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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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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3answers
2k 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 ...
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5answers
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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. ...
28
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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 ...
21
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1answer
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What did Hilbert actually want for his second problem?

When I read about the historical background of Gödel's incompleteness theorems, it is often mentioned that he was essentially responding to Hilbert, who was trying to prove the consistency of (...
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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. ...
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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 ...
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Is this visual analogy to Gödel's incompleteness theorem accurate?

Today I was trying to explaining the Gödel's theorem to a layman, I've drawn a figure similar to the one below and said that: A truth is a consequence of the axioms (with the axioms also being truth)...
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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' ...
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6answers
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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 ...
13
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733 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 ...
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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 ...
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1answer
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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 ...
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637 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
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2answers
336 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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2answers
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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 ...
10
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1answer
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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 ...
9
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3answers
533 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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1answer
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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 ...
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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
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3answers
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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
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491 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 ...
7
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3answers
232 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
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2answers
105 views

Are there any coherent, complete, mathematical systems that do not imply the existence of the infinite?

Basically, are there any systems(complete with axioms) that do not imply the existence of an infinity? Is it possible to construct a mathematical system without infinity?
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1answer
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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., ...
7
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1answer
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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 ...
7
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1answer
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Primitive recursion and $\Delta^0_0$

Until recently I assumed that primitive recursive relations are exactly $\Delta^0_0$ (i.e. bounded) ones, but I learned they're different (the former is a proper superclass of the latter). I have ...
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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
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1answer
230 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 ...
6
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3answers
162 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 $PA'...
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5answers
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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
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3answers
391 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
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3answers
232 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
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278 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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1answer
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Given any set of consistent axioms, is it always possible to find a model for these axioms in ZFC set theory?

If not, are there any conditions under which there must be a model under ZFC theory? Alternatively, is there any set of axioms for which this does hold true? If so, can we drop some of the axioms and ...
5
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2answers
123 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
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1answer
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What does “dual statement” mean exactly in category theory?

I have long been confused about this notion. I know that for a statement within a single category, forming the dual statement is just reversing every arrows. But what about a statement concerning ...
5
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263 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 (...
5
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2answers
215 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 ...
5
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1answer
438 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 ...
5
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2answers
95 views

Meta proof-searching

Suppose you have a particular theory (ex: $ZFC$) in which you want to prove a statement $\phi$. One can attempt to find a proof of $\phi$ that can be verified, but another tactic can be to find a ...
4
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1answer
49 views

Discussion on AC and countable unions of countable sets

My question stems from the comments following Asaf Karagila's answer here : Why can't you pick socks using coin flips? At some point there was a discussion about whether or not a countable union ...
4
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1answer
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Quantification over the set(?) of predicates

When learning set theory and logic, one fact that popped up a handful of times was that we did not quantify over predicates. To quote the notes I took in class on the axiom of unrestricted ...
4
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0answers
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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
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2answers
899 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
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3answers
338 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 ...
3
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4answers
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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 ($\...
3
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1answer
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Help understand theorem that any set of first order sentences satisfied by N has a model that's a strict superset of N.

I saw the following theorem (in Computational Complexity book by Papadimitriou, p. 111) : Theorem : If $\Delta$ is a set of first-order sentences such that $N$ $\vDash \Delta$, then there ...
3
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1answer
531 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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1answer
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Formalizing the meta-language of First order Logic and studying it as a formal system

We've a formal system say First order Logic, we reason about it in our meta-language using our meta-logic. We study its properties as a mathematical object. We prove theorems like group theory. This ...