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 ...

learn more… | top users | synonyms

59
votes
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 ...
34
votes
3answers
1k views

What is your method?

Many young, and not so young, mathematicians struggle with how to spend their time. Perhaps this is due to the 90%-10% rule for mathematical insight: 90 pages of work yield only 10 pages of useful ...
29
votes
5answers
1k 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. ...
25
votes
3answers
939 views

Rejecting infinity

I've heard about mathematicians who defend a strictly finite conception of mathematics, with no room for infinity. I wonder, how is it possible for these people to do this? Are there any concepts that ...
15
votes
5answers
683 views

Definition of definition

I was wondering if there is a good way to "define" what definition means exactly in mathematics. Since the answers may be subjective or philosophical, I want to ask only for references on this topic. ...
13
votes
6answers
3k views

Why bother with Mathematics, if Gödel's Incompleteness Theorem is true?

OK, maybe the title is exaggerated, but is it true that the rest of math is just "good enough", or a good approximation of absolute truth - like Newtonian physics compared to general relativity? How ...
12
votes
6answers
978 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 ...
12
votes
2answers
605 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 ...
11
votes
1answer
349 views

Formalizing metamathematics

I am reading historical/philosophical stuff on the concept of "metamathematics" and am by now quite confused. Several questions emerged, but they are probably somehow confused and interrelated, I ...
10
votes
4answers
460 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. ...
10
votes
2answers
286 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 ...
8
votes
6answers
1k views

Is It True that We Can Never Be Sure of Validity of a Mathematical Proof?

The reason I ask this is because difficult mathematical proofs are just not plain self-evident. You would need a few years of intensive study before you can get to the point of understanding the ...
8
votes
5answers
1k views

Common misconceptions about math

YARFMO (Yet another reposting from Mathoverflow) ;-) The more you know about math the more you find conceptions previously thought correct to be false: 1.) math is not as exact as many believe - in ...
8
votes
3answers
1k views

Proposition vs Theorem

What is the distinction between a proposition and a theorem? How do people decide which of the 2 to use in, say, textbooks? Somehow I think proposition sounds less serious... Thanks.
8
votes
2answers
890 views

Definition of “non-constructive proof”

I was wondering if it is possible to define exactly what a non-constructive (nc) proof is. I have often seen the concept associated with the use of principles such as the axiom of choice or the law of ...
8
votes
1answer
970 views

Fiction “Division by Zero” By Ted Chiang

Fiction "Division by Zero" By Ted Chiang I read the fiction story "Division by Zero" By Ted Chiang My interpretation is the character finds a proof that arithmetic is inconsistent. Is there a ...
8
votes
1answer
709 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 ...
7
votes
3answers
356 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 ...
7
votes
3answers
219 views

Is there a way to prove that a theorem has no elementary proof? Or to prove that something may have no proof?

Recently I was trying to prove something, more or less elementarily, but eventually started going in circles. My prof said that the proof involves mathematical tools that I've not seen yet, and that ...
7
votes
1answer
136 views

Are there any strongly axiomatizable logics that are not compact?

I mean here a logic in the sense of a language and semantics. By strongly axiomatizable I mean strongly sound and strongly complete. So I'm basically asking if there is a particular deductive system ...
6
votes
6answers
891 views

Why some people don't like proofs by contradiction [duplicate]

Possible Duplicate: Are the “proofs by contradiction” weaker than other proofs? I have been active on this site for two months and on a few occasions I noticed that some people ...
6
votes
3answers
149 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 ...
6
votes
1answer
106 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., ...
5
votes
5answers
647 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 + ...
5
votes
3answers
330 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 ...
5
votes
3answers
217 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 ...
5
votes
2answers
80 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 ...
5
votes
1answer
319 views

Mathematical structures

Preamble: My previous education was focused either on classical analysis (which was given in quite old traditions, I guess) or on applied Mathematics. Since I was feeling lack of knowledge in 'modern' ...
4
votes
4answers
382 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 ...
4
votes
0answers
82 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 ...
3
votes
2answers
581 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, ...
3
votes
2answers
122 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, ...
3
votes
1answer
246 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 ...
3
votes
2answers
91 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
votes
2answers
82 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 ...
3
votes
0answers
918 views

New branches of math? [closed]

I have been wondering if math would be more enjoyable, if one was able to start a new field and come up with all the definitions, methods, etc. rather than starting where someone else ended. ...
2
votes
2answers
787 views

Is Gödel's theorem invalid? [closed]

Right now I've skim through Gödel's theorem is invalid by Diego Saá on arXiv (freely available). As it seems very plausible, I ask for any references and scrutinizations of the paper.
2
votes
2answers
242 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
votes
1answer
222 views

Truth and undecidability

I believe this is more of a philosophical question. Given a consistent theory T and a statement S independent of T. Can S be true or false in T? (I don't see any contradiction with that) I read that ...
2
votes
1answer
123 views

question about Godel numbering

I have a question about Godel numbering, it is trivial but I would like to know how can you know the length of an expression through its Godel number. ¿? I think you can use a recursive function but ...
2
votes
1answer
93 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 ...
2
votes
1answer
109 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 ...
2
votes
1answer
274 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 ...
2
votes
1answer
41 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 $n,m \in \mathbb{N}$: ...
2
votes
3answers
66 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 ...
2
votes
4answers
182 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 ...
2
votes
2answers
150 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 ...
2
votes
0answers
43 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 ...
2
votes
0answers
229 views

Statements about logic (=“metalogic”(??)) [closed]

Sometimes there are statements about logic e.g. "That's not logical" and I can neither prove nor disprove a statement about logic with no definition for logic itself. It's just a negation and it's ...
1
vote
2answers
81 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 ...