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Questions tagged [formal-systems]

A formal system is broadly defined as any well-defined system of abstract thought based on the model of mathematics. (Def: http://en.m.wikipedia.org/wiki/Formal_system)

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What is the simplest formal system falling prey to Gödel's incompleteness theorems?

What is the the simplest formal system falling prey to Gödel's incompleteness theorems? Is the answer different for the first and second theorems? Is the answer Q for the first theorem and PRA for ...
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Metamathematic: Cover the case if X=Y

I want to formalize: "If X is less than Y, Then U is equal to Y ", and have been told that $$ \bf [\forall V \sim X=(Y+V)]U=Y $$ does not cover the case X=Y. Therefore I have rewritten it as $$ \bf [\...
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Is it possible to formalize all mathematics in terms of ordinals only?

Our experience shows that all finitary mathematical objects could be encoded using the natural numbers, and all operations on those objects could be expressed in terms of a few basic operations on ...
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Can I effeciently check whether the inverse of a semantic function exists?

I'm relatively new to the fields of formal semantics/systems/languages or even model theory and therefore I miss some knowledge and experience. I try to boil the question down to the core essence of ...
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Given a formal system show that a variable contains more of one symbol than an initial part of that variable

Given a formal system[of 4 symbols: 0, 1, ( , ) ] with rules: You may write down 0 or 1 at any time. if strings s and t have been written down, you may write down (st). write ⊢s to mean that s can ...
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Recursive definitions in formal logic

A binary tree $T$ is either A single vertex, or A graph formed by taking two binary trees, adding a vertex, and adding an edge directed from the new vertex to the root of each binary tree. Suppose ...
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Is there a definition of the existential quantifier which does not imply the axiom of choice?

The definition of the existential quantifer given in Bourbaki's Theory of Sets is $$(\exists x)R \iff (\tau_x(R)\mid x)R.$$ Here $x$ is a letter, $R$ is a relation, and $(\tau_x(R)\mid x)R$ means ...
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How can Godel's theorem apply to every formal system?

How can Godel's theorem apply to any formal system of logic, if the truth of the theorem itself is only relative to the axioms and rule of inference that were used to generate it. In other words ...
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Analysis and formal proofs.

Ever since I started learning formal logic I've had these kind of doubts: Is analysis ever studied in a completely axiomatic/formal proofy way? What I mean is, given a set of axioms and inference ...
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Peano Arithmetic: How would this formalized statement be correct?

Using Peano Axioms I have formalized the following: x is the square of an odd prime number For some odd prime number x' , x is its square IF x is some odd prime number, THEN x is the square of x' IF ...
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Connection between Algebraic structures and Formal Systems

I am a college student with very little mathematics background (up to Calculus 152 at Rutgers University), but have become increasingly interested in computing and mathematics in the last year. I am ...
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Can there exist a definition of 'multiplication' in Presberger Arithmetic?

According to Wikipedia, Presberger Arithmetic is the first-order theory of the natural numbers with addition. It can be proved to be consistent, complete, and decidable. Though it contains no axioms ...
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What's wrong with this proof that ZFC is consistent

If we take ZFC minus all it's axioms, we can easily prove that ZFC's first axiom is independent (because all statements are independent in a formal system without any axioms). We can then prove that ...
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Will assuming an undecidable statement result in a consistent system?

If you assume that an undecidable statement in a consistent axiomatic system is true (or false), will that new system also be consistent? For example, does $\mathsf{ZF}$ being consistent imply that ...
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Rational numbers and leibniz law

Leibniz law says $a = b \implies f(a) = f(b)$. Unfortunately this law seems to fail for rational numbers e.g. ${1 \over 2} = {2 \over 4}$ but $numerator({1 \over 2}) \neq numerator({2 \over 4})$. I ...
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Order of Parentheses is Irrelevant: Metatheorem?

Here was shown by induction that the order of parentheses is irrelevant when associativity is verified. Question: Would this be a metatheorem about the formal language (say, of ZF) where the ...
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Is there a formal system which is generally accepted, and how is it called?

I read the question at Gödel's Incompleteness Theorem -- meta-reasoning "loophole"? about Gödel's incompleteness theorem. My question is little about the contents of that other ...
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Proving the rules of a complicated game are well defined

What strategy could one use to formally model a game and prove that the rules do not lead to any self contradiction? A major example that comes to mind is Magic the Gathering. The card ...
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Logical systems and formal proof

Is there any good book dealing with various formal systems and a book for formal proofs. Or atleast some good notes. This page on wikipedia also says: 'This article needs attention from an expert in ...
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What do the ZFC axioms look like in terms of subset?

I'm reasonably familiar with the ZFC axioms. I know they can be formalized in many different ways. However, I usually see them presented in terms of the elementary "propositional calculus primitives" $...
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Incompleteness of formal systems as opposed to completeness of a non-formal theory

I have read that Gödel's incompleteness theorem does not apply to real closed field theory. But the incompleteness theorem applies only to formal systems, that is systems whose alphabet of symbols and ...
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Strong logical system without principle of explosion

Are there some logical systems strong enough to contain theorems of first/second order Peano Arithmetic but constructed in such way that principle of explosion does not hold for them?
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Regular Expression of alternative 0's and 1's?

