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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Numbers/cardinal numbers [closed]

so i had two classes this semester abstract algebra and philosophy of math. I enjoyed them both but there is one thing that i have been trying to understand all the semester and just can't get ...
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How do I view latex source code in answers or questions as rendered form or images? [migrated]

Is it normal to see only the latex source code in questions and answers where they are typed? My perspective: https://i.gyazo.com/1682f307c1ef8eb53f95ed1868afe3fe.png Is there any way to render this ...
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51 views

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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41 views

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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66 views

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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1answer
32 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 ...
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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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135 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 ...
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76 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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50 views

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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86 views

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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Establishing First Order Logic and basic results with PRA

I am just beginning my study of mathematical logic (I’ve worked through the first 7 chapters of Kleene’s Introduction to Metamathematics) and like many others who are studying FOL for the first time, ...
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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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131 views

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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139 views

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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95 views

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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125 views

Bourbaki: Quantification Expansion

I'm reading Theory of Sets where existential quantification is defined: a) $$ \exists x R = (\tau_x(R)|x)R $$ and I have questions regarding the expansion for an example relation: b) $$ \exists y \...
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345 views

In what sense do isomorphisms "preserve all logical properties"?

My background in mathematical logic, model theory, etc. is patchy, so I'm looking for a clearer way to think about this. (Edit: I'm not asking what an isomorphism is, I'm asking how to formalize the ...
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Why do we not let the disjoint union function union joint sets?

The disjoint union function is strictly defined as a binary operator that unions two disjoint sets with the strict expectation of both sets not equaling one another (well they are disjoint, so ...
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1answer
123 views

How meta is today's metamathematics? [closed]

I've recently been interested in the study of metamathematics, as mathematical theories about mathematical theories has quite the charm of novelty to it from the perspective of an undergrad. I recall ...
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Is it possible to list all hidden lemmas of a proof?

I'm studying Imre Lakatos' Proofs and Refutations for my master's thesis. Currently I address the concept of hidden lemmas, which I understand to mean unstated assumptions of a mathematical proof, ...
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Substitution of Atomic Sentences in Propositional Logic [duplicate]

Kleene states Theorem 1. on p. 14 that if we have a formula $E$ of atomic variables $p_1,p_2,\cdots,p_n$, then we may substitute in formulas $A_1$ for $p_1$, $A_2$ for $p_2$ etc... to get $E^*$ such ...
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100 views

Rank-Induction Principle (following Van Dalen's Logic and Structure, 4th edition)

This post contains questions on understanding Van Dalen's proof of the Rank-Induction principle and questions concerning its wording and presentation. Please don't feel obliged to answer everything. ...
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101 views

If solutions to an expression are just another expression, why are they considered solutions?

Questions in mathematics are posed as: Solve for y = yadda yadda yadda The yadda yadda yadda is some expression involving variables and integrals and so forth. “...
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54 views

Can the existence of some positive integers satisfying some equations ever be independent of ZFC?

In a subject like number theory there are often open questions along the lines of 'Are there positive integers $a_1 , \ldots , a_n $ satisfying the following (finite number of) equations'. For example ...
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124 views

Course-of-values induction, according to Kleene IM (1952)

I'm having troubles with exercise *162a in "Introduction to Metamathematics", by S.C. Kleene, 1952. The ask is to prove: (1) $ \vdash A(0) \& \forall x [\forall y (y \leq x \Rightarrow A(...
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What is the intuitive reason why there is no recursively related notation-system which gives a name to every constructive ordinal?

The Question: I found this claim in Douglas Hofstadter's book 'Gödel, Escher, Bach'. The author says that this result was proven by Church and Kleene. Correct me if I am wrong, but I interpret this ...
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156 views

In what formal system is Godel's Incompleteness Theorem (or similar statements of undecidability) proven?

For example, consider the proof using Rosser's trick as shown on wikipedia. https://en.wikipedia.org/wiki/Rosser%27s_trick#The_Rosser_sentence That proof isn't inside the arithmetical theory T, but in ...
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Why is everything geometrical modeled on $\Bbb R$?

