Questions tagged [meta-math]

Meta-theory is the term for the theory in which mathematics is formalized (often PA, ZFC or similar theories). Meta-mathematical statements are statements which are evaluated at the level of the meta-theory rather than the theory. This tag is for questions regarding meta-mathematical theories, and related topics.

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78 views

Physics and Riemann hypothesis [closed]

I was reading the article "Quantum physics sheds light on Riemann hypothesis" from Bristol University (http://www.bristol.ac.uk/maths/research/highlights/riemann-hypothesis/) and stopped ...
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40 views

Error in Kleene “Introduction to Metamathematics” parentheses lemmas?

In the classic book, Introduction to Metamathematics by Steven Kleene, Lemma 2 of Section 7 (chapter 2), seems to me to be false. I am wondering if I am missing something. Here is the context: The ...
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111 views

What does it really mean for a model to be pointwise definable?

(Note: I'm only an amateur in logic, so I'm sorry for any weird terminology or notation, or excessive tedious details. Most of what I know is from Kunen's Foundations of Mathematics.) I'm trying to ...
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Introduction to Metamathematics by S C Kleen. Help with exercise *135b needed.

I'm having trouble with exercise *135b in Introduction to Metamathematics by S. C. Kleene. The ask is to show that: $\vdash 0<a^{'}$. Here is how I would do it. Assume $a=b$. With Axiom 17 and ...
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73 views

Simultaneous / blended develop logic and set theory? [duplicate]

My goal right now is to gain a deep understanding of how to talk about mathematical objects formally. The presentation of how to do this in most books is generally to "assume some basic set ...
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68 views

Borders of set theory

I came across the following question in a discussion about set theory and model theory with a friend of mine. It is purely intrinsic. In some text books on set theory, e.g. the german text book by ...
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Is there an open mathematical conjecture which has been shown not to be provable in Peano Arithmetic before (possibly) being proved true?

There are several mathematical statements $\varphi$ about the natural numbers which are known to be true and known not to be provable in Peano Arithmetic (c.f. Gödel's incompleteness theorem, Paris-...
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48 views

qualify on metalanguage will cause what?

Add quantificational expressive power to metalanguage will cause what? In Teller's A Modern Formal Logic Primer, for example, (∀I) { [ Mod ( I , X ) & Mod (I , Y) ] -> Mod ( I, W ) }, which ...
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38 views

Are Real Numbers a Formal System?

I don't know a lot of mathematics but I have noticed that every branch of Mathematics has the same structure: some axioms (For example in Geometry might be Euclid's Axioms, in Probability might be ...
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51 views

Does “equivalence” of sets of axioms imply equality of the resulting theories?

For a set of sentences $A$, denote by $\langle A\rangle$ the set of sentences which can be derived from $A$ in some formal system. Define the equivalence relation $\sim$ on the set of finite sets of ...
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56 views

Is there a point at which you can no longer formally define something in math?

I'll admit this is a bit of a vague question, but I'm having trouble actually formulating it. I understand definitional systems can have recursive definitions however this can turn into nonsense ...
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45 views

Complex Constants

During my studies in mathematics a lot of people who are concerned about the communication of mathematics say that the term "imaginary numbers" is a misleading term. Maybe because these numbers are ...
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89 views

Formulation of constructible power in terms of Gödel operations; Going through metatheory

I am studying with Jech's Set Theory. He states this: For every transitive set $M$, $$\operatorname{def}(M) = \operatorname{cl}(M \cup \{M\}) \cap \mathcal{P}(M)$$ where $\operatorname{cl}$ ...
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Question of differential of undetermined coefficient

How to solve it by the method of undetermined coefficient. $$(D^3-D^2+3D+5)y = e^x \cos x$$ As CF of it is $$c_1e^{-x} + e^x ( c_2 \cos2x + c_3 \sin2x)$$ and $e^x$ is common in both CF and RHS of ...
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Why do models of ZF which are not $\omega$-models have non-standard formulas whose length is “infinitely large natural numbers”?

In his popular book Set Theory: An Introduction to Independence Proofs, Kunen gives the following definitions on the bottom of page 145: Let $\mathcal{A} = \lbrace A, E \rbrace$ be a structure for ...
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72 views

Understanding the role of Metatheories through an example

I've been studying for a course in set theory and I still have some problems in understading clearly the relation metatheory\theory. Being more specific, I'll present an example: If we choose $\...
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260 views

Formalized proof for any (already proven) theorem of Mathematics?

