questions which merely seek to apply logical or formal reasoning to other areas of mathematics should not use this tag. Consider using one of the following tags as well, if they fit the question: (model-theory), (set-theory), (computability), and (proof-theory). This tag is not for logical puzzles, use (puzzle).

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How to formalize $P: \zeta \rightarrow \eta iff ...$ into a formula using First-Order language for set-theory?

In Ebbinghaus' "mathematical logic" Chapter X which is about decidability and enumerability, there is a definition: $P:\zeta \rightarrow \eta$ iff P started with $\zeta$ eventually stops, ...
peter's user avatar
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ZFC construction of tuple-based cartesian product for arbitrary finite number

Question: Can the finite, tuple-based Cartesian product of sets be defined for an arbitrary number of sets, using only sets and NOT relying on classes or other non-existing non-set objects in ZFC? Key ...
AMI's user avatar
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0 answers
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Each creative set is contained in some simple set [closed]

Prove that each creative set is contained in some simple set(Exercise 6.2.8 of Computability Theory of Barry Cooper) These are the definitions: $W_e = \{x| \phi_e(x) \ halt\}$ and $\phi_e$ is e-th ...
SyHoMadara's user avatar
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A confusion about Karnaugh Map

Consider the following four variable Boolean function: $$F(A,B,C,D)=\sum(0,2,3,5,7,8,9,10,11,13,15)$$ If I show you the map, then what I get is: I have marked the Essential Prime Implicants with a ...
M.Riyan's user avatar
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28 views

Proof of countable Borel uniformization

Corollary 13.8 of these notes says that the Borel sets are closed under Borel functions whose fibers are all countable. Trying to digest that proof, but there's a step I'm unable to follow. I'll ...
user1510061's user avatar
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0 answers
23 views

bijection between n and k letter alphabet [closed]

What is bijection between strings over n and k letter alphabet, k,n positive integers ? I'm about to construct a bijection for sentences and binary strings, but I've failed . Can I construct a ...
Jan Pax's user avatar
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1 answer
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Generalization (semantic) proof for Hilbert-Calculus

I constructed this lean proof to the Generalization Rule in the Hilbert System. I have marked the passage where I am not sure if I overdo. It may not be proper language but I want it to be better ...
God's user avatar
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Propositional logic: Finding the maximum length of an implication chain

Particular definitions, Let $\mathcal{L}$ be a language of a propositional logic. For each sentence $\psi$ in $\mathcal{L}$, denote by $T_{\psi}$ the set of all truth assignments $\mathfrak{U}: \...
Pexx's user avatar
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How to construct a propositional logic formula to enforce a certain relation?

Disclaimer: this is a question from a homework but i am really struggling with it and would like to get some help. I have a set of students $S$, set of Time-slots $T$,set of cookies $P$ and the ...
fady abo swees's user avatar
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2 answers
93 views

What is wrong with my Prenex normal form? [closed]

I had this equation $$∃yG(y)→¬(∀xF(x,y)↔G(y)).$$ Then I turned it to $$∀y(¬G(y)) \lor ∀x(F(x,y))\land ¬G(y) \lor ∃x(¬F(x,y))\land G(y).$$ Then I turned it to Prenex normal form: $$∀k∀s∃m(¬G(k) \lor ...
Пчеловод's user avatar
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1 answer
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The definition of inconsistency in a logical theory

A classical definition of inconsistency is the theory leads to contradictory statements Shoenfield: Mathematical Logic defines this on page 42 that A theory T is inconsistent if every formula of T ...
Gergely's user avatar
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Unsure about soundness of proof about transitivity.

sorry in advance for any imprecision, missing tags etc but this is my first question. I'm reading Daniel J. Velleman 'How To Prove It' and i'm having a hard time understanding how to write a sound ...
Luca 's user avatar
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Would solving the Liar's Paradox allow for self-meta-logics?

By self-meta-logic, I mean a logic that is its own meta-logic; a logic that can talk abouts its models. If I understand correctly, this is impossible (for "interesting" logics at least) due ...
user110391's user avatar
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Is there a "tree of derivations" construction?

