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I know, thanks to some clarifications received from a user of this site, the definition of a model. When evaluating the cardinality of a model by taking the interpretations of all the constants, functions and $k$-ary relations ($k\ge 0$) into account, what do we consider the interpretation of a $k$-ary relation and, in particular, the interpretation of a 0-ary relation, i.e. a proposition without any argument?

For example for the theory $\{R_1,R_2,\ldots,R_n,...\}$ read as $\{$it rained today, it rained yesterday, it rained the day before yesterday, it rained the day before the day before yesterday,...$\}$, is the cardinality of the considered model $|M|=|\{$it rained today, it rained yesterday, it rained the day before yesterday, it rained the day before the day before yesterday,...$\}|=\aleph_0$ or $|M|=|\{0,1\}|=2$ according to the truth values assigned $R_1,R_2,\ldots$? I thank you anybody for any clarification!

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In keeping with Mr. Karagila's answer to your previous question, the cardinality of your example model is 2, not $\aleph_0$.

In general, the interpretation of a $k$-ary relation can be seen as a subset of the cartesian power $M^k$. In particular, suppose that $R$ is a $k$-ary relation in the language $\mathcal L$ and that $M$ is the domain of some given $\mathcal L$-model. Then we can write $R^M\subseteq M^k$ for the interpretation of $R$ in that model, so that $R$ holds of a $k$-tuple $(m_1,\ldots,m_k)$ if and only if $(m_1,\ldots,m_k)\in R^M$. Using this notion to explain interpretation of relation symbols, a $0$-ary relation symbol should hold true only of the empty tuple.

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  • $\begingroup$ Perfectly understood. Thank you very much! $\endgroup$ Jan 31, 2015 at 22:03
  • $\begingroup$ I'm always weirded out when people write "Mr. Karagila" (it's even worse when you get emails with Dr. or Prof. Karagila, though). $\endgroup$
    – Asaf Karagila
    Jan 31, 2015 at 22:06
  • $\begingroup$ @AsafKaragila: I received an e-mail from a student who called me a dr. two months after my getting my master's degree. Better yet, a friend of mine had a misprint on some (semi-official) documents for his (master's) defense, due to which they said that he is already a professor. $\endgroup$
    – tomasz
    Feb 1, 2015 at 3:15
  • $\begingroup$ I have searched and compared several resources on the definition of universe, whose cardinality (cfr. math.stackexchange.com/questions/1127863/…) is used to define the cardinality of a model. Therefore, if our model "reads" $\{R_1,R_2,...\}$ as $\{$it rained today, it rained yesterday,...$\}$ with no elements of a universe involved in $k$-ary relations with $k\ge1$, I'm inclined to think that the cardinality of such a model would rather be 0... $\endgroup$ Feb 1, 2015 at 14:05
  • $\begingroup$ ...if I've correctly understood what a universe is. I had completely misunderstood what $M$ is in Asaf's answer believing that it coincides with the universe. I hope to be corrected if I'm wrong... I thank everybody! $\endgroup$ Feb 1, 2015 at 14:06

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