# Tag: Stars

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- What does {aaa, aab, baa}* equal to? November 6, 2024The answer should be formulated as a property of the lengths of consecutive blocks of identical letters. So, it can't be given using some algorithm or recursive methods. Verification of belonging to the language must be a linear condition. I can easily describe some properties of such words. But how do I come up with […]простой человек
- Let A, B, and C be sets. Show/prove (Not with venn diagram) that (A-B)-C = (A-C) - (B-C) November 5, 2024I tried to prove this by using logical equivalencies. I attached the picture of my proof. In my proof I stated that (A-B)-C is equal to A and NOT (A and (B or C)) and (A-C) - (B-C) is also equal to A and NOT (A and C) and NOT (B and (B OR C)) […]Samet Cimen
- The Proof of the Model Existence Lemma and Gödel's Incompleteness Theorem [duplicate] November 5, 2024Model Existence Lemma: given a consistent set of sentences $S$ in the language $L$, there is a model of $S$. I'm sure this is just a silly confusion. The usual proof of the Model Existence Lemma requires us to construct a complete, consistent, witnessing extension of $S$. If $S$ was to contain Peano Arithmetic $(\textbf{PA})$, […]Sam
- Confused about a possible asymmetry between $\land$ and $\lor$ in proof theory November 5, 2024Currently, I believe two things (please correct me if I'm wrong): $\def\f{\phi} \def\y{\psi} \def\LLL{\mathcal{L}} \def\S{\Sigma}$ given sentences $\f,\y$ in the language $\LLL$, and a theory $\S$, we have that $\S \vdash \f\land \y$ if and only if $\S\vdash \f$ and $\S\vdash \y$. Proof: for the $(\Rightarrow)$ direction, simply use the deduction rule $$\frac{\f\land\y}{\f}$$ or any […]Sam
- The difference between biconditional and equivalence [duplicate] November 5, 2024My question is simple is the Tautology $p\Leftrightarrow q$ and $a \equiv b $ the same ?bigbng
- Symbolising "Every team in league S has exactly one rival in league S" November 5, 2024Given Predicates: T(x, y): Team x plays in league y. R(x, y): Team x is a rival of team y. Note: R is symmetric: R(x, y) = R(y, x). Statement to Translate: Every team in league S has exactly one rival, that is also in league S. My Solution Attempt: $$\forall x \left( T(x, S) […]utterdisaster
- Why must $S$ be extended to a complete set of sentences to prove the Model Existence Theorem November 5, 2024Model Existence Theorem: let $S$ be a consistent set of sentences in the language $L$. Then $S$ has a model. The proof usually extends $S$ to a completely consistent set $T$ of sentences with witnesses. I understand why the extension $T$ needs to have witnesses: in order for statements of the form $\exists x\phi(x)$ to […]Sam
- "Spectrum" of a Heyting category November 5, 2024Fix a Heyting algebra $H$. By Stone duality for distributive lattices, we know that we can embed $H$ into the lattice of open sets of the space $\mathrm{Spec}(H)$ given by the set of all prime filters on $H$. Then, we also know that the co-Heyting algebra $H^{op}$ embeds into the lattice of closed sets of […]safsom
- Translating syllogisms [duplicate] November 5, 2024Are there tables of set-theoretic equivalences to syllogistic formulas? Or is there a standard way of translating a syllogism into the languages of first-order logic and set theory?inkd
- Can the standard model $M$ of arithmetic be described as that for which any $m\in M$ is of the form $s(\dots s(0)\cdots )^M$? November 5, 2024Can the standard model $M$ of arithmetic be defined (up to isomorphism) as that for which any element $m$ of $M$ is the semantic meaning of $$s(\cdots s(0)\cdots )$$ for a finite number of applications of $s$?Sam

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