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What is the importance of first order language in logic?

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How do I prove $\vDash \ (\forall x \varphi \rightarrow \psi)\leftrightarrow \exists x (\varphi \rightarrow \psi)$ with $x$ not free in $\psi$?

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#logic #first-order-logic
585
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Are we in a nonstandard model of arithmetic?

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#model-theory #first-order-logic #nonstandard-models
80
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Find all submodels of $(\mathbb{N} ,+, \cdot , 0)$ that contain $2$ and $5$

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#logic #model-theory #first-order-logic
609
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How do I axiomatize algebraically closed fields?

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#abstract-algebra #logic #field-theory #first-order-logic
990
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Could the natural numbers be unique after all?

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#logic #set-theory #model-theory #first-order-logic #philosophy
50
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I am getting two answer for this 1st order logic question.

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#discrete-mathematics #predicate-logic #first-order-logic #quantifiers
1.4k
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Infinitely many axioms of ZFC vs. finitely many axioms of NBG

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#logic #soft-question #set-theory #first-order-logic #philosophy
67
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How to define a function in first order logic that return different values each time?

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#first-order-logic
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How to think about theories that prove their own inconsistency?

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#logic #soft-question #first-order-logic #philosophy
586
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How to correctly write arguments in Predicate Logic?

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#logic #first-order-logic #predicate-logic #logic-translation
4.7k
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Can someone please explain the difference between first/second/third order logic, using examples?

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#logic #predicate-logic #first-order-logic #higher-order-logic
93
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Problem with understanding proof of lemma 6.7 in Manin.

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#logic #first-order-logic
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Domains in First-order logic

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#elementary-set-theory #logic #first-order-logic
94
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Problem with understanding proof of Proposition 7.3 in Manin's 'A Course in Mathematical Logic'

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#logic #first-order-logic

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