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 ...

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Is there a standard name for using a function application, rather than a variable, as a summation index, as in $\sum_{f(x)}$?

I am trying to find out whether there is a standard notion of generalizing indexing such as $\sum_i$ to function applications as in $\sum_{f(x)}$. Intuitively, the latter means "iterating over all ...
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58 views

Dependence of axioms of Zermelo set theory

I'm reading a book about Zermelo–Fraenkel set theory, and I was told that there are a lot of dependence relation among the commonly used axiom system (in the question What axioms does ZF have, ...
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49 views

Are Euclid's Axioms Non-Logical or Logical

This question may seem trivial, but I recently became aware of the distinction between the two types of axioms: Logical and non-logical. What category does Euclid's fall under? I would assume they ...
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3answers
203 views

Is there an area of study regarding why certain mathematical definitions are useful?

Often in my studies I'll come across an definition, which I understand, and but don't necessarily see why the particular definition was chosen to be studied. For example, the topological axioms ...
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26 views

For someone who loves mathematics - the more theoretical, the better - what type of engineering would you recommend?

I'm thinking there must be a type of engineering where math is strongly involved. What type of engineering would you suggest?
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2answers
95 views

How to show Universal Quantifier distributes over implication?

How to show Universal Quantifier distributes over implication? I've tried to no avail to show $\forall x(P(x) \implies Q(x))$ is equivalent to $\forall x(P(x)) \implies \forall (Q(x))$ but it seems no ...
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1answer
50 views

What is the difference from a theorem and a meta-theorem?

I'm confused about what a meta-theorem exactly is and if a meta-theorem can be used to prove a theorem. To illustrate my confusion i give an example. Given the three statements: Every vector space ...
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4answers
174 views

Meaning of symbols $\vdash$ and $ \models$

I'm confused about the use of symbols $\vdash$ and $ \models$. Reading the answers to Notation Question: What does $\vdash$ mean in logic? and What is the meaning of the double turnstile symbol ...
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2answers
199 views

Good source on current 'views and thoughts' on mathematics

Recently I've become interested in the history of the way people think about mathematics. What I mean by this is for example how Godels proof basically put an end to the whole school of thought ...
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1answer
49 views

Model-theoretic question about language of field theory.

Let $\mathscr{L}=\{+,·\}$ be the language of the theory of fields. Let $\phi$ be a sentence in this language. Show, using the compactness theorem of first-order logic, that if $\phi$ holds in finite ...
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1answer
75 views

Most general mathematical framework

One can think of the same mathematical object in many different ways. For example take $\mathbb{R}$. One can think of this as (assume necessary hypotheses and so on) As a group. As a one ...
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3answers
216 views

Where can I learn about Mathematical Philosophy?

This is a very vague question, but a question nonetheless. I am becoming increasingly more interested in what can be vaguely categorized as Mathematical Philosophy, or more specifically perhaps, ...
5
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2answers
143 views

Has the Gödel sentence been explicitly produced?

I do not pretend to know much about mathematical logic. But my curiosity was piqued when I read Hofstadter's Gödel, Escher, Bach, which tries to explain the proof of Gödel's first incompleteness ...
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1answer
77 views

Where can I find an ontology of algebraic structures?

A group is a monoid where every element admits an inverse, A ring is a monoid under multiplication that distributes over a commutative group A field is a ring whose non-zero elements form a group ...
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4answers
419 views

The standard role of intuitive numbers in the foundations of mathematics

In my career I've been formed mostly in the formal side of mathematics, that is, standard set theory and every classical branch of mathematics that uses set theory. However, I am not quite sure about ...
3
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2answers
115 views

Can a non-classical logic be used as a meta-logic to develop classical logic?

I have read much about non-classical logics such that paraconsistent logics , relevance logics , substructural logics , non-monotonic logic and so on. I think that the meta-logic logicians use to ...
3
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2answers
133 views

General view of Theorems

I'm trying to see almost all theorems ( at least the non-existential ones ) as affirming that some formula ( mostly of first-order logic language ) is a logical consequence of other formulas. So, ...
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1answer
43 views

First-Order Logic: Non-Normal Model of Sentences True in all Normal Models?

