Mathematics Stack Exchange
Mathematics Stack Exchange

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.

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.

333 questions
33
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4 answers

What exactly is an equation?

It seems to me an equation, in an abstract sense, must always involve some varying quantities where the varying quantities belong in some space (set, algebraic structure, what have you). In order to make precise the phrase, "vary quantities", it…
user1054
10
votes
3 answers

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 site which seemed intriguing. "I believe that in…
tcmtan
  • 1,883
10
votes
1 answer

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 formal proof the fiction can't come true? (I don't…
jerr18
  • 415
8
votes
6 answers

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 topics and the proofs. The problem is that for a…
Graviton
  • 2,292
5
votes
3 answers

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 the Riemann hypothesis. A secondary goal seems to…
user23211
5
votes
4 answers

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 the objects in any model of set theory is to me…
echoone
  • 1,975
2
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0 answers

What is the proof system of meta-mathematics?

After having completed a few courses in logic, even at a graduate level, at no point has it been discussed what the proof system for proving statements about formal systems actually is. It seems that using regular set-theoretic mathematics is…
Tony
  • 1,044
1
vote
1 answer

Theorem XIV Corollary in Kleene's Introduction to Metamathematics

In Kleene's IM, the Corollary to Theorem XIV in §60 states: If a class can be enumerated (allowing repetitions) by a general recursive function, it can be enumerated (allowing repetitions) by a primitive recursive function where A set or class…
Mircea Baja
  • 133
1
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0 answers

Method for calculating exponents and beyond with logarithmic growth provided a memorized set of smaller problems

This question is very hard to word so I'm sorry about that, but here goes a try. With Addition Let's assume I have all addition facts from 1-10 memorized. When doing $125+126$ i will employ these steps $5+6 = 11$ $2 + 2 + 1 = 5$ carrying the one…
Zachiah
  • 123
1
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2 answers

Isn't Math basically a matter of combinatorics?

As I discover the foundations of mathematics, I begin to understand that it is a matter of arbitrarily defining axioms and combining them - arriving at what we call theorems. Having said that, it looks to me the following: We could fix a finite…
Alexandre Tourinho
  • 809
1
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0 answers

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. ((a + b) * c) + 7 Is this true for all expressions used in mathematics? What insights can…
2080
  • 150
1
vote
1 answer

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, then $\phi$ is general recursive.
BENG
  • 1,105
1
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2 answers

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 works. As we know, mathematics is about…
Archival
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1
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0 answers

Where to put the dot at the end of a sentence when using cases-figure?

Often I use the cases-figure at the end of a sentence and I never know where to put the dot at the end of the sentence. I think, there are two options. Option 1: I can put the dot in the last case: I get a sentence that looks…
user34632
1
vote
0 answers

how often do you think about mathematics?

As a person who is serious about the subject, how often do you think about mathematics? I assume most of you are graduate students or PhD students and would consider yourselves mathematicians. In other words, you are serious about your subject. …
Nażysław Zbyłutowicz
  • 709
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