What is the easiest example of an infinite chain in a Lindenbaum algebra for the propositional calculus?
Does there exist an infinite antichain in a Lindenbaum algebra?
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$\begingroup$ What for do you need an infinite chains in Lindenbaum algebra? $\endgroup$– borgCommented May 22, 2012 at 11:21
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$\begingroup$ To have better understanding of this object. $\endgroup$– MarkNeuerCommented May 22, 2012 at 11:30
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3$\begingroup$ An infinite chain is $\{p_1,p_1\land p_2,\ldots,p_1\land\ldots\land p_n,\ldots\}$. An infinite antichain is $\{p_1,\lnot p_1\land p_2, \lnot p_1\land\lnot p_2\land p_3,\ldots\}$ where $p_i$ are the atoms. $\endgroup$– ApostolosCommented May 22, 2012 at 11:52
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$\begingroup$ @m.woj What for do you need an explanation of the reason? $\endgroup$– MJDCommented May 22, 2012 at 13:43
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1 Answer
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You need to have an infinite supply of variables, because the algebra for classical propositional logic in any finite number of variables is finite.
So say the variables are $\{A, B, C, \ldots \}$. Then there is an:
Infinite chain: $A \vdash A \lor B \vdash A \lor B \lor C \vdash \cdots $
Infinite antichain: $\{A, B, C, \ldots\}$.
Apostolos said as much in a comment while I was typing this, so I will make it community wiki.