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.

344 questions
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7 answers

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 there some kind of normalization going on, or some…
Greg L
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88
votes
10 answers

Is mathematics just a bunch of nested empty sets?

When in high school I used to see mathematical objects as ideal objects whose existence is independent of us. But when I learned set theory, I discovered that all mathematical objects I was studying were sets, for example: $ 0 = \emptyset $ What…
Vinicius L. Deloi
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64
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4 answers

Why can we use induction when studying metamathematics?

In fact I don't understand the meaning of the word "metamathematics". I just want to know, for example, why can we use mathematical induction in the proof of logical theorems, like The Deduction Theorem, or even some more fundamental proposition…
183orbco3
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48
votes
2 answers

How do you go about doing mathematics on a day to day basis?

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 ideas. A venerable mathematician once described his…
Jon Bannon
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39
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8 answers

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. So I am looking for references which answer the…
student
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39
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4 answers

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. The question is interesting to me and I would be…
user23211
38
votes
4 answers

What is the difference between a proposition and a theorem?

What is the difference between a proposition and a theorem? How do people decide which of the two to use in, e.g., textbooks? Somehow I think "proposition" sounds less serious.
M. M.
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36
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3 answers

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 they use to replace results that have to do with…
user12205
35
votes
4 answers

How are metalogic proofs valid?

I'm reading some materials on mathematical logic. I wonder how we can "prove" metalogical properties (soundness, completeness, etc.)? As at this point, the proof system has not been verified yet. Isn't this a chicken-and-egg question (we may then…
A.Stone
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34
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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
23
votes
6 answers

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 do we know that our "approximation" is the right…
Adrian
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23
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1 answer

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 (Peano) Arithmetic. One textbook I read a while ago…
Kevin
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3 answers

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' maths, I have some time now while doing my PhD…
SBF
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18
votes
4 answers

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 ($\models$)? I see that: the turnstile symbol $ \vdash…
Emilio Novati
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17
votes
2 answers

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 theorem by using an invented formal system called…
StrangerLoop
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