Let $\mathcal{L}$ be a first-order language with quantifier $\forall$, connectives $\neg$ and $\rightarrow$, a two-place predicate $E$ and a one-place function symbol $f$. There are no other ...
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68 views

Maths, esp. Godel, and poetry

At the risk of being an interloper: I'm a poet with a bit of mathematical training. Right now I've got a grant from the Arts Council of Northern Ireland to write a collection (loosely) based on the ...
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1answer
55 views

What exactly is a property?

How is a property $P$ formally defined in mathematics? I mean for example if $f$ is a morphism from an object $X$ to $Y$ in some category, then somehow I feel that "has codomain $Y$" is too broad to ...
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1answer
46 views

Are all theorems of minimal arithmetic theorems of a given theory?

I am working on some metamathematics revision and the following question came up. Let the theory $R_0$ be axiomatized by the following axiom schemata which hold for all $n,m \in \mathbb{N}$: ...
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2answers
117 views

Use of propositional logic connectives in the meta-language

I have a doubt that might seem a bit confusing so i will try to explain it the clearer i can. Suppose we have an expression "A o B" in the meta-language, where 'o' refers to those logical ...
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2answers
96 views

What is gained by internalizing LST (the language of set theory)?

I'm reading up on Gödels constructible universe L in the book "Constructibility" by Devlin, and by comparing his text with texts like Kunen and Jech, there is one thing in particular that he's doing ...
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3answers
78 views

Satisfiability Proof Question

Exercise: Prove that $\Gamma\models A$ iff $\Gamma\cup\{\neg A\}$ is not satisfiable. Proof: We must prove two clauses: $\Gamma\models A\Rightarrow \Gamma\cup\{\neg A\}$ is not satisfiable ...
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1answer
129 views

Do metatheoretic results carry between mutually interpretable theories?

If two theories A and B are mutually interpretable, in the sense of there existing a translation procedure from A to B and from B to A, does it follow that whatever metatheoretic results (e.g., ...
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4answers
278 views

Stats is not maths?

How mainstream is the claim that stats is not maths? And if it's right, how many people don't agree? Given that it's all numbers, taught by maths departments and you get maths credits for it, I ...
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165 views

Is there a useful application of Peano arithmetic?

If there is, can someone provide an example of how Peano arithmetic can be used to solve a real-world problem? If not, can someone provide an example of any axiomatic system other than ZFC that can ...
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4answers
544 views

Clarification of a remark of J. Steel on the independence of Goldbach from ZFC

On page 424 of the following paper: S. Feferman, Harvey M. Friedman, P. Maddy and John R. Steel, ``Does Mathematics Need New Axioms?'' The Bulletin of Symbolic Logic, Vol. 6, No. 4 (Dec., 2000), pp. ...
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256 views

The mathematics of mathematical knowledge

It's been many years since I did any real mathematics but last night after pondering the process involved in my mathematical journey I had an idea about the abstraction of how mathematical analysis ...
2
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1answer
153 views

question about Gödel numbering

I have a question about Gödel numbering, it is trivial but I would like to know how can you know the length of an expression through its Gödel number. ¿? I think you can use a recursive function but ...
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5answers
714 views

Is mathematics the only language that is not subject of interpretation?

Do you know any other "language" that is used by people except mathematics and is not subject of interpretation? By subject of interpratation I mean e.g. that 1 000 000 people will undertand that 1 + ...
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1answer
88 views

A Simpler Characterization of Inductive Definitions?

While reading appendix A of John Harrison's "Handbook of Practical Logic and Automated Reasoning" a somewhat advanced theorem is appealed to as a prerequisite for characterizing when an inductive ...
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397 views

Meaning and example(s) of Qiaochu's quote.