The reals naturally arise when discussing limiting processes of rational numbers like trying to compute roots or $\pi$. Similarly I have read that axioms of Euclidean geometry (I mean those not ...
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1answer
128 views

What is the difference inbetween $\leftrightarrow$, $\iff$, and $\equiv$? [duplicate]

What is the difference inbetween $\leftrightarrow$, $\iff$, and $\equiv$ ? Especially when used in the same text. My thought about that was that it has something to do with object language and meta ...
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91 views

A question about undecidable sentences with purely mathematical contents

I'm struggling with the idea that the continuum hypothesis does indeed have a purely mathemathical/set theoretical meaning, but is neither provable nor disprovable in ZFC (according to Gödel and Cohen)...
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65 views

A question on the Substitution for predicate letters theorem as in Kleene Introduction to Metamathematics (1952)

I'm self studying Kleene's IM (1952). On page 149 Kleene defines a name form thus: "Let $x_1, ... x_n$ be distinct variables, and $A(x_1, ... x_n)$ a formula. When we are interpreting $A(x_1, ... ...
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Questions about foundations of mathematics

It seems to me that in trying to create a foundation of mathematics, mathematicians are trying to create a formal system that models the language used in what I view as common-sense, real mathematics. ...
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Axioms and Definitions [duplicate]

After a few years of university mathematics I just realised that I do not really know the difference between an axiom and a definition. With some "research" on the interwebs (mathexchange, ...
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Proof verification- Am not comfortable enough with the concept of absoluteness to know if my proofs are correct.

Let $M$ be a transitive proper class model of $ZFC$. I want to prove a few absoluteness results: $(\omega^\omega)^M$ = $\omega ^\omega \cap M$ By the way when this happens do we say $\omega^\omega$ ...
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102 views

Can it be shown that there exists no finite proof of CH from second-order set theory?

As is well known, all models of (full) second order set theory (e.g., ZFC2) are quasi-isomorphic. This implies (or at any rate: has been taken to imply) that CH is "decided" by second-order ...
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84 views

Why doesn't this show that first-order Peano arithmetic is consistent?

SOME PRELIMINARIES: Predicate logic is consistent and complete. In other words, (i) for a closed formula $F$ in predicate calculus with equality and functions, $\vdash F$ if and only if $\,\vDash F$ (...
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Existence Introduction on Kleene's Introduction to Metamathematics

Existence introduction in Kleene's IM is stated as $A(t) \vdash \exists x A(x)$. The way the notation $A(t)$ in introduced by Kleene (page 78), make me think that all occurrences (instances) of t in $...
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133 views

Physics and Riemann hypothesis [closed]

I was reading the article "Quantum physics sheds light on Riemann hypothesis" from Bristol University (http://www.bristol.ac.uk/maths/research/highlights/riemann-hypothesis/) and stopped ...
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1answer
62 views

Error in Kleene "Introduction to Metamathematics" parentheses lemmas?

In the classic book, Introduction to Metamathematics by Steven Kleene, Lemma 2 of Section 7 (chapter 2), seems to me to be false. I am wondering if I am missing something. Here is the context: The ...
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166 views

What does it really mean for a model to be pointwise definable?

(Note: I'm only an amateur in logic, so I'm sorry for any weird terminology or notation, or excessive tedious details. Most of what I know is from Kunen's Foundations of Mathematics.) I'm trying to ...
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1answer
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Introduction to Metamathematics by S C Kleene. Help with exercise *135b needed.

I'm having trouble with exercise *135b in Introduction to Metamathematics by S. C. Kleene. The ask is to show that: $\vdash 0<a^{'}$. Here is how I would do it. Assume $a=b$. With Axiom 17 and ...
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Simultaneous / blended develop logic and set theory? [duplicate]

My goal right now is to gain a deep understanding of how to talk about mathematical objects formally. The presentation of how to do this in most books is generally to "assume some basic set ...

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