My question stems from the following observation: in Logic we construct a series of finitary syntactical structures by means of trees. First we have the base constants, variables, and function symbols,...
Sho's user avatar
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Maximum cardinality of a collection of sets of integers with pairwise finite intersections [duplicate]

Let $\mathcal{C}$ be a collection of subsets of $\mathbb{N}$ such that any two distinct $U,V\in\mathcal{C}$ have finite intersection $U\cap V$. Then what can be said about the cardinality of $\mathcal{...
SmileyCraft's user avatar
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Is there a syntactic "higher-up" to derivations?

The question is likely to be illposed, but I was wondering whether there was some known syntactic "higher-up" to the notion of derivation of semi-exoteric use. By this I mean: when working ...
Sho's user avatar
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6 votes
3 answers
173 views

"You cannot engage in argument unless you rely on the principle of non-contradiction"

A very good SEoP article "Aristotle on Non-contradiction" by Paula Gottlieb makes two interesting claims, one after the other: Claim 1: "Anyone asking for a deductive argument for PNC [...
logiclearner's user avatar
-5 votes
1 answer
68 views

Why does "p⊃q" instead of "q⊃p" mean "if p then q"? [closed]

The “if-then” sentence is called a conditional and will be symbolized as (p ⊃ q). The part to the left of the horseshoe is called the antecedent (what comes before), and the part to the right of the ...
陈海斌's user avatar
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3 answers
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Do we need not need truth values to describe models in predicate logic?

I am reading Kirby's Model theory. In section 3.3, the following definition of interpretation is given: $\textbf{Definition 3.4 (Interpretation of formulas)}$ Let $\varphi$ be a formula of $L$ and $\...
Brian's user avatar
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4 votes
2 answers
355 views

Can proof by contradiction be used disprove a statement?

I'm currently taking an introductory proofs class. The textbook that we are using justifies proof by contradiction by stating that the statement $P$ is logically equivalent to the statement $(\neg P\...
Learning Math's user avatar
2 votes
1 answer
104 views

Show law of excluded middle is valid

Example 1.12 from David Marker's An Invitation to Mathematical Logic (p. 10) shows $(\phi \vee \neg \phi)$ (that is, the law of excluded middle (LEM)) is valid in finitary model-theoretic FOL: For any ...
silly-little-guy's user avatar
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1 answer
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How do you construct a While loop using propositional dynamic logic?

I am learning some modal logic from a textbook (Modal Logic from Blackburn, De Rijke and Venema) and they have an example about propositional dynamic logic (PDL). Here they explain that if $\pi$ is a ...
UnrulyTank's user avatar
1 vote
0 answers
59 views

Proof of Deduction Theorem in Hilbert-Calculus, Beginner friendly

Deduction Theorem: Let $ \Sigma$ be an axiom system, and let $ \psi, \varphi$ be $ L$-formulas. Then the following always holds: $ \Sigma, \psi \vdash \varphi \iff \Sigma \vdash \psi \rightarrow \...
God's user avatar
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0 answers
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Proof of the General Deduction Theorem

Theorem: For all axiom systems $\Sigma$ and all formulas $\varphi, \psi_1, \ldots, \psi_n$, it holds that: $ \Sigma, \psi_1, \ldots, \psi_n \vdash \varphi \iff \Sigma \vdash \psi_1 \to \ldots \to \...
God's user avatar
  • 19
2 votes
1 answer
137 views

Existence of a model of $\mathsf{ZFC}$ which has the internal standard model of arithmetic coincide with the real standard model of arithmetic

$\mathsf{ZFC} + \operatorname{Con}(\mathsf{ZFC})$ can prove that there is a model $M$ of $\mathsf{ZFC}$ which has the natural numbers structure $\mathbb{N}^M$ as the standard model of arithmetic ...
Pointwise's user avatar
3 votes
2 answers
223 views

Do "truly" infinite proofs exist?