Let $L$ be the language of $0$'s and $1$'s in alternate positions, where $$ L = \{ \epsilon, 0, 1, 01, 10, 01010,\ldots\}. $$ Is $(0)*$ + $(1)*$ a valid regular expression that represents this ...
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Difference between Hilbert's program and Russell & Whitehead's Principia Mathematica

Can someone explain to me the difference between Hilbert's program and Russell & Whitehead's Principia Mathematica? I know both of them wanted to reduce the mathematics into a set of axioms and ...
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prove this theorem $\vdash (\exists x_i (A\to B)\to (A\to \exists x_i B))$

Here is my thought to prove the theorem we should get $\{\exists x_i (A \to B), A\} \vdash \exists x_i B $ then, I don't know how to process...
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True or false? If $\eta$ is an explicitly defined incomputable number, then no formal system can pin down the value $\eta$ to arbitrary precision.

Let $\eta$ denote an explicitly defined incomputable real number (the bounty text is faulty, and does not mention incomputability of $\eta$). Then I think that no (recursively ennumerable) formal ...
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How can a proof by formula induction in a formal language be formalized?

From a set of not-so-rigorous lecture notes on Metalogic: Formulas of $L$: (i) Each sentence letter is a formula. (ii) If $A$ is a formula, then so is $\neg A$. (iii) If $A$ and $B$ are ...
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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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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?
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semantics(truth) vs formal system?

my first question is can we just define semantics in logic and not define a formal system ? why do we need a formal system to prove a proposition when for example we know the proposition is true ? <...
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Intuition for the choice of background (set) theory

Problem From the formalist point of view, any mathematical statement should ultimately be an assertion about the derivability of a certain formula in a certain formal system, call it the background ...
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How do you go about formalizing a concept?

I am reading Godel Escher Bach. In the first few chapters, the author shows what a formal system is, and gives examples that eventually lead to a typographical formal system of strings that represents ...
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how to prove : there are an infinite number of points on the circle

I think the follow problem is equal to the problem set 1.16.(a) in Principles of Mathematical Analysis (walter ruldin), And we take (a, b) in $R^2$, X in $R^i$ how to prove : there are an infinite ...
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How many undecidable statements are there in ZFC?

There are several statements known to be undecidable in ZFC, with the continuum hypothesis probably being the most "popular" one. Is it known how many undecidable statements are there in ZFC? I.e. ...
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What does “calculus” mean?

"calculus" and "formal system" From http://en.wikipedia.org/wiki/Propositional_calculus#Terminology a calculus is a formal system that consists of a set of syntactic expressions (well-...
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theory, theorems and axioms

According to Wikipedia In mathematical logic, a theory (also called a formal theory) is a set of sentences in a formal language. Usually a deductive system is understood from context. An element $...
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Are axioms assumed to be true in a formal system?

In a logical system, there is assignment of truth values to the sentences in the language, and axioms are assigned the true value. A logical system is a formal system. In a formal system, there is ...
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Are there formal systems that are not logical systems?

From WIkipedia A logical system or, for short, logic, is a formal system together with a form of semantics, usually in the form of model-theoretic interpretation, which assigns truth values to ...
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What do “completely described” and “complete formalisation” mean?

From Wikipedia In mathematics, an axiomatic system is any set of axioms from which some or all axioms can be used in conjunction to logically derive theorems. A mathematical theory consists of ...
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What are the differences between a collation and a rule of formation?

I'm a beginner in mathematical logic, and currently studying(myself, without any colleague, which is sad and so asking in here) basics of formal system. Before asking a question, I'll introduce my ...
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Are there any formal theories that are not axiomatic?

Are there any formal theories that are not axiomatic? Could you give me some examples?
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How do definitions work in Martin-Lof type theory?

The classical viewpoint is that we can found mathematics by specifying a formal system $F$ whose theorems are precisely those of ZFC. However, since $F$ has essentially no support for the concept of a ...
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Precisely, what is a primitive recursive definition?

If I understand correctly, the language of primitive recursive arithmetic has a distinguished function symbol $S$, together with a function symbol for each primitive recursive definition (hereafter ...
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If $T$ proves any incorrect $\forall$-rudimentary sentence, then $T$ is inconsistent

A theory $T$ in the language of arithmetic is called $\omega$-inconsistent if for some formula $F(x)$, $\exists x F(x)$ is a theorem of $T$, but so is $\neg F(n)$ for each natural number $n$. ...
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Prove the admissibility of $\Gamma, A \vdash_N B $, from $\Gamma \vdash_N B$

(This is an assignment) To prove: $$\frac{\Gamma \vdash_N B}{\Gamma, A \vdash_N B}$$ ($\Gamma, A = \Gamma \cup \{A\}$) I have what I think is a proof for this. I would like and be grateful for ...
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Certain sequents as inference rules

Fix a signature $\sigma.$ Then a coherent formula is a first-order formula built using only $\{\wedge,\vee,\top,\bot,\exists\}.$ See the link for more information. Furthermore, by a "special" ...
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Textbook on Basics of Formal Systems

Whilst trying to learn more about logic I came across Smullyan's Theory of Formal Systems on Google Books. What I liked about the book was how clearly it managed to describe (on pages 3-5 in chapter ...
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$\beta$ - conversion and $\alpha$-reduction problem in $\lambda$-calculus

Here is an expression that I am trying to reduce and my operations so far: $$((\lambda x.(x (\lambda z.zy))) (\lambda z.\lambda y. zy) )= (x (\lambda z.zy))[x \to \lambda z.\lambda y. zy ] = ((\...
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Is this $\beta$-reduction well defined?

Would it be possible to apply $(\lambda x.\lambda y. x)$ to the argument $y$? It seems to me that this must not be possible as it would give a different answer if applied to a constant, call it $\...
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From a foundational point of view, what's a good specification language for formal systems?

Suppose you and I are writing a book that develops (classical) mathematics from scratch. In the course of writing this book, we'll need to define a variety of formal systems, like a system for ...