# 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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### Validity of alternate interpretations of mathematical definitions?

There are two ways of viewing definitions of mathematical properties that I am comfortable with. By property, I mean a statement about one or more sets, that forms a proposition when the subjects are ...
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### 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 ...
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### What is meant by the term „set“ when talking about models of ZFC?

A model $\langle M,R\rangle$ of ZFC is a set $M$ together with a binary relation $R$ on $M$. My question is: what exactly do we mean by saying $M$ is a set, since it somehow comes from „outside“ the ...
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### Other important results of class/set distinction?

Most explanations of classes vs sets motivate the discussion via Russell's paradox. This feels like a kind of "gotcha technicality" to me, in the sense that one might reasonably prove ...
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### Arguments for Primitive Recursive Arithmetic + \epsilon_0 Induction being "True"

Gentzen presented a proof of the consistency of PA. This proof can be formalized in PRA (Primitive Recursive Arithmetic) + "Transfinite Induction up to $\epsilon_0$". In order to accept ...
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### Origin of quote “In order to simplify…”

I am looking for the origin of a quote found in a mathematics textbook and attributed to Mandelbrot. I'm convinced that a math book we used in high school had this quote in it: In order to simplify, ...
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### Are there cases where a flawed proof seems correct?

A mathematical proof is known to be wrong when either of the following is found: a flaw in the logic (including perhaps unwarranted assumptions); or a counterexample. Wikipedia has an extensive list ...
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### How does formalization work in mathematics?

I would be extremely grateful is someone could review/comment/complement my reasoning and understanding of formalization in mathematics. Let $T$ be a mathematical theory, say real analysis. $T$ is ...
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### Kleene: sets of parentheses and proper parentheses

The following is from Introduction to Metamathematics by Steven Kleene. In Lemma 2 of section 7 of chapter 2, it seems ambiguous what constitutes a “set of parentheses.” Here are some relevant ...
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### Is proving metatheorems circular logic? [duplicate]

I am currently learning mathematical logic, and I came across a dilemma. In proving metatheorems (theorems about formal systems), almost all the proofs for said metatheorems used mathematics (...
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### Are there infinitely many metalogics?

Given the definitions of material implication, logical implication, and what a tautology is, we can prove: \mathcal B\text{ logically implies }\mathcal C\text{ if and only if }(\mathcal B\rightarrow\...
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### Difference between implies and "turnstile" symbols (→ and ⊢) [duplicate]

According to Wikipedia's list of logic symbols: A → B means A → B is false when A is true and B is false but true otherwise. ...
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### Is there a mathematical system that is: complete, consistent, and decidable?

I know very little about modal logic (only some set theory) in mathematics, but I am aware that there exists a completeness theorem, incompleteness theorem, and the axiom choice, and that maths is not ...
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### What is the link between interpretability hierarchy and consistency strength

I am trying to understand this definition https://plato.stanford.edu/entries/independence-large-cardinals/#IntHie of Interpretability Hierarchy and how it relates to the concept of Consistency ...
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### A question about the seventh formative criteria in Bourbaki's Theory of Sets

CF7. Let $\boldsymbol{A}$ be a relation (resp. a term) in a theory $\mathcal{T}$, and let $\boldsymbol{x}$ and $\boldsymbol{y}$ be letters. Then $(\boldsymbol{y}|\boldsymbol{x})\boldsymbol{A}$ is a ...
1 vote
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### Are the naturals really a subset of the real numbers? [duplicate]

Ok, so this question seems obvious, right? But what I mean is the numbers in the way they are logically / axiomaticaly defined in the foundations of Mathematics. As far as I know, the naturals are in ...
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### If a metatheory proves consistency of a theory, will consistency of the metatheory suffice?

I'm going to call a metatheory reasonable if it's consistent and whenever there exists a proof for a sentence $\phi$ from a theory $T$ the metatheory proves $T\vdash \phi.$ Suppose that some ...
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### why Gödel use $[R(n);n]$ in his introduction?

in "on formally undecidable propositions of principia mathematica and related systems"'s introduction, Gödel used the notation $R(n)$ and $[R(n); n]$ to state the unprovable formula meaning &...
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### Meta-Definition of Convergence

So, just recently I realized that the idea of convergence is not "all encompassing"... Let me explain. I thought that the topological definition of convergence was the most basic one in the ...
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### Why isn't math proofs just a computer trial and error?

I already asked a similar question, but I recently began a course of Logic and it gave me not an answeat but a refination of my question, which I redefine here. My thinking is the following: Suppose ...
1 vote
122 views

### What formal logic has the smallest metatheory?

I'm currently studying A Concise Introduction to Mathematical Logic (Third Edition) by Wolfgang Rautenberg. What sparked my interest in logic is my interest in foundations in general, so I was ...
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### Definition by generalized recursion, Peter Hinman's Foundations of mathematical logic

I'm reading the Foundations of mathematical logic by Peter Hinman, and something is unclear to me in his applications of theorem 1.2.14. Here's the theorem: Let $\mathcal{X}=(X,X_0, \mathcal{H})$ be ...
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### how universal is Conway's game of life? is it reasonable to expect that a technological alien civilization would recognize, say, a glider?

This is a philosophical one, so apologies if it's not appropriate. I can think of several reasons that Conway's Game of Life would be rediscovered by any mathematically inclined biological life forms. ...
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### Can metalogic and model theory be formalized?

All of mathematics formulated using ZFC can be "formalized" in the sense that each statement could be translated into a logical string, and each proof can be translated into a formal proof. ...
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### 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 ...
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### Is there a category (or rather a mathematical theory) for which we know a lot about, but not whether its object class is empty or not?

this is a bit of a vague question so let me describe a bit what motivates it: Yesterday I was reading the Wikipedia article about perfect numbers, where I find the section https://en.wikipedia.org/...
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