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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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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Why do we have different sets of axioms? (metamathematics reference request)
For example, ZFC and ZF.
I have come across the notion of pure and applied mathematics, and how the development of the former can (and is usually intended to) lead to the furtherance of the latter. In ...
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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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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 ...
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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 ...
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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 ...
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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 ...
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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\...
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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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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 $\...
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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
...
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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 ...
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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 ...
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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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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 ...
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Demonstrate a nondenumerable set of consistent extensions of $\mathbf{Q}$ that are pairwise inconsistent.
Given that for any $n \in \mathbb{N}$, there are $2^n$ consistent, axiomatizable extensions of $\mathbf{Q}$ that are pairwise inconsistent, show that there is a nondenumerable set of such consistent ...
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Show a model of $\mathbf{Q}$ where this is false $\forall x \forall y (x \cdot y = y \cdot x) $
Show that there are models of $\mathbf{Q}$ where the following sentence is false
\begin{align*}
\forall x \forall y (x \cdot y = y \cdot x) \\
\end{align*}
For reference, the ten finite axioms of $\...
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Computability and Logic by Boolos et all Problem 17.1
Show that the existence of a semirecursive set that is not recursive implies that any consistent, axiomatizable extension of $\mathbf{Q}$ fails to prove some correct $\forall$-rudimentary sentence.
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Isn't Math basically a matter of combinatorics?
As I discover the foundations of mathematics, I begin to understand that it is a matter of arbitrarily defining axioms and combining them - arriving at what we call theorems.
Having said that, it ...
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Is the universal quantifier redundant?
Whenever we use the string
$(\forall x)P(x)$
We are using a meta variable, in this case $x$, which stands for any object in the reference set.
However, the semantics of the symbol $\forall$ indicate ...
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