# Tagged Questions

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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### 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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### 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 ...
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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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### 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 ...
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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 (\$\...
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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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### 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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### 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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### 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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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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### 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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### 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 ...
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### 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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### 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 ...
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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.