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2
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0answers
36 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?
2
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1answer
60 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 ...
2
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2answers
132 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 ...
3
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2answers
86 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 $n$. ...
0
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0answers
32 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 ...
1
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1answer
49 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 ...
2
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0answers
56 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" ...
3
votes
2answers
84 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
138 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 ] = ...
0
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1answer
55 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 ...
3
votes
1answer
112 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
vote
1answer
54 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 $11011,$ ...
8
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2answers
232 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 ...
1
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0answers
76 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 ...
2
votes
2answers
118 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 ...
4
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1answer
193 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 ...
1
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3answers
102 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 ...
6
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2answers
195 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, ...
8
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1answer
175 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 ...
4
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3answers
334 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 ...
2
votes
3answers
247 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 ...
1
vote
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 ...
1
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1answer
148 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 ...
1
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2answers
311 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 ...
3
votes
2answers
598 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
vote
1answer
262 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 ...
2
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0answers
177 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: $$ ...
1
vote
1answer
117 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 ...
2
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2answers
349 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) ...
3
votes
1answer
238 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 ...
0
votes
2answers
122 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 ...
3
votes
1answer
101 views

What is the minimal axiomatization of a set of structures?

I wonder what the minimal axiomatization of a set of structures mean? I came across this term from Wikipedia: For a theory $T\in A,$ let $F(T)$ be the set of all structures that satisfy the ...
0
votes
1answer
34 views

Are interpretation and formalization inverse to each other?

I am not sure if I understand this correctly. Please correct me. In a formal system, an interpretation is a mapping from its formal language to one of its structures ie models. an ...
3
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0answers
249 views

Understanding definition of conservative extension of a theory

From Wikipedia In mathematical logic, a logical theory $T_2$ is a (proof theoretic) conservative extension of a theory $T_1$ if the language of $T_2$ extends the language of $T_1$; every ...
2
votes
3answers
736 views

Hofstadter's TNT: b is a power of 2 - is my formula doing what it is supposed to?

If you've read Hofstadter's Gödel, Escher, Bach, you must have come across the problem of expressing 'b is a power of 2' in Typographical Number Theory. An alternative way to say this is that every ...
3
votes
1answer
122 views

Axiom and concept

Is there a concept called "concept" defined in a formal system? Can concepts always be treated as axioms? Can axioms always be used to define concepts? For example, in ZFC set theory, I think the ...
4
votes
2answers
247 views

How does a recursive definition fit into a formal proof?

I understand a proof as a series of statements that are either axioms or follow from previous statements by a small set of rules of inference. I understand a recursive definition to be something like ...
1
vote
0answers
70 views

Differential fields and rings

If one is to compute the derivative of $$ y=3x+2 $$ by $$ \frac{\mathrm{d}(3x+2)}{\mathrm{d} x} $$ Would I be working with differential fields? Since differential fields is a first-order ...
4
votes
1answer
132 views

Computing square roots and calculus

If one were to verify that $$ \sqrt{2} < 3 $$ would the underlying formalisation require a logic more expressive than first-order? Or, is FOL sufficient since real numbers can be formalised in ...
1
vote
1answer
211 views

Expressing P = NP as a first order formula

I want to express P = NP in a completely formal way. My first try: There exists an algorithm A and a polynomial bound p such that for all input i, A(i) = true iff i is a satisfiable formula and ...
5
votes
5answers
231 views

Periodic sequences on finite alphabet

Let $\Sigma=\{A,B,C\}$ be an alphabet, and let $\Sigma^{\mathbb{N}}$ be the set of infinite sequences on $\Sigma$ (ie $ABCBCCCBABC...$). By outside conditions, I have several subsequences that are ...
1
vote
1answer
103 views

Understanding recursion in λ calculus

In recursion for λ calculus, I was wondering why the following two are equal (λx.g (x x)) (λx.g (x x)) g ((λx.g (x x)) (λx.g (x x))) How shall I understand g ((λx.g (x x)) (λx.g (x x)))? ...
0
votes
1answer
199 views

The way that a regular expression describes a regular language

A formal language is a set of words in some alphabet. It may be defined as being generated by a formal grammar or as being recognized by an automaton. For a regular language, it can also be described ...
2
votes
2answers
358 views

Can a formal language always be generated by a formal grammar?

A formal language is often defined by means of a formal grammar. I wonder for a formal language if there is always a formal grammar that generates the language? Does this answer have something to do ...
0
votes
3answers
185 views

Is a regular expression a string or a set of strings?

Quoted from Introduction to the Theory of Computation by Sipser, a regular expression is defined as: Say that R is a regular expression if R is a for some a in the alphabet $\Sigma$, ...
2
votes
2answers
168 views

Accessible formal specification and explanation of First Order Logic?

I am trying to get good at proofs by working through How To Prove It. Unfortunately I am very bothered by the fact that I do not understand all the formalities in First Order Logic + set theory ...
1
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1answer
157 views

What formal mathematical models exist for digital hardware?

What formal mathematical models exist for digital hardware? I am familiar with several non-formal models that are used as the basis of several hardware description language simulators and ...
1
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1answer
61 views

How to show that a set does not contain a specific string

If I have a set $S$ defined as the smallest set $S$ over an alphabet $A=\left\{ \star, \urcorner,(,), a_0,a_1, \dots \right\}$ ( $S\subseteq \cup_{k \in \mathbb{N}} A^k$) satisfying: $\bullet \ a_0, ...
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vote
2answers
152 views

Informal Equivalents of Mathematica “Set” and “SetDelayed”

How would one distinguish between what is meant by Mathematica's "Set" and "SetDelayed" functions in informal mathematical notation? Is there a way to make this distinction any any reasonably standard ...
2
votes
1answer
208 views

Generating all words in a language from CFG

I have a non-ambiguous context-free grammar. Is there some standard algorithm to create list of all the words in the language the CFG defines? This can be done with an abvious brute-force search by ...