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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 ...
Ben123's user avatar
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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 ($...
ac15's user avatar
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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\...
M. Sperling's user avatar
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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 ...
space_kale's user avatar
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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\...
RataMágica's user avatar
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2 answers
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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,...
Feri's user avatar
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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 ...
Inquisitor's user avatar
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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 ...
user700974's user avatar
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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 ...
Vivek Joshy's user avatar
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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 ...
btshepard's user avatar
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3 answers
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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 ...
Laska's user avatar
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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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Do Tarski's axioms apply to higher dimensions?

I came across the wonderful fact that the theory of Euclidean geometry in 2 dimensions is complete, consistent and decidable — as shown by Tarski's axiomatization. I know very little about this. My ...
Alex's user avatar
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What subset of mathematics can be done using only formulae with no more than one occurence of each variable?

When I was a kid learning algebra for the first time, I remember finding it unintuitive that each instance of a given variable had to refer to the same thing. "Why can't I set the first ...
M. Sperling's user avatar
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Can ZFC be proven from weak systems using consistency of those systems?

Tl;dr Can we take a weak system $A_0$ then show $$A_0 + Con(A_0)\implies Con(A_1)), \space A_1 + Con(A_1)\implies Con(A_2)), \space A_2 + Con(A_2) \implies \dots$$ terminating in ZFC? My understanding ...
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How does one prove that constructive type theory is isomorphism-invariant?

In his paper Structuralism, Invariance and Univalence (pdf), Steve Awodey makes the following claim about constructive type theory: The system of type theory has the important property that any ...
Penelope Clairmont's user avatar
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Why is it specified in the 2nd incompleteness theorem that a system of arithmetic cannot prove its own consistency?

I have seen in places I have read about Godel's incompleteness theorem that the second incompleteness theorem can be summarized as saying: No axiomatic system with sufficiently strong arithmetic can ...
Nikolas Koutroulakis's user avatar
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Is this an example of an unprovably true statement in set theory? (A Corollary to the axiom schema of separation)

In ZFC, the axiom of separation is given as follows. Let $\phi(x, w_1,\ldots, w_n)$ be a wff formula in FOL, for free variables $w_1,\ldots,w_n$. Then, $$\forall w_1,\ldots,w_n\forall A \exists B\...
Nikolas Koutroulakis's user avatar
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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 ...
Iain's user avatar
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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 ...
Mircea Baja's user avatar
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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 ...
Dave The Minion's user avatar
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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 ...
BENG's user avatar
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Do there exist "genuinely maximal" theories $\text{Th}(\mathcal M)$?

Given a language $\mathcal L$ and an $\mathcal L$-structure $\mathcal M$, the theory of $\text{Th}(\mathcal M)$ is defined to be the set of $\mathcal L$-sentences $\phi$ for which $\mathcal M \vDash \...
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Confusions regarding the metatheory

Kunen's Set Theory (2011) raised some uncomfortable questions regarding my understanding of the metatheory. Let me present my questions in in terms of two perspectives, in both of which we use $\...
God bless's user avatar
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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, ...
Ruben Verborgh's user avatar
3 votes
2 answers
176 views

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 ...
Promethèus's user avatar
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1 answer
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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 ...
Cyrus's user avatar
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1 answer
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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 (...
Kang Won Lee's user avatar
5 votes
3 answers
299 views

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. ...
Elliott's user avatar
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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 ...
user avatar
4 votes
1 answer
215 views

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 ...
Matteo Casarosa's user avatar
1 vote
1 answer
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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 ...
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1 vote
0 answers
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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 ...
P. Grewe's user avatar
2 votes
1 answer
99 views

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 ...
subrosar's user avatar
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2 answers
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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 &...
blahblah's user avatar
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4 votes
1 answer
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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 ...
Davi Barreira's user avatar
3 votes
1 answer
177 views

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 ...
Alexandre Tourinho's user avatar
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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 ...
God bless's user avatar
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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 ...
comsenwar's user avatar
1 vote
1 answer
161 views

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. ...
neph's user avatar
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1 answer
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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 ...
Zachiah's user avatar
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4 votes
1 answer
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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/...
jgrk's user avatar
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How to prove the consistency of a collection of axioms?

Is there a way to prove the consistency of some chosen axioms? In the two senses following: In each mathematical logic book, there is a special kind of deduction system, which include some logical ...
Michael's user avatar
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