# Tagged Questions

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### Has the Gödel sentence been explicitly produced?

I do not pretend to know much about mathematical logic. But my curiosity was piqued when I read Hofstadter's Gödel, Escher, Bach, which tries to explain the proof of Gödel's first incompleteness ...
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### The standard role of intuitive numbers in the foundations of mathematics

In my career I've been formed mostly in the formal side of mathematics, that is, standard set theory and every classical branch of mathematics that uses set theory. However, I am not pretty sure about ...
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### Can a non-classical logic be used as a meta-logic to develop classical logic?

I have read much about non-classical logics such that paraconsistent logics , relevance logics , substructural logics , non-monotonic logic and so on. I think that the meta-logic logicians use to ...
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### General view of Theorems

I'm trying to see almost all theorems ( at least the non-existential ones ) as affirming that some formula ( mostly of first-order logic language ) is a logical consequence of other formulas. So, ...
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### Are all theorems of minimal arithmetic theorems of a given theory?

I am working on some metamathematics revision and the following question came up. Let the theory $R_0$ be axiomatized by the following axiom schemata which hold for all $n,m \in \mathbb{N}$: ...
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### Use of propositional logic connectives in the meta-language

I have a doubt that might seem a bit confusing so i will try to explain it the clearer i can. Suppose we have an expression "A o B" in the meta-language, where 'o' refers to those logical ...
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### Satisfiability Proof Question

Exercise: Prove that $\Gamma\models A$ iff $\Gamma\cup\{\neg A\}$ is not satisfiable. Proof: We must prove two clauses: $\Gamma\models A\Rightarrow \Gamma\cup\{\neg A\}$ is not satisfiable ...
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### Is there an “end”? [closed]

This question may seem silly, but I nevertheless think that it is worth wondering over: is mathematics itself finite? As I understand, mathematics is a study of form and existence under constraints, ...
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### Do metatheoretic results carry between mutually interpretable theories?

If two theories A and B are mutually interpretable, in the sense of there existing a translation procedure from A to B and from B to A, does it follow that whatever metatheoretic results (e.g., ...
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### Is there a useful application of Peano arithmetic?

If there is, can someone provide an example of how Peano arithmetic can be used to solve a real-world problem? If not, can someone provide an example of any axiomatic system other than ZFC that can ...
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### Clarification of a remark of J. Steel on the independence of Goldbach from ZFC

On page 424 of the following paper: S. Feferman, Harvey M. Friedman, P. Maddy and John R. Steel, Does Mathematics Need New Axioms?'' The Bulletin of Symbolic Logic, Vol. 6, No. 4 (Dec., 2000), pp. ...
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I have a question about Godel numbering, it is trivial but I would like to know how can you know the length of an expression through its Godel number. ¿? I think you can use a recursive function but ...
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### A Simpler Characterization of Inductive Definitions?

While reading appendix A of John Harrison's "Handbook of Practical Logic and Automated Reasoning" a somewhat advanced theorem is appealed to as a prerequisite for characterizing when an inductive ...
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### Is Gödel's theorem invalid? [closed]

Right now I've skim through Gödel's theorem is invalid by Diego Saá on arXiv (freely available). As it seems very plausible, I ask for any references and scrutinizations of the paper.
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### Are there formal systems that can not be proved to be complete or incomplete?

I'm reading GEB and was thinking about this. Are there any formal systems where the proof of their completeness/incompleteness is unprovable? This question could go ad infinitum. Could there be any ...
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### If $T$ proves $\operatorname{Con}(ZFC)$, is $T$ at least as strong as set theory?

I am looking for either a proof of counterexample of this: Lemma: Let $\pi$ be a faithful interpretation of $PA$ into $ZFC$, and let $PA'$ be the image of $PA$ under $\pi$. If there is a $T$ with ...
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### An “internal” condition on $T$ so that for the standard provability predicate, $T$ proves $\text{Pf}(\underline S)$ implies $T$ proves $S$?

This is probably quite basic, but I'd like to make sure I got this right. Regarding the proof of Goedel's first incompleteness theorem, say that we have $T$ containing $PA$ effectively axiomatizable ...
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### A theorem about inductive inference

In the book 'Introduction of the theory of Statistics' by Mood,Graybill,Boes (third edition)on page 220 (Chapter 6 on Sampling) you can read: 'Inductive inference is well known to be a hazardous ...
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### Definition of “non-constructive proof”

I was wondering if it is possible to define exactly what a non-constructive (nc) proof is. I have often seen the concept associated with the use of principles such as the axiom of choice or the law of ...
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### Truth and undecidability

I believe this is more of a philosophical question. Given a consistent theory T and a statement S independent of T. Can S be true or false in T? (I don't see any contradiction with that) I read that ...
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### Formalizing metamathematics

I am reading historical/philosophical stuff on the concept of "metamathematics" and am by now quite confused. Several questions emerged, but they are probably somehow confused and interrelated, I ...
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### Statements about logic (=“metalogic”(??)) [closed]

Sometimes there are statements about logic e.g. "That's not logical" and I can neither prove nor disprove a statement about logic with no definition for logic itself. It's just a negation and it's ...