An assumption underlying this earlier question was the existence (and greater expressive strength) of infinite proofs in logics like $\mathcal{L}_{\omega_{1}^{CK}, \omega}$ (based on, for example, the ...
NikS's user avatar
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1 vote
2 answers
175 views

How do you define universal and existential quantification rigorously and non-circularly?

Universal and existential quantification are necessary to build math from set theory. But I can't think of any way to define them besides the traditional intuition: that is, $(\forall x\in S)[p(x)]$ ...
Mathemagician314's user avatar
-1 votes
2 answers
72 views

Do quantified statements without domains mean anything? What is one to do when it seems impossible for there to be any domain? [duplicate]

I used to find quantified statements without domains perfectly sensible, but now I can hardly grasp them. Such statements are in introductory texts, proof assistants, and many other places. When a ...
interested's user avatar
-3 votes
1 answer
31 views

Define if the statement: ∀x(A(x) ≡ B(x)) ~ (∀x A(x) ≡ ∀x B(x)) is always true

There is a statement: $$∀x(A(x) \equiv B(x)) \sim (∀x A(x) \equiv ∀x B(x))$$ that seems to be true in all models, but unfortunately I can't think of an idea to proove that, except this, but I am not ...
Artkol's user avatar
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6 votes
2 answers
163 views

Demonstrating that an Induction Hypothesis is too weak.

My background is in Computer Science. Proofs by induction are ubiquitous here, especially in programming languages. As a running example, let us take proving things about the natural numbers. To prove ...
Suraaj K S's user avatar
3 votes
0 answers
119 views
+50

The Hanf number of higher-order logic

The reduction of second-order logic (SOL) to a $\Pi_1^1$-fragment on the Stanford Page of Higher-Order Logic (Section 4) is as follows: From this we get that the Hanf and Löwenheim number of SOL is ...
SJe967's user avatar
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-1 votes
2 answers
98 views

Unsure about a logic question by Brilliant.org

For context: I wanted to try out brilliant.org for a while after seeing it on numerous sponsored videos by educational YouTubers. I decided to try its level 1 logic course and everything seems to make ...
Singaporean Hermit's user avatar
2 votes
1 answer
95 views

Reference Request: A general approach to forcing, and some clarification

I've wanted to understand forcing for quite some time now, but I wish to understand it in the general sense of formal logic, rather than just the specific case of set theory, which I find quite ...
Joseph_Kopp's user avatar
4 votes
1 answer
225 views

Any way to show that second-order logic is incomplete by means other than using Gödel's incompleteness theorem?

I know a way to show second-order logic is incomplete (in the sense of full semantics). the proof is as follows. the second-order arithmetic $\mathbf{Z}_2$ or the second-order peano arithmetic $\...
Rosser's user avatar
  • 43
-1 votes
1 answer
63 views

(p, q) ∈ S only if p ∧ q is a tautology. Assessing the relation S. [closed]

This is the first time I come across a question that combines relations and logic. It is said that: “Let $𝑆$ be a relation, which is defined on the set of propositional formulas as follows $$ \left(�...
matinaros's user avatar
3 votes
0 answers
78 views

Axiomatizability of inclusion over any powerset [duplicate]

Is class $(\mathcal{P}(A),\subseteq)$ for arbitrary $A$ axiomatizable? One of the assignments I got was to prove that it is not, but a friend of mine suggested this structure is a complete atomic ...
Rikimaru's user avatar
0 votes
0 answers
73 views

Best tool for proving that a (big) first order model is actually the model of some (big) first order theory. [migrated]

I have to check that a (big: several hundreds axioms) first order logic theory is satisfiable. There are various authomated provers for first order logic (e.g. Vampire) that could tell me if the ...
kataph's user avatar
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0 votes
0 answers
35 views

$x\lt Sy\iff x\lt y\lor x=y$ can be deduced from Robinson arithmetic $Q$?

