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.
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Platonistic interpretation of Gödel (theorem 14.2, I in Kunen).
On page $41$ in Kunens "Set Theory An Introduction to Independence Proofs", after proving
If $\phi(x)$ is any formula in one free variable, $x$, then there is a sentence $\psi$ such that
$$...
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Theorem 5.6, chapter IV in Kunens Set Theory
Boldface-letters (i.e. $\mathbf{A}$,$\mathbf{V}$,etc.) indicate a class.
The following is an excerpt from Kunens "Set Theory An Introduction to Independence Proofs" (theorem 5.6, $\S5$ of ...
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Why doesn't $RCA_0$ prove $\Sigma^0_1$-comprehension?
Answer: because that's $ACA_0$, alright, but:
Friedman et al.'s 1983 "Countable algebra and set existence axioms" has [verbatim, including old terminology and dubious notation]:
Lemma 1.6 ($...
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How do we escape infinite regress when discussing truth in mathematical logic? [duplicate]
In (first-order) logic, I understand that there are two notions of the truth of a sentence $\phi$ in a theory $T$:
Syntactic truth: $T\vdash \phi$ if $\phi$ is provable from $T$,
Semantic truth: $T\...
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Understanding Metatheory and the Broader Picture of Foundational Set Theory
So I'm trying to put together a clearer picture of what is going on when we study set theory. I'll describe my current picture which I'd appreciate some feedback on, and I'll ask some specific ...
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If set theory only contains the notions of “set” and “is a member of” as primitives, how can an axiom of set theory refer to a “formula”?
It's said that the primitive concepts of set theory are those of "set" and "membership", then all axioms of set theory must begin with "Let $A$ be a set" or "Let $x\...
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Meaning of "theorem of a system"
The following excerpt is from page 357 of Logic: The Laws of Truth by Nicholas Smith:
Given a system of proof - say, the tree method for GPLI - we call propositions that can be proven using that ...
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Math that does not have infinity
I am not a mathematician. So I am not even sure if what I am asking is logically coherent. But I do have some application-based curiosity that I would like to enlighten myself about. I will first pose,...
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does deductive completeness implies semantic completeness
i wanted to understand godel's $ \ \bf completeness \ $ theorem, so while doing some research on google i found this wikipedia page " https://en.wikipedia.org/wiki/G%C3%B6del%...
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Isn't the Compactness theorem in propositional logic trivial?
I am learning propositional logic via a script.
The compactness theorem is presented as: " Let S be a set of propositional formulas. If each finite subset of S is satisfiable, then
S is ...
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What is a "class model" exactly?
In the literature about set theory, one encounters the words "set-model" and "class-model" which I have difficulties to understand. Here is my viewpoint :
One starts with a ...
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What is the most primitive notion in mathematics?
I had a recent conversation with a professional mathematician about the status of relations, functions and predicates. I was arguing that it seems intuitive (to me at least) to classify them in this ...
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What is the categorical setting for higher-level real analysis?
A lot of disciplines in higher-level mathematics can be summarized by describing what objects they study and in what setting they are studied in. For example,
Topology is the study of topological ...
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What does it mean for one theorem to depend on another?
Recently, there is a happy result by some high-schoolers: a proof of Pythagoras by using trigonometry without using circular reasoning i.e. $\sin^2A + \cos^2A = 1$. Good for them, hurray!
But it got ...
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Type Theory as a Meta-Language for Logic
I am unsure which StackExchange site is the most appropriate for this question, but I believe this site is the most appropriate.
My current project involves rigorously proving all the mathematical ...
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