# Questions tagged [metalogic]

For questions related to metalogic. It is the study of the metatheory of logic. Whereas logic studies how logical systems can be used to construct valid and sound arguments, metalogic studies the properties of logical systems. Logic concerns the truths that may be derived using a logical system; metalogic concerns the truths that may be derived about the languages and systems that are used to express truths.

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### Questions on metalogic and consistency

We know from Godel's theorems that interesting logics can not prove their own consistency, so whenever we want to prove the consistency of such a logic $L$, we need to take a step back, place this ...
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### Are soundness and completeness a part of proof theory, model theory or something else?

I have a question that I hope can clarify the scopes of model theory and proof theory. I have the following naïve understanding of the two areas (please correct me if I'm wrong): Model theory is ...
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### Logical consequence (⊨ and ⊢) — A statement? A judgement? A meta-statement?

I'm trying to understand the terminology used in mathematical logic and I'm confused about the distinctions between statement, proposition, judgement, claim, and meta-variants of those. In particular, ...
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### What types of axioms retain consistency of ZFC under additional axioms about its consistency?

My question is pretty simple. If ZFC does not prove that it is not consistent, then can we add the axiom to ZFC that it proves it is not consistent and is consistent and achieve equiconsistency? I ...
• 553
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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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### 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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### Valid form and true premises makes an argument sound, but do 'premises' mean P, Q, R,... or 'what comprises the antecedent'?

Suppose we have an argument 'Disjunctive Syllogism' as below: $$P\lor Q \\{\sim}P \\∴Q.$$ which essentially means $$\big((P\lor Q)\; \&\; {\sim} P\big) \to Q.$$ Its truth table: row P Q P$\lor$Q ~...
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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 ...
1 vote
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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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### Finitary reasoning and the distinction between mathematical and metamathematical theorems

Godel has argued that Skolem's finitism was responsible for his failure to prove completeness, despite having all the components of a proof. One can challenge this argument with the objection that ...
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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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