We can define new predicate symbol "$\lt$" in Robinson arithmetic $Q$ by adding the axiom $x\lt y\iff\exists z (x+Sz=y)$.With this definition, $Q$ can prove that (1) $\lnot(x\lt0)$ because ...
smooth manifold's user avatar
1 vote
1 answer
55 views

Question About Axiom Group 5 in Enderton's Logic Textbook

In Enderton's Textbook "Introduction to Mathematical Logic", he specifies 6 Axiom Groups on Page 112 Chapter 2.4 which are used in conjunction with their generalizations, a set of hypothesis ...
NiraDoesMath's user avatar
3 votes
2 answers
178 views

Number of countable models of DLOWE with an increasing order-isomorphism.

Recently I tried to answer this question by taking the DLOWE and an ascending sequence of constants ($c_1 < c_2 < c_3 \cdots$) construction and encoding it in a finite language with a finite ...
Greg Nisbet's user avatar
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0 votes
0 answers
49 views

Reference Request: Second Order ZFC

I've seen a fair bit of discussion about second order ZFC (Second order ZFC, intuition required, How is second-order ZFC defined?, A question about Second-Order ZF and the Axiom of Choice, A question ...
daRoyalCacti's user avatar
0 votes
1 answer
70 views

On the Definition of Simultaneous Substitution of Terms in FOL formula

Consider a firsr-order language $L$. Let $\varphi$ be a formula, $t$ a term and $x$ is a variable. We define the formula $\varphi[t/x]$, which denotes the formula obtained by substituting $t$ in place ...
Ray's user avatar
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2 votes
0 answers
43 views

Is there a realistic threshold for formula complexity which might indicate consistency?

Many arguments in favour of the consistency of $\mathsf{ZFC}$ and similar systema point to their empirical usage throughout the years without the apparition of any purported inconsistencies. While I ...
Sho's user avatar
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2 votes
0 answers
91 views

Example of a (true) statement in PA which cannot be proved in PA and cannot be proved in ZFC?

I'm aware of a number of statements (for example, Con(PA), and those given by Goodstein's theorem and the Paris–Harrington theorem) which are stateable in Peano Arithmetic (PA), but not provable ...
Ollie Taylor's user avatar
0 votes
1 answer
57 views

How can I ensure that $x \iff p \land q$ here?

Say I am writing a Boolean formula in which only $\land$ is between clauses, and only $\oplus$ is between literals in the clauses. How can I make it such that an assignment of variables satisfies the ...
Princess Mia's user avatar
  • 3,235
2 votes
3 answers
144 views

formalize "there exists a function f such that f(a)=b" into first-order formula [closed]

I want to know how to formalize the sentence "there exists a function f such that f(a)=b" into a first-order formula. It should not be $\exists f f(a)=b$ because first-order logic does not ...
peter's user avatar
  • 197
3 votes
1 answer
51 views

Problem characterizing Lindenbaum algebras

Given a first order type $\tau$, we use $S^\tau$ to denote the set of $\tau$-sentences, i.e. formulas with no free variables. Let $T = (\Sigma, \tau)$ a theory, with $\Sigma$ the set of axioms in the ...
lafinur's user avatar
  • 3,565
0 votes
1 answer
58 views

Completeness for existential second-order logic

I find in this link that existential second-order logic is compact. On the other hand, I find in this link that existential second-order logic is not complete. Here is a quote: A "naturally ...
user1868607's user avatar
  • 6,075
1 vote
0 answers
85 views

Does Peano Arithmetic implicitly describe a set?

Apologies if my language is incorrect. My question is that in the axiomatizations of PA I have seen, they implicitly posit a 'set' of natural numbers and then describe which elements belong to it. To ...
Orion Jordan's user avatar
2 votes
0 answers
70 views

References and Explanations for Hugh Woodin "On the Mathematical Necessity of the Infinite", Finite Peano Arithmetic (FPA)

In this lecture (I will only ask about the first 20 minutes), Woodin makes several interesting statements, that I would like more explanation on. @6:08, Woodin outlines the axioms of FPA (the axioms ...
D.R.'s user avatar
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