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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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Is there an ordering of logical systems defined by reductions?

I am aware of the lambda cube which gives an ordering to several variants of the lambda calculus. My intuition says that this ordering should have the following property: For logics $A,B\in\lambda\...
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Why do mathematicians generally write definitions in “declarative” rather than “imperative style?

In programming, we can make the distinction between declarative / functional and procedural / imperative programming. The distinction is not exact, but nevertheless meaningful. One major difference ...
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Definitions in metamathematics

In model theory, the satisfiability relation $ \vDash$ between a model $M= (D,f)$ and a set of formulas tells us when a formula $\varphi$ is true or not in the model ("interpretation") $M$. This ...
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Can any result on metric spaces be applied to metrizable spaces?

Can you give me a result (theorem, lemma, proposition) that doesn't hold on a metrizable space, but does on a metric space? This question is a bit vague, because you could easily say "A metric space ...
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Formalizing the deduction theorem in the metatheory

Here is the deduction theorem, in the "$\Leftrightarrow$" version (I'm considering it for first order logic): $$\Delta \cup \lbrace A \rbrace \vdash \lbrace B \rbrace \Longleftrightarrow \Delta \vdash ...
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Metamathematics and the foundations of mathematics

I have some really big doubts about what is the real starting point of all (formal) mathematics. For example: when I search on internet or study texts about the foundations of mathematics such as ...
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Is there a computable and complete “probabilistic” theory of arithmetic?

Let $\mathbb T$ be a probability distribution over complete and consistent theories of first order arithmetic that contain $PA$. Additionally, we will require that for any sentence $\phi$ in the ...
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62 views

Defining the complex numbers as the algebraic closure of the real numbers [closed]

It is of course possible to define the complex numbers as a quotient ring of real polynomials: https://math.stackexchange.com/a/1083130/359302. But is it possible to prove all of their properties -- ...
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Example of a $Δ_1$-formula that is not $Δ_0$ (arithmetical hierarchy).

In the arithmetical hierarchy, the class of $Δ_1$-formulas is defined as the intersection of $Σ_1$- and $Π_1$-formulas. It is obvious that every $Δ_0$-formula is $Δ_1$, but not every $Σ_1$- or every $...
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Can you recommend literature - easy/gentle/for self-study/introductory… - for the following topics…?

I am looking for literature that is as self-explanatory, easy, gentle, readable to the beginner, suitable for self-study, etc.. as possible, in the following fields. (I mean the mathematical part as ...
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Classification of Proof Techniques

A few introductory papers and Wikipedia contain incomplete lists of proof techniques, e.g.: direct by induction contraposition contradiction Are there classifications of proof techniques that ...
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85 views

Arithmetic systems without Induction

It's often said that AC is a controversial axiom and so often in my math classes when it is used a brief comment is made to the effect of "we can prove this without Zorn's Lemma but it's more work". ...
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Is there such a thing as ordering algebraic theories and, secondarily, their theorems?

Often, after learning a new definition, I find myself wondering what the "simplest" thing I can say now is, and the next "simplest" and so on. I do the same for structures as well. It seems like it ...
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Given two formulas $A$ and $B$, if $A$ follows from $B$ and $B$ follows from $A$ then is it true that $A$ and $B$ are equivalent?

This is true if $A$ and $B$ are statements, but formulas are statements too, so I expect the answer to be yes. But let's consider this simple example: Formula A: $$\sum_{k=0}^{n-1}x^k=\dfrac{1-x^n}{...
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Are results of relative consistency metatheorems?

Suppose that $S, T$ are two theories in the language of set theory, and suppose I prove - using relativization of concepts, for example - that $\operatorname{Con}(S) \rightarrow \operatorname{Con}(T)$....
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n-consistency - provability/truth of $\Sigma^0_n$ and $\Pi^0_{n+1}$ -formulas; n-consistent extensions, etc.

