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Difference between Hilbert program and Russel & Whitehead's Principia Mathematica

May some one explain me what is difference between Hilbert program and Russel & Whitehead's Principia Mathematica? I know both of them wanted to reduce the mathematics into a set of axioms and ...
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1answer
44 views

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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1answer
181 views

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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1answer
69 views

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 $B$ ...
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111 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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2answers
52 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?
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5answers
79 views

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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0answers
88 views

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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3answers
294 views

How do you go about formalizing a concept?

I am reading Godel Escher Bach. I love it. 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 ...
3
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1answer
125 views

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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2answers
185 views

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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0answers
27 views

Do automatons and formal grammars belong to formal systems

A formal system consists of a formal language, a formal grammar that generates the language, a set of axioms, and a set of inference rules. Are automatons also formal systems? Is an automaton ...
3
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1answer
136 views

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 ...
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1answer
39 views

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 ...
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0answers
49 views

Meaning of abstractness and concreteness

Do abstractness and concreteness mean for formal systems and their models respectively? Do they relate to how big the theory is? For example, the theory of rings is richer than the theory of ...
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3answers
89 views

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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2answers
83 views

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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1answer
37 views

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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1answer
56 views

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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42 views

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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1answer
78 views

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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2answers
167 views

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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2answers
103 views

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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41 views

The role of verifiable computing in the formalization of mathematics

I've been thinking about this for a while, and it seems to me that mathematics "works" because (in principle) we can to check proofs very quickly, even though the discovery of that proof may have ...
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1answer
53 views

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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0answers
58 views

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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2answers
115 views

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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2answers
178 views

$\beta$ - 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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1answer
58 views

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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1answer
114 views

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 ...
1
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1answer
55 views

In a string rewriting system, does rewriting consume the string?

In string rewriting system, does rewriting 'consume' the string? For instance, suppose $101$ is written down and there's a rule $1x1 \rightarrow 11x11,$ we can apply this rule to write down $11011,$ ...
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2answers
286 views

How do we derive new inference rules?

I've been toying with a system of inference rules for propositional logic. I can use the system to prove theorems; but my question is, can I use the system to obtain new inference rules? Here are the ...
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0answers
83 views

A question about the analogy between formal systems and Turing machines

It is well known the analogy between formal systems and Turing machines. If I am not wrong, you can code any formal system of language L in first order logic into a Turing machine, and there is a ...
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2answers
121 views

Would it be possible to concoct a “harmful” axiom?

Suppose I run an automated theorem prover. It begins with the axioms of ZFC, and using a random number generator, it proves more theorems, and it runs for two days. At the end of the second day, it ...
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1answer
246 views

Understanding the syntactical completeness

A formal system is syntactically complete if for each sentence (closed formula) $\varphi$ either $\varphi$ or $\lnot \varphi$ is provable. A formal system is semantically complete if every ...
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3answers
114 views

Are PA and ZFC examples of logical systems?

Wikipedia says 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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2answers
218 views

Symbols and terminology for distinguishing derivability from sequents

First some definitions to make it clear what I'm talking about: A deductive system is a set $J$ of judgments together with a set $R$ of inference rules each of the form $$ j_0 \leftarrow j_1, ...
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1answer
203 views

Some different versions of completeness of a formal system, of a logic and of a theory

I have seen people talking about Gödel's complete theorems and Gödel's incomplete theorems on math.SE. I am curious what they are really about, so I try to understand the meaning of completeness ...
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3answers
372 views

Aftermath of the incompletness theorem proof

This is somewhat of a minor point about the incompletness theorem, but I'm always a little unsure: So one proves that there is a formula which is unprovable in the theory of consideration. Okay, at ...
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3answers
286 views

Why can't we use memoization to parse unambiguous context-free grammars in linear time?

This is a follow-up question to Why is it hard to parse unambiguous context-free grammar in linear time? I know that Parsing Expression Grammars (PEG) can be parsed in linear time using a packrat ...
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1answer
69 views

Satisfiability problem for FOL[<,R]

Let FOL[<,R] be the fragment of first-order logic enriched with two relational symbols < and R and the first-order axioms that say: < is a strict partial order and R is an irreflexive and ...
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1answer
179 views

Definition and meaning of “Proof Schema”, “Class Sign”

I'm a newbie in advanced mathematics, and I'm trying to understand Godel's theorem. I came across these two words which I couldn't understand clearly. "Proof Schema" and "Class-Sign" Can anybody ...
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2answers
364 views

Why is it hard to parse unambiguous context-free grammar in linear time?

From this question, I gather that whether unambiguous CF grammar can be parsed in linear time is an open problem. I'd like to know what the major roadblocks to achieve this are. That is, what made the ...
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2answers
684 views

Determining if a grammar can be converted to LL(1)/LL(k)

(This is a cross-post of http://cstheory.stackexchange.com/questions/11676/determining-if-a-grammar-can-be-converted-to-ll1-llk in the hopes of gaining a wider audience.) I'd like to know if there is ...
1
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1answer
335 views

Differences between the formal grammar, formation rules and automaton for a formal language

Added: A formal grammar is a set of formation rules for strings in a formal language. Formation rules are rules for describing which strings of symbols formed from the alphabet of a formal language ...
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0answers
231 views

Summation notation and a negative sign of some elements

Having sequence like $$ \beta_1 \cos\theta_1 + \beta_2 \cos\theta_2 + \beta_3 \cos\theta_3 + \dots + \beta_n \cos\theta_n$$ it is possible to present it using summation notation as follows: $$ ...
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1answer
127 views

How it is possible that sound system is inconsistent

We have some sound deduction system (every provable sentence is true), which has property of principle of explosion and some theory T described in that system. Lets assume that theory T is ...
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2answers
405 views

How it is posible that $\omega$-inconsistency does not lead to inconsistency

After wikipedia: Theory is $\omega$-inconsistent if, for some property P of natural numbers, T proves P(0), P(1), P(2), and so on (that is, for every standard natural number n, T proves that P(n) ...
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1answer
279 views

What are metatheory, metalanguage and meta-…

I have been reading the Wiki articles for metatheory and metalanguage, but not sure if I have understood what they are about. Some accessible examples may help clarify a bit, I guess. Do metatheory ...
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128 views

Does a formal system having inference rules imply that it is a logic system?

From Wikipedia Formal systems in mathematics consist of the following elements: A finite set of symbols (i.e. the alphabet), that can be used for constructing formulas (i.e. finite ...