Questions tagged [logic]
Questions about mathematical logic, including model theory, proof theory, computability theory (a.k.a. recursion theory), and non-standard logics. Questions which merely seek to apply logical or formal reasoning to other areas of mathematics should not use this tag. Consider using one of the following tags as well, if they fit the question: (model-theory), (set-theory), (computability), and (proof-theory). This tag is not for logical puzzles, use (puzzle).
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Is the transfer principle useless?
In this MathOverflow thread, a comment states:
Any proof using a transfer principle can be rewritten without it, so in some sense it can't play an "essential" role in a proof.
Is there a known ...
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Is there a definition for stronger independence of a proposition concerning an axiomatic system.
I wonder if a definition like the following already exists.
If we have an axiom set $A,ドル can we define the independence of $\alpha$ in the situation where we are adding a new axiom $\alpha$ to $A$ ...
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Proof of completeness theorem in first-order logic
I am starting to program a proof assistant in Python. The proof assistant will be based in FOL. I might be interested in proving the soundness and completeness theorem for first-order logic in my ...
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Does truth not exist in ZFC, or is it merely not definable?
I am having a lot of trouble with the concept of Tarski's undefinability theorem as it relates to set theory.
Tarski's undefinability theorem says that there is no formula $Tr$ on the natural numbers ...
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Syntactic consequence implies semantic consequence [duplicate]
To give a slight background, I am not familiar with any theory of modern logic. I'd been reading Greenberg's book on Geometry where he claims the following.
Suppose we have a statement in the formal ...
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What counts as an argument, technically speaking?
A := It rains.
B := I should pick an unbrella.
It rains, therefore I should pick an umbrella.
p1) A
c) B
Consider the above premise and conclusion. Initially I'd ...
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Is there a complete proof system for deriving some axiom schema from others?
Say that an axiom schema is an algorithm $A$ that produces a family of first-order statements (I believe we can recursively enumerate all the algorithms that produce well-formed sentences). Given ...
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What is Gödel's argument for why his proof for a single system applies to all systems
I'm having trouble understanding how Gödel extrapolates from a consistent formal system to any formal system.
For reference, his First Incompleteness Theorem states:
Any consistent formal system $F$ ...
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Ordinal definability of a minimal witness in L(R, Hom)
Recall that for a transitive set $X,ドル $L(X)$ is class of sets constructible from $X\cup\{X\}$. For some reason I need to show that every element of $L_\alpha(X)$ is ordinal definable over $L_\alpha(X)$...
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Relation between Tarski's conception of truth and (implicit) Axioms
I am trying to understand the original paper of Tarski Concept of Truth in the formalized languages, as printed in his collected works.
I have read introductory texts from Shoenfield Mathematical ...
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What does $\forall x\exists x P(x)$ mean? [duplicate]
Is this even a valid formula syntactically?
Implies "for all $x$ there is an $x$" that the $\forall x$ is redundant, because obviously there is an $x$? Or is the variable in the new context (...
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The necessity of introducing CwF for the categorical semantics of Martin-Löf theories?
This question involves two simple subquestions regarding the intuition of necessity of introducing CwF (Category with families) as categorical semantics of Martin-Löf theories. (Note that I'm not ...
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Guessing the outcome of a coin toss with a probability greater than 0.5
I stumbled across an answer about The envelope paradox which states that:
Let Player 1 write two different numbers on two slips of paper. Then player 2 draws one of the two slips each with probability ...
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Arguments vs Material conditionals? [closed]
An argument has premises $p$ and one conclusion $c$. Can I write the argument as $p \to c,ドル in the form of Material conditional?
What are the differences between arguments and material conditionals?
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What is it we are actually doing when we assume a statement $A$ for the sake of argument?
A very simple question has been bugging me for a very long time now: what is it we are actually assuming when we assume $A$ during natural deduction proofs? (i.e. write "Suppose $A$...")
Or '...