# 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.

212 questions
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### Are the HBL derivability conditions necessary for Gödel's incompleteness theorems? (For Löb's theorem?)

I am currently working with 'proof theory and logical complexity', a monograph on proof theory. In the exercises of the first chapter one was asked to prove Löb's theorem (https://en.wikipedia.org/...
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### An alternative formulation (or corollary) of Tarski's theorem? [Or just a typo?]

In my proof theory monograph (proof theory and logical complexity, Girard from '87) there is an exercise 1.5.4. on page 78 called 'Tarski's theorem'. It says: "Show that there is no truth predicate ...
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### Why the 'natural' consistency proof of PA cannot be carried out $\textbf{in}$ PA

In my proof theory monograph there is this exercise: "The natural proof of PA cannot be carried out in PA. Why? (This proof consists in showing that all theorems of PA are ture.)" Apparently, by '...
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### Questions in proof theory (PRA-provability of EA-theorems, Girards book from '87)

I am working through the above mentioned book, 'proof theory and logical complexity, volume 1', with some trouble here and there. I would be glad if someone can help me with some of the exercises, ...
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### Questions in proof theory (interpretation of PRA in PA, Girards book from '87)

I am working through the above mentioned book, 'proof theory and logical complexity, volume 1', with some trouble here and there. I would be glad if someone can help me with some of the exercises, ...
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### Questions in proof theory (Definition of an interpretation of one theory in another, Girards Book from '87)

I am working through the above mentioned book, 'proof theory and logical complexity, volume 1', with some trouble here and there. I would be glad if someone can help me with some of the exercises, ...
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### Is induction something we take on faith?

I understand that in mathematics and logic we can continue to reduce things to simpler axioms, principles, and so on, and we have to "stop" at some point otherwise we're just going on forever. We ...
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### ZFC-Infinity+PA: Does it prove Con(PA)?

We define the theory ZFC-Infinity+PA as follows. We start with the axioms of ZFC-Infinity. Next we assert that there is a model of arithmetic $(\mathbb N, 0, S, +, \times)$. Next, for every axiom of (...
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### Resources like “How to solve it” by Polya

In How to Solve It, G. Polya describes methods of problem solving. I'm looking for more resources discussing the meta-level of how math is done.
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### Is there a countable computably saturated model of ZFC that correctly solves the halting problem?

We say that a model of ZFC $M$ correctly solves that halting problem if for every turing machine $T$, $T\text{ halts} \iff M \models T \text { halts}$. Is there a countable computably saturated model ...
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### Misleading formulation in “which area is greater?” question [closed]

Questions involving area comparison in geometric figures often ask "which area is greater?". See for example, Which area is larger, the blue area, or the white area? and Is the blue area greater than ...
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### 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 ...
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### 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 ...
### What does a proof that $\mathcal{N}$ is a model of $PA$ look like?
What does a proof that $\mathcal{N} \models PA$ (where $\mathcal{N}$ is the structure $($$\mathbb{N}$ $, 0, 1, S, +, \times, \leq)$, $\models$ is the satisfaction relation, and $PA$ is first-order ...