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%TCIDATA{Created=Mon Dec 07 18:54:39 2009}
%TCIDATA{LastRevised=Mon Dec 07 21:18:11 2009}
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\ \ \ \ \ \ \ \ \ \ \ \ \ \ \ \ \ \ \ \ \ \ \ \ \ \ \ \ \ \ Matematikai
logika zh.
\ \ \ \ \ \ \ \ \ \ \ \ \ \ \ \ \ \ \ \ \ \ \ \ \ \ \ \ \ \ \ \ \ \ \ \ \ \
2009.12. 08.
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1. Formaliz\'{a}lja a megadott $\mathcal{L}$ nyelven a k\"{o}vetkez\H{o}
defin\'{i}ci\'{o}t: \ Az adott $f(n,x)$ f\"{u}gv\'{e}nysorozat egyenletesen
konverg\'{a}l az adott $F(x)$ -hez a sz\'{a}megyenesen. ($\varepsilon >0$
-ra, $n>N$ -re $\left| f(n,x)-F(x)\right| <\varepsilon $). Sorolja fel \'{e}%
s jellemezze a felhaszn\'{a}lt nem-logikai konstansokat.
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2. a) Hat\'{a}rozza meg a k\"{o}vetkez\H{o} formula egy er\H{o}s Skolem form%
\'{a}j\'{a}t:
\qquad \qquad \qquad $\ \ \ \ \ \ \forall x(\forall yRxy\rightarrow \exists
y\rceil (Qxy\rightarrow Lxy))$
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b) Mutassa meg, hogy az ekvivalencia rel\'{a}ci\'{o} elm\'{e}lete (axi\'{o}m%
\'{a}k: reflexivit\'{a}s, szimmetria, tranzitivit\'{a}s) nem komplett.
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3. a) Fogalmazza meg \'{e}s bizony\'{i}tsa be a kompakts\'{a}gi t\'{e}telt
(szemantikai v\'{a}ltozat)
\ \ \ b) Fogalmazza meg az analitikus f\'{a}k teljess\'{e}gi t\'{e}tel\'{e}t.
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4. Mit \'{e}rt\"{u}nk
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a) elm\'{e}leten a szintaktik\'{a}ban?
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b) axiomatiz\'{a}lhat\'{o} elm\'{e}leten?
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c) egy strukt\'{u}ra elm\'{e}let\'{e}nek nem-standard modellj\'{e}n?
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5. Mely ir\'{a}nyokban igazak az al\'{a}bbi k\"{o}vetkeztet\'{e}sek? \'{A}ll%
\'{i}t\'{a}sait indokolja r\"{o}viden!
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a) $\Sigma $ egy elm\'{e}let a szemantikai \'{e}rtelemben \ $\Leftarrow
?\Rightarrow \;\Sigma $ egy elm\'{e}let a szintaktikai \'{e}rtelemben
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b) $\beta $ t\'{e}tele a Peano aritmetik\'{a}nak \ $\Leftarrow ?\Rightarrow
\;\beta $ igaz a term\'{e}szetes sz\'{a}mok szok\'{a}sos modellj\'{e}ben
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c) $\models \exists x(\alpha \rightarrow \beta )\,$\ $\Leftarrow
?\Rightarrow \;\models \exists x\alpha \rightarrow \exists x\beta $
\ \ \ ahol $\alpha $ \'{e}s $\beta $ tetsz\H{o}leges, de r\"{o}gz\'{i}tett
formul\'{a}k.
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6. Adjon p\'{e}ld\'{a}t
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a) komplett elm\'{e}letre
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b) nem els\H{o}rend\H{u} tulajdons\'{a}gra.
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