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229 votes
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For each first order sentence $\phi$ in the language of groups, define : $$p_N(\phi)=\frac{\text{number of nonisomorphic groups $G$ of order} \le N\text{ such that } \phi \text{ is valid in } G}{\...
Dominik's user avatar
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142 votes
6 answers
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I'd heard of propositional logic for years, but until I came across this question, I'd never heard of predicate logic. Moreover, the fact that Introduction to Logic: Predicate Logic and Introduction ...
Alex Basson's user avatar
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50 votes
15 answers
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Why is this true? $\exists x\,\big(P(x) \rightarrow \forall y\:P(y)\big)$
Mats's user avatar
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44 votes
6 answers
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Suppose we have a line of people that starts with person #1 and goes for a (finite or infinite) number of people behind him/her, and this property holds for every person in the line: If everyone ...
Færd's user avatar
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40 votes
5 answers
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Do I get this right? Gödel's incompleteness theorem applies to first order logic as it applies to second order and any higher order logic. So there is essentially no way pinning down the natural ...
M. Winter's user avatar
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38 votes
6 answers
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Noson Yanofsky is a theoretical computer scientist at Brooklyn College. He presents the following argument on pages 329-330 of his book The Outer Limits of Reason, published by the MIT Press. The set ...
simple jack's user avatar
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38 votes
9 answers
3k views

The competition has ended 6 june 2014 22:00 GMT The winner is Bryan Well done ! When I was rereading the proof of the drinkers paradox (see Proof of Drinker paradox I realised that $\exists x \...
Willemien's user avatar
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35 votes
2 answers
1k views

Is there a first-order formula $\phi(x) $ with exactly one free variable $ x $ in the language of ordered fields together with the unary function symbol $ \exp $ such that in the standard ...
Dominik's user avatar
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34 votes
3 answers
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I'm having trouble understanding the necessity of the Axiom of Choice. Given a set of non-empty subsets, what is the necessity of a function that picks out one element from each of those subsets? For ...
J Hanson's user avatar
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31 votes
1 answer
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In propositional logic, there are truth tables. So you can check if the logical structure of your argument is, not correct per se, but if it's what you intended it to be. In predicate logic, I have ...
user2901512's user avatar
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30 votes
4 answers
11k views

I understand the difference between free and bound variables, but what are free variables actually useful for? Can't you use quantifiers to express everything that you would want to express with both ...
Guildenstern's user avatar
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30 votes
1 answer
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I am reading Peter Smith's An Introduction to Gödel's Theorems. In chapter 10, he defines "baby arithmetic" $\mathsf{BA}$ to be the zeroth-order version of Peano arithmetic ($\mathsf{PA}$) ...
WillG's user avatar
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30 votes
2 answers
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I'm reading through Hindley and Seldin's book about the lambda calculus and combinatory logic. In the book, the authors express that, though combinatory logic can be expressed as an equational theory ...
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29 votes
3 answers
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Or more importantly, is it independent of the axiom of choice. The compactness theorem states the given a set of sentences $T$ in a first order Language $L, T$ has a model iff every finite subset of $...
Mr X's user avatar
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