# Tagged Questions

In the philosophy of mathematics, intuitionism, or neointuitionism (opposed to preintuitionism), is an approach where mathematics is considered to be purely the result of the constructive mental activity of humans. That is, given a set of axioms, the only propositions that can exist are those that ...

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### Is there a simple example of how the law of the excluded middle can be inapplicable?

Why does a logic system not use the law of the excluded middle? I studied non-classical logic (intuitionistic and modal) where double negation can't be removed and the law of excluded middle can't be ...
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### Compound interest coumpounded n time per year formula. $A=P\left(1+\frac{r}{n}\right)^{nt}$ intuition behind it.

I know that the compound interest formula for the interest compounded annually is given by $$A=P(1+r)^t$$ I know the intuition behind it. But why the compound interest formula for the interest ...
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### Is intuitionistic naive set theory consistent?

I'm asking because the usual argument that a set either belongs in itself or not doesn't apply. I did a quick search and it appears that the logic is also required to be contraction free. If it's ...
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### How is the double negation translation similar to CPS in functional programming languages?

In Wikipedia's Double-negation translation article, I found that any formula in classical logic has its double negation as its intuitionist equivalent: It is also possible to define φN by ...
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### Contraposition in intuitionistic logic?

I read that contraposition $\neg Q \rightarrow \neg P$ in intuitionistic logic is not generally equivalent to $P \rightarrow Q$. If this is right, in what case can this contraposition ...
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### $(\forall x,\, p\vee q(x))\leftrightarrow p\vee\forall x,\, q(x)$

Consider the logical formula $(\forall x,\, p\vee q(x))\leftrightarrow p\vee\forall x,\, q(x)$ where x does not appear free in p. This formula is not derivable in intuitionistic logic, but it is in ...
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### Equivalence between $\neg\neg\bot$ and $\bot$ in intuitionistic logic.

As the title says, why are those two equivalent? I can find a simple derivation (using natural deduction) of $\bot$ from $\neg\neg\bot$, but i fail at proving the other implication.
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### Example of an interesting theorem that fails in intuitionistic set theory but is classically valid?

I'm interested in intuitionistic set theories at the moment. I know that lots of principles imply LEM and so fail intuitionistically, and also a few basic principles - linear ordering of ordinals, for ...
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### Goedel's completeness theorem in and/or for intuitionistic first order logic

Warning: I am neither a logician nor a set theorist, just curious about foundations of classical and intuitionistic mathematics. Therefore it might well be that the things to come are plain wrong, and ...
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### What is the purpose of defining the negation of a proposition A as A $\rightarrow \bot$?

I know that this definition is unrelated to the law of the excluded middle, but as a beginner in logic (I've studied the first half of Chiswell and Hodges' Mathematical Logic), the use of the name ...
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### How is Goedel's 1st incompleteness theorem related to the Axioms of a theory

i am thinking of various connections and formulations of Goedel's 1st incompleteness theorem. Apart from connections to Turing's Halting Problem and Algorithmic Complexity Theory, i am looking for ...
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### Axiom of Choice - Type Theory (Proof)

Background In Intuitionistic Type Theory (p. 27-28), Martin Löf provides a proof of the axiom of choice that is constructively valid. This version is considerably weaker than the ordinary set theory ...
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### Minimal difference between classical and intuitionistic sequent calculus

Consider propositional logic with primitive connectives $\{{\to},{\land},{\lor},{\bot}\}$. We view $\neg \varphi$ as an abbreviation of $\varphi\to\bot$ and $\varphi\leftrightarrow\psi$ as an ...
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### Axioms based on $\leftrightarrow, \lor, \bot$ for propositional intuitionistic logic?

Propositional intuitionistic logic can be axiomatized based on $\;\to, \land, \lor, \bot\;$, with modus ponens $$\text{from }\; \phi \;\text{ and }\; \phi \to \psi \;\text{ infer }\; \psi$$ as the ...
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### How can some statements be consistent with intuitionistic logic but not classical logic, when intuitionistic logic proves not not LEM?

I've heard that some axioms, such as "all functions are continuous" or "all functions are computable", are compatible with intuitionistic type theories but not their classical equivalents. But if they ...