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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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    What for do you need an infinite chains in Lindenbaum algebra?2012-05-22
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    To have better understanding of this object.2012-05-22
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    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.2012-05-22
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    @m.woj What for do you need an explanation of the reason?2012-05-22

1 Answers 1

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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.