As everybody knows, there are thousands and thousands of theorems in mathematics having a proof that is accepted without any doubt because it follows impeccable and rigorous mathematical reasoning. I ...
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50 views

Necessary and sufficient conditions on antecedent for deduction theorem

The deduction theorem for classical predicate logic states that for any set of wffs $\Gamma$ , any closed wff $A$, and any wff B, if $\Gamma\cup\{A\}\vdash B$, then $\Gamma\vdash A\implies B$. The ...
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77 views

Is it possible that there is an axiom we have not discovered yet?

For example, the completeness of real numbers was discovered around 1800 and now $\mathbb{R}$ is considered to be a Dedekind-complete ordered field. Is it possible that a new axiom about $\mathbb{R}$ ...
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30 views

(Functional) space of all lambda types (algorithms)?

The most simplest notion of the algorithm is some kind of function with input and output. Input and output can be very sophisticated mathematical objects (not only numbers), but the is irrelevant, ...
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112 views

A finitist Gödel second incompleteness theorem

Can we prove by finitist means (such as with $\text{Con}(\text{ZFC}) \to \text{Con}(\text{ZFC + CH})$; see Kunen's Set Theory, p.8) that $\text{ZFC} \vdash \text{Con}(\text{ZFC}) \Rightarrow \text{ZFC}...
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93 views

Shoenfield's approach to the metamathematics of forcing

Cohen's approach to the metamathematics of forcing (see Kunen IV.5.1, new edition) involves talking about unspecified finite subsets $\Omega, \Lambda \subset \text{ZFC}$. Apparently, this is not very "...
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Non-string symbolic languages (e.g. diagrams) as “formal languages”

A formal language is a concept that is fundamental to computer science, logic, and metamathematics. It is defined as a subset $\mathcal L$ of the set $S^*$ of sequences of symbols from a set $S$, ...
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195 views

Where do new mathematicians come from? [closed]

It appears to me most research in mathematics requires close studying of the work done by the previous generation, so this seems to rely on mathematicians keeping track of and recording the new ...
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123 views

Are there projects that visualize how proofs relate to each other (similar to what the Paperscape Project does for publications)?

The Mathematics Genealogy Project lists mentoring relationships between mathematicians, the Paperscape Project visualizes which publications are "close" to each other (by analyzing citations and ...
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122 views

Estimating meta-mathematical properties of conjectures

Given a formalized system of mathematics (a set of axioms and deduction rules), is it possible to estimate meta-mathematical properties of conjectures in this system such as the number of deduction ...
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18 views

Example of logic (concept) that can not be embedded (expressed) in Isabelle/HOL?

Tutorial for Isabelle/HOL https://isabelle.in.tum.de/doc/tutorial.pdf states that: HOL can express most mathematical concepts, and functional programmingis just one particularly simple and ...
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1answer
71 views

What alternatives are there to the generalized continuum hypothesis?

Since the generalized continuum hypothesis is independent of ZFC, we can adopt its negation as an axiom. However, this would not be a very "nice" axiom, since all it does is assert that there is at ...
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89 views

Confusion about soundness of sequent calculus

I'm currently reading a pdf textbook called Sets, Logic, Computation An Open Introduction to Metalogic Remixed by Richard Zach, and it covers sequent calculus LK and (partially) proves its soundness. ...
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86 views

Is a variable/arbitrary element a precisely defined mathematical object? [duplicate]

I have been wondering about the formalism of what 'exactly' a variable is, and its role in proofs in mathematics. I have seen a few questions here, such as this one: Is there a way of defining the ...
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30 views

Why are summation/product operators not much welcomed as being part of closed form expressions, even if they have finite terms?

I read somewhere that in most closed form expressions, which are expressions used to calculate a certain outcome with only finite terms, summation/product operators are mostly neglected. I can ...
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Theory of arithmetic equiconsistent with ZF

I am wondering if there is an extension of PA by additional axioms, but without extending its language, which is equiconsistent with ZF. In particular I am wondering if there is a relatively short ...
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62 views

How can I be sure an expression cannot be further simplified?