I happen to come across this page http://math.uchicago.edu/~chonoles/quotations.html which contains some beautiful quotes by various mathematicians and I came across Qiaochu's quote as claimed by the ...
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714 views

What branch of mathematics is most needed in the industry or how one can make living with mathematics (apart from teaching)?

If you learn carpentry or programming you have the clear options of becoming a carpenter or programmer. But what if you learn mathematics? I know that there are some financial institutions out there, ...
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308 views

Are there formal systems that can not be proved to be complete or incomplete?

I'm reading GEB and was thinking about this. Are there any formal systems where the proof of their completeness/incompleteness is unprovable? This question could go ad infinitum. Could there be any ...
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1answer
122 views

Separation of mathematics and metamathematics

I recall reading that it's important to separate mathematics and metamathematics. What exactly does this mean, and why is it so? I understand that this question may make no sense without more ...
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2answers
88 views

Does a “solution map” system like this already exist?

I was doing this simple Calc 1 problem and it took me forever to get it right and it was embarrassing. I could see that the problem was easy but I just couldn't 'see' what I was doing. I couldn't ...
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1answer
875 views

How to address mistakes in published papers?

I have recently discovered some mistakes in a published maths article. I have contacted the author pointing out politely my concerns, but I got no specific answer, just a "polite" one, that the ...
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3answers
222 views

Has anyone ever tried to develop a theory based on a negation of a commonly believed conjecture?

I know that plenty of theorems have been published assuming the Riemann hypothesis to be true. I understand that the main goal of such research is to have a theory ready when someone finally proves ...
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3answers
153 views

If $T$ proves $\operatorname{Con}(ZFC)$, is $T$ at least as strong as set theory?

I am looking for either a proof of counterexample of this: Lemma: Let $\pi$ be a faithful interpretation of $PA$ into $ZFC$, and let $PA'$ be the image of $PA$ under $\pi$. If there is a $T$ with ...
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90 views

An “internal” condition on $T$ so that for the standard provability predicate, $T$ proves $\text{Pf}(\underline S)$ implies $T$ proves $S$?

This is probably quite basic, but I'd like to make sure I got this right. Regarding the proof of Goedel's first incompleteness theorem, say that we have $T$ containing $PA$ effectively axiomatizable ...
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2answers
252 views

Is there a way of defining the notion of a variable mathematically?

I know that the notion of "set" is one that cannot be defined mathematically since it is the fundamental data type that is used to define everything else (and the definition which says that "sets" are ...
2
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1answer
370 views

Mathematical structures and signature

From Wikipedia: In mathematics, a structure on a set, or more generally a type, consists of additional mathematical objects that in some manner attach (or relate) to the set, making it easier ...
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8answers
3k views

Why are all the interesting constants so small?

A quick look at the wikipedia entry on mathematical constants suggests that the most important fundamental constants all live in the immediate neighborhood of the first few positive integers. Is ...
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6answers
1k views

Geometric proof of existence of irrational numbers.

It is easy, using only straightedge and compass, to construct irrational lengths, is there a way to prove, using only straightedge and compass, that there are constructible lengths which are ...
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5answers
2k views

Why is it considered unlikely that there could be a contradiction in ZF/ZFC?

EDIT: No answer addresses the "bottleneck" question. It's not surprising to me because the question is vague. But I would like to know whether that is indeed the reason, or perhaps something else. ...
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1answer
317 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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356 views

Are “axioms” in topology theory really axioms?

If I understand correctly, axioms are those statements that we assume to be true, instead of proving to be true. I have seen that in topology theory, various axioms of countability and separation ...
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157 views

A theorem about inductive inference

In the book 'Introduction of the theory of Statistics' by Mood,Graybill,Boes (third edition)on page 220 (Chapter 6 on Sampling) you can read: 'Inductive inference is well known to be a hazardous ...
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648 views

How rare is it that a theorem with published proof turns out to be wrong?

There is a story I read about tiling the plane with convex pentagons. You can read about it in this article on pages 1 and 2. Summary of the story: A guy showed in his doctorate work all classes of ...