I’m trying to understand first-order-logic and have this simple question. Given the following predicates: $Thing(t)$, which states that $t$ is a thing; $Word(w)$, which states that $w$ is a word; and $HurtsYouMoreThan(x,y)$, which states that $x$ hurts you more than $y$, I need to create a first-order-logic statement that says “There is nothing that hurts […]

On the Wikipedia page for Hilbert style axioms, in the “Logical axioms” section, it gives the axioms to manipulate universal quantifiers : $Q5. \forall x(\phi)\rightarrow \phi[x:=t] $ $Q6. \forall x(\phi \rightarrow \psi)\rightarrow (\forall x(\phi) \rightarrow \forall x (\psi) )$ $Q7. \phi\rightarrow \forall x (\phi) $ where x is not free in $\phi$ and then says […]

Prove that no zero-order theory (i.e. propositional calculus, without quantification) or first-order theory can describe the “connected graph” (i.e. from any point one can reach each other point in finite steps). The only weapon I know in these situations is the compactness theorem: so I would like to prove that 1) such a theory is […]

I realize from the answer to this post that the fallacy in my “proof” of “ZF is inconsistent” was that I was not considering that there are models with non-standard integers. However now I think I developed an actual deduction of $T \vdash \text{Cons} T$ for any sufficiently powerful theory $T$ thus implying by Godel’s […]

Finite Ramsey theorem: $ \def\nn{\mathbb{N}} $ For any $e,k,r \in \nn$, there exists a least natural number $m=R(e,r,k)$ so that, for any set $M$ with cardinality at least $m$, with each of the $e$-sets of $M$ coloured with one of $r$ colours, there exists a subset $H$ of $M$ with cardinality $k$ so that all […]

In set theory, we have the following: Observation 0. Let $X$ denote a set. Let $A$ and $B$ denote subsets of $X$. Then if $A$ has at least one element, $B$ has at most one element, and $A \subseteq B$, then $B \subseteq A$. Rehashing this into logical language, we have: Observation 1. $$\frac{\exists a […]

In first order logic we often convert prenex normal form statements to Skolem normal form statements to eliminate the existential quantifier: $\exists$x$\forall$y$\exists$z$\phi$(x,y,z) becomes $\forall$x$\phi$(a,x,f(x)) Where a is a ‘Skolem constant’ and f is a ‘Skolem function’ Because we remove all the existential quantifiers, we can drop all quantifiers and consider all variables implicitly universally quantified […]

If I have a first-order theory $T$ with a constant symbol $c$ in its language, does this implicitely imply that I have to include the following axiom into $T$? $$\exists x[x=c]$$ More generally, for an n-ary function symbol $f$, do we have to include the following axiom? $$\forall y_1,…,y_n\exists x[f(y_1,…,y_n)=x]$$

Is there a first order theory $T$ in the language of rings such that its finite models are exactly the fields $\Bbb F_p$ with $p$ prime (but no $\Bbb F_q$ with $q$ a proper prime power is a model of $T$)? EDIT: Since this question turned out to be trivial, I asked if it is […]

What is the best and quickest way to find out if a set of formulas in first order logic is inconsistent? I really have no idea how to do that. As an example the $\forall x \exists y \forall z$ $ \phi$ is inconsistent with $\exists x \neg \exists y \exists z$ $ \phi$ but it is […]

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