I am not even sure what I mean here, saying "simplification". I have been taught this for many years, but I realize I actually have no idea. For instance, consider: $$ \tag1 \frac {\sqrt{x} + \sqrt{...
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104 views

Shortest FOL deduction for $\forall x \forall y(x = y \to y = x)$

In Enderton's A Mathematical Introduction to Logic (second edition, SECTION 2.4 - A Deductive Calculus, page 122), he provides a worked example shown below. Show that $\vdash\forall x\forall y(x=y\...
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Who introduced the term torsion first and why [duplicate]

In algebra, an element of finite order is called a torsion, and a subgroup consisting of all torsions of a group is call a torsion subgroup. And I just got curioused about the term 'torsion'; who ...
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2answers
49 views

Are predicate or function symbols with 3+ places actually used in mathematical logic?

Let $n$ be a nonnegative integer. The language of first-order logic includes the following symbols : predicate symbols with $n$ places: $P^n_0$, $P^n_1$, $P^n_2$, $\dots$ function symbols with $n$ ...
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64 views

Second Metamathematics Book Suggestion

I worked through Kleene’s Introduction to Metamathematics and am interested where we stand in the field 70 years later. What are some good text suggestions in modern metamathematics to follow this ...
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46 views

Gödel's Second Theorem and the Consistency of Robinson's System

In Kleene, Introduction to Metamathematics, $\S42$ (end), it is shown that if formal number theory is simply consistent, then its consistency cannot be proven formally within the system. In other ...
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Resource Metalogic: Proving that a theorem can (not) be deduced from given axioms in a certain number of steps

One of the reasons automated theorem proving systems haven't caught up with humans yet might be that they have no intuition about resource exploration vs exploitation and other evolved heuristics, due ...
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Importance of predicates similar to the form $(Ex)(y)R(a_1,…,a_n,x,y)$ where $R$ is general recursive

In Introduction to Metamathematics (Kleene) there are a few theorems pertaining to predicates of the the form $$(Ex)R(a_1,...a_n,x) \,\,\,\,\,\,(x)(Ey)R(a_1,...a_n,x,y)\,\,\,\,\,\,(Ex)(y)(Ez)R(a_1,......
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46 views

Completeness theorem: validity in some non-empty domain

The completeness theorem in my text says that if a formula $G$ is valid in the domain of natural numbers, then $G$ is provable in the predicate calculus. The corollary also says that then $G$ is ...
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Can all math expressions be converted into trees?

Operators usually have different notations (prefix, infix, postfix, ... ?), but expressions using them can all be transformed into expression trees. E.g. ...
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Questions on the primitive recursiveness of ${\mu}y_{y<z}R(x_1,…x_n,y)$

In Introduction to metamathematics (Kleene), The function ${\mu}y_{y<z}R(x_1,...x_n,y)$ is shown to be primitive recursive in $R$ (where ${\mu}y_{y<z}R(x_1,...x_n,y)$ is the smallest $y<z$ ...
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24 views

Trivial question on general recursive functions

I pretty sure this is true but I couldn't find it stated in my text, so I just wanted to verify it. Is the following true? If $\phi$ is general recursive in $\Psi$ and $\Psi$ is general recursive, ...
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34 views

What are the conclusions we can draw from Kleene's Recursion Theorem regarding computability?

Kleene's Recursion Theorem in his Introduction to Metamathematics $\S66$ is written Theorem XXVI: For any $n\geq0$, let $\textbf{F}(\zeta;x_1,...,x_n)$ be a partial recursive functional, in which ...
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79 views

Proof vs Demonstration

In metamathematics, what convention is used to distinguish an informal proof about the system and a formal proof in the system? I'd like to reserve "proof" for the object language, what should I ...
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90 views

is there some statement equivalent to consistency of $ZFC$ in $ZFC$?

In my studies I learn the strengthened finite Ramsey theorem is equivalent to consistency of peano arithmetic and is a sentence of the language of arithmetic. I wanna a example of set theory
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94 views

What is the difference between a definition and an equivalence class?

In what way is 'the definition of $x$ is $y$' ($x:=y$) not the same as '$x$ is equivalent to $y$' ($x=y$)? I can find no justification for making the distinction aside from 'it feels right'. It seems ...
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49 views

Is Base Unit Geometry Lacking?

This article somewhat on the rationality of pi defines some geometric terms differently, such that there are no perfect circles, and no irrational numbers, there are only approximations. At least, ...
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81 views

Can we use metatheory of set theory for proof statements in set theory?

I wanna know if is possible or have examples of theorems in set theory, for example $\beta$, that have a demonstration of forme $Cons(ZFC)\Rightarrow \beta$ but $\beta$ is independent of axioms of $...

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