Questions tagged [first-order-logic]

For questions about formal deduction of first-order logic formula or metamathematical properties of first-order logic.

First-order logic contains, in addition to all symbols and rules of , the universal $\forall$ and existential $\exists$ quantifiers. These satisfy $\neg\forall xP(x)\iff\exists x\neg P(x)$ and $\neg\exists xP(x)\iff\forall x\neg P(x)$. First-order logic is used to build up the axioms of most set theory formulations, including Zermelo–Fraenkel.

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Is there a 0-1 law for the theory of groups?

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}{\text{number of nonisomorphic groups of order} \le…
Dominik
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**Ended Competition:** What is the shortest proof of $\exists x \forall y (D(x) \to D(y)) $?

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 \forall y (D(x) \to D(y)) $ is also a theorem. I…
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Is $ \pi $ definable in $(\Bbb R,0,1,+,×, <,\exp) $?

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 interpretation of this language in $\Bbb R $ (where $ \exp $…
Dominik
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How do we know what natural numbers are?

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 numbers that we think of in everyday life? First…
M. Winter
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What is the purpose of free variables in first order logic?

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 bound and free variables? I am a little…
Guildenstern
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Axiom of Choice: What exactly is a choice, and when and why is it needed?

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 instance, take a look at following paragraph from…
J Hanson
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Why can't we formalize the lambda calculus in first order logic?

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 in first order logic, the lambda calculus cannot.…
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How does Gödel Completeness fail in second-order logic?

So a while ago I saw a proof of the Completeness Theorem, and the hard part of it (all logically valid formulae have a proof) went thusly: Take a theory $K$ as your base theory. Suppose $\varphi$ is logically valid but not a theorem. Then you can…
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Is $\Bbb R$ definable in $(\Bbb C,0,1,+,*,\exp)$?

Is there a first-order formula ϕ(x) with exactly one free variable $x$ in the language of fields together with the unary function symbol $\exp$ such that in the standard interpretation of this language in $\Bbb C$ (where $\exp$ is interpreted as the…
Dominik
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Do we have to prove how parentheses work in the Peano axioms?

One thing that has bothered me so far while learning about the Peano axioms is that the use of parentheses just comes out of nowhere and we automatically assume they are true in how they work. For example the proof that $(a+b)+c = a+(b+c)$. Given…
user537069
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Is the compactness theorem (from mathematical logic) equivalent to the Axiom of Choice?

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 $T$ has a model. So for any natural number $n,…
Mr X
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Predicate vs function

In logic, what is the difference between a predicate and a function? To be specific, I am just interested in First Order Logic. Thanks!
yoyostein
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With this definition of completeness, Gödel's Incompleteness result seems not surprising, so why it was back then?

According to wikipedia a theory (i.e. a set of sentences) is complete iff for every formula either it, or its negation, is provable. On the other side, a logic is complete iff "semantically valid" and "provable" are the same. The first notion of…
StefanH
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Does "every" first-order theory have a finitely axiomatizable conservative extension?

I've now asked this question on mathoverflow here. There's a famous theorem (due to Montague) that states that if $\sf ZFC$ is consistent then it cannot be finitely axiomatized. However $\sf NBG$ set theory is a conservative extension of $\sf ZFC$…
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Is the negation of a non-theorem a theorem?

I don't know if that is something obvious or if it is a dumb question. But it seems to be true. Consider the non-theorem $\forall x. x < 1$. Its negation is $\exists x. x \geq 1$ and is a theorem. Is this always true? I couldn't find a…
Rafael Castro
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