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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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Assigning truth values to unpredicated statements

Note: any time a complete statement is represented using a letter, the letter will be underlined. Background (not required): In order for a single logical theory to encompass all of mathematics à la ...
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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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1answer
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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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1answer
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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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106 views

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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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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1answer
59 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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1answer
58 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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1answer
42 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
39 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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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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57 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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1answer
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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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1answer
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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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1answer
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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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1answer
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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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52 views

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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2answers
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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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63 views

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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1answer
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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
109 views

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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1answer
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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 ...
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Does specifying which variables depend on which other variables strengthen arithmetic?

For a set of universally quantified variables $\{a_1,\dots,a_n\}$, let $\exists_{a_1,\dots,a_n} x. s$ mean that there exists $x$ depending on $a_1,\dots,a_n$ (and no other universally quantified ...
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1answer
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Do models (in logic) contain elements?

The material I'm reading from is located at: https://pdfs.semanticscholar.org/e508/b945c5c95fb4ac5810a180536be3b6292743.pdf I'm confused how a model can contain elements.... I thought that a model ...
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1answer
52 views

Set theory that proves that if its consistient, is only proves true things about arithmetic [closed]

Is there a computable ω-consistent set theory $Q = ZFC + T$ (for some set of statements $T$), such that for every statement $s$ in the language of arithmetic, $Q$ proves $Con(Q) \implies (\mathbb N \...
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Where to put the dot at the end of a sentence when using cases-figure?

Often I use the cases-figure at the end of a sentence and I never know where to put the dot at the end of the sentence. I think, there are two options. Option 1: I can put the dot in the last case: ...
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How quickly can EFA define things, asymptotically?

EFA is theory of arithmetic. For a number $l$, we define $EB(l)$ as the largest $n$ such that there is a predicate $\phi$ with $l$ or less symbols such that $EFA \vdash \forall x. \phi(x) \iff x = n$,...
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Is there a weak set theory that can prove that the natural numbers is a model of PA?

$ZFC$ proves that $\mathbb N$ is a model of $PA$. Even $ZF$ does. How weak can go? In particular, is there some weak set theory that proves that $\mathbb N$ is a model of $PA$, but does not proof ...
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1answer
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Is the style of _Scott 1967_ outdated in discussing continuum hypothesis in a probability space?

Here is the article I am considering: Scott, D. Math. Systems Theory (1967) 1: 89(see here), an article named A proof of the independence of the continuum hypothesis. He shows that CH is not true in ...
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1answer
138 views

Exotic schemes of implications, examples

We are all familiar with schemes of implication like: $A\rightarrow B $ $A\iff B $ Or even more complex structures like a collection of three statements any two of which imply the third one. Are ...
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1answer
63 views

Maximum possible reputation? (NOT a meta question)

I had the following idea about the reputation system of MSE, that led to a math question: Suppose a certain user on MSE gains an average of $+200$ reputation per day - the daily maximum. Suppose ...
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3answers
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Should axioms be seen as “building blocks of definitions”?

This question is about the difference between a definition and an axiom. However, it does not address the following point: Whenever we define something, this is often written as a series of axioms. ...
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How to expand second-order ZFC to include classes?

The system of second-order $ZFC$, presented in Shapiro, "Foundations without Foundationalism", is formulated in second-order logic and includes the usual axioms of extensionality, foundation, pairs, ...
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3answers
105 views

Good resources on the intersection of probability theory and logic from a foundations/philosophical perspective?

What are some good books, courses, or online resources for probability theory that highlights differences between classical, frequentist, Bayesian, epistemic etc.? I majored in philosophy and am now ...