# 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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### Is there a sense in which a formal theory can be more or less “metamathematically aware”?

For example, in classical logic, because of the law of the excluded middle, all propositions must be true or false, so in order to prove that a proposition is actually independent of the theory, we ...
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### Artificial intelligence methods in mathematics

Are there aritficial intelligence methods in mathematics, automatic theorem discovery and proving? Google gives results in the opposite direction - mathematical methods of AI. Are there applications ...
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### Connection regularity in measure theory and approximation in premeasure

In the measure theory lecture, we defined a measure-theoretic content as follows: $\mu: \mathscr{C} \rightarrow [0,\infty]$ with the property being additive on disjoint sets and that the empty set ...
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### For someone who loves mathematics - the more theoretical, the better - what type of engineering would you recommend?

I'm thinking there must be a type of engineering where math is strongly involved. What type of engineering would you suggest?
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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 ...
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### 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 ...
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### Could the classical Ramsey numbers be successors of primes?

Let $R(n)$ denote the classical Ramsey-number, that is, the smallest natural $N$, for which every blue-red edge-colouring of $K_N$ admits a monochromatic $K_n$. Consider the conjecture: $R(n)-1$ is ...
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### Could mathematical reasoning be non-axiomatic?

"Mathematics is not a deductive science—that's a cliché. When you try to prove a theorem, you don't just list the hypotheses, and then start to reason. What you do is trial and error, experimentation, ...
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### Is there a standard name for using a function application, rather than a variable, as a summation index, as in $\sum_{f(x)}$?

I am trying to find out whether there is a standard notion of generalizing indexing such as $\sum_i$ to function applications as in $\sum_{f(x)}$. Intuitively, the latter means "iterating over all ...
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### Dependence of axioms of Zermelo set theory

I'm reading a book about Zermelo–Fraenkel set theory, and I was told that there are a lot of dependence relation among the commonly used axiom system (in the question What axioms does ZF have, exactly?...
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### Intuitonism and metamathematics.

There are various reasons why one would want to reject the law of the excluded middle when doing "normal" mathematics, which I won't get to here, but accepting those, does the same reasoning hold when ...
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### What is the study of systems called?

Before I begin, i'll admit this isn't a maths question, however it is intrinsically a part of maths, and systems such as logic, set theory, and of course mathematics, will have been studied by many ...
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### Metamathematic: Cover the case if X=Y

I want to formalize: "If X is less than Y, Then U is equal to Y ", and have been told that $$\bf [\forall V \sim X=(Y+V)]U=Y$$ does not cover the case X=Y. Therefore I have rewritten it as  \bf [\...
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### Standard model of the Meta theory in the proof of arithmetic incompleteness

I have a question about the incompleteness of arithmetic. I wonder in which meta theory we are reasoning. For example, if we are in ZFC, then there are some non standard models of ZFC. As far as I ...
I am wondering about the use of the word separablity in two different areas of mathematics, namely algebra and topology. In topology, we call a topological space $(X,\mathscr{T})$ if it contains a ...