I am facing difficulties with the following exercise. (It is 1.5.9. from 'proof theory and logical complexity', Girard, '87) (i) T is $\textbf{n-consistent} \ (n>0)$ if any $\Sigma^0_n$ - theorem ...
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recursive inseparability of the two Gödelnumber-sets: theorems and 'anti-theorems' of EA

Here again one of my more or less basic proof-theoretic questions, working through Girards monograph from '87. This is about exercise 1.5.10. - "recursive inseparability", on page 80. It is this: ...
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Ordering between formal theories by provability of consistency

I am studying proof theory with Girard's monograph from '87 ('proof theory and logical complexity'). 1.5.6. is an exercise called 'ordering between theories'. It reads as follows: " (i) Let $\...
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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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105 views

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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67 views

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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62 views

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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62 views

Ambiguity in abbreviation $a<b$

I am studying mathematical logic and metamathematics and I have encountered formalization of number theory as a formal system. There, the following abbreviation is used: $a<b$ stands for $\exists t(...
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43 views

Truth values give unique answer

I wanted to ask the proof for uniqueness of answer given by truth tables. I am reading Kleene's "Introduction to Metamathematics" Chapter 6 Section 28 on evaluation and consistency. There he ...
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1answer
46 views

What is substitution simultaneously?

I have been reading Kleene "Introduction to Metamathematics" and found out that, even though he has been using a notion of "substituting simultaneously", he has never defined it. On page 78 he says "...
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1answer
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Converse of substitution for propositional letters

I would like to get some advice in understanding the following theorem found in Kleene's "Introduction to Metamathematics" chapter VI section 25. Theorem: Let $\Gamma$ be propositional letter ...
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3answers
60 views

Interpretation of generality introduction rule

I have been reading Kleene's "Introduction to Metamathematics" Chapter 5 Section 24 where it is stated that $A(x) \vdash \forall xA(x)$ is a deduction rule. I was wondering on the interpretation of ...
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Strongest axiomatic systems accepted as consistent by math community

As I understand it, most math is done implicitly within $ZFC$, but sometimes stronger systems are used--for instance, the initial proof of Fermat's Last Theorem used Grothendieck universes, which ...
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Are “Discovery Systems” still not viable in mathematics?

I am currently reading Why did AM run out steam?, an article regarding Douglas Lenat's Automated Mathematician (AM). AM is an early example (from 1976) of a "discovery system" - a system that attempts ...
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Beth's theorem through Robinson Lemma

I found this exercise (5.6.1) in "Introduction to model theory and to the metamathematics of algebra" of Abraham Robinson with and Can someone help me to solve it or, alternately, give me a ...
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60 views

Alternatives for Integers in fundamental mathematics?

I think this may be a very stupid question but here goes: I have a basic understanding of complex numbers and know that you can raise a number to a complex power, etc. But it seems to me that regular ...
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Can truth be defined by a 'same-level' language according to Tarski's undefinability theorem?

I have been learning about Tarski's undefinability theorem. My current understanding is that you need a 'meta-language' to define truth in a language (let's call this language 'A'). But could the same ...
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What do the Incompleteness theorems really say about the inexhaustibility of mathematics.

It seems that Godel himself believed that the incompleteness theorems seem to imply the inexhaustibility of mathematics; since he states you can simply add the consistency statement of the system as a ...
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Questions about the statement “Every number can be specified by less than twenty words.”

This is really an interesting question, though I do not know how to word it in a mathematical way. I am glad if one can help me to reword it mathematically. A friend of mine comes up with this ...
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How is this geometrical invariant called in English?

I am studying affine and projective geometry and I have encountered some invariant: the cross ratio, which in Italia is called "birapporto" and another one which I do not know the name of in English. ...
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Are Mathematicians Pluralists About Math?

This has been rangling around my head for awhile. With the death of Hilbert's program via Gödel's Incompleteness Theorems (and the prior damage done to Logicism via Russell's Paradox), have ...
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Metaphysical/psychological aspects of describing a formal language (mentioned in Bourbaki)

In the introduction to Bourbaki vol. 1, we read: "It goes without saying that the description of the formalized language is made in ordinary language, just as the rules of chess are. We do not ...
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Can second order peano arithmetic prove that first order peano arithmetic is sound? [closed]

Can second order peano arithmetic prove that first order peano arithmetic is sound? Note that I'm not just talking about its axioms, but also its theorems.
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Can two different models of arithmetic have non-comparable views of peano arithmetic?

For a given model of arithmetic $M$, we say that models view of peano arithmetic, $V(M)$, is $\{\phi : M \models (PA \vdash \phi) \}$. For example the view of the standard model is $\{\phi : PA \...
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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 ...
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1answer
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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 ...
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What does it means for a metatheory to be finitary?

In a finitary metatheory it is claimed that object variables of the formal language are generated by finitary methods. What does this finitary method mean? Also all the object variables of a formal ...