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I know that it is provable that the free boolean algebra on countably many generators is incomplete. For the sake of concreteness, let's call the generators $p_1, p_2, p_3,...$ and refer to them as "basic formulas". I have been looking for a concrete example of a subset which lacks either a least upper bound or a greatest lower bound. In fact, I almost gave up; I've been trying to do this for months.

However, I stumbled upon this page: http://thue.stanford.edu/bool.html

The author claims that in the free boolean algebra on countably many generators (FBACMG), "any set $X$ of variables has a least upper bound if and only if the set is finite."

This seems wrong to me: consider the set of basic formulas. If I am not mistaken, it has no upper bounds except the tautology, and thus the tautology must be its least upper bound.

Am I wrong, or is the site wrong? And if the site is wrong, is it possible to describe a subset of the FBACMG which provably has no least upper bound or no greatest lower bound? If not can I prove that it is impossible to describe such a set?

I know that it is provable that the free boolean algebra on countably many generators is incomplete. For the sake of concreteness, let's call the generators $p_1, p_2, p_3,...$ and refer to them as "basic formulas". I have been looking for a concrete example of a subset which lacks either a least upper bound or a greatest lower bound. In fact, I almost gave up; I've been trying to do this for months.

However, I stumbled upon this page: http://thue.stanford.edu/bool.html

The author claims that in the free boolean algebra on countably many generators (FBACMG), "any set $X$ of variables has a least upper bound if and only if the set is finite."

This seems wrong to me: consider the set of basic formulas. If I am not mistaken, it has no upper bounds except the tautology, and thus the tautology must be its least upper bound.

Am I wrong, or is the site wrong? And if the site is wrong, is it possible to describe a subset of the FBACMG which provably has no least upper bound or no greatest lower bound? If not can I prove that it is impossible to describe such a set?

I know that it is provable that the boolean algebra on countably many generators is incomplete. For the sake of concreteness, let's call the generators $p_1, p_2, p_3,...$ and refer to them as "basic formulas". I have been looking for a concrete example of a subset which lacks either a least upper bound or a greatest lower bound. In fact, I almost gave up; I've been trying to do this for months.

However, I stumbled upon this page: http://thue.stanford.edu/bool.html

The author claims that in the boolean algebra on countably many generators (BACMG), "any set $X$ of variables has a least upper bound if and only if the set is finite."

This seems wrong to me: consider the set of basic formulas. If I am not mistaken, it has no upper bounds except the tautology, and thus the tautology must be its least upper bound.

Am I wrong, or is the site wrong? And if the site is wrong, is it possible to describe a subset of the BACMG which provably has no least upper bound or no greatest lower bound? If not can I prove that it is impossible to describe such a set?

I know that it is provable that the free boolean algebra on countably many generators is incomplete. For the sake of concreteness, let's call the generators $p_1, p_2, p_3,...$ and refer to them as "basic formulas". I have been looking for a concrete example of a subset which lacks either a least upper bound or a greatest lower bound. In fact, I almost gave up; I've been trying to do this for months.

However, I stumbled upon this page: http://thue.stanford.edu/bool.html

The author claims that in the free boolean algebra on countably many generators (FBACMG), "any set $X$ of variables has a least upper bound if and only if the set is finite."

This seems wrong to me: consider the set of basic formulas. If I am not mistaken, it has no upper bounds except the tautology, and thus the tautology must be its least upper bound.

Am I wrong, or is the site wrong? And if the site is wrong, is it possible to describe a subset of the FBACMG which provably has no least upper bound or no greatest lower bound? If not can I prove that it is impossible to describe such a set?

I know that it is provable that the boolean algebra on countably many generators is incomplete. For the sake of concreteness, let's call the generators p1,p2,p3... and refer to them as "basic formulas". I have been looking for a concrete example of a subset which lacks either a least upper bound or a greatest lower bound. In fact, I almost gave up; I've been trying to do this for months.

However, I stumbled upon this page: http://thue.stanford.edu/bool.html

The author claims that in the boolean algebra on countably many generators (BACMG), "any set X of variables has a [least upper bound] if and only if the set is finite."

This seems wrong to me: consider the set of basic formulas. If I am not mistaken, it has no upper bounds except the tautology, and thus the tautology must be its least upper bound.

Am I wrong, or is the site wrong? And if the site is wrong, is it possible to describe a subset of the BACMG which provably has no least upper bound or no greatest lower bound? If not can I prove that it is impossible to describe such a set?

I know that it is provable that the boolean algebra on countably many generators is incomplete. For the sake of concreteness, let's call the generators $p_1, p_2, p_3,...$ and refer to them as "basic formulas". I have been looking for a concrete example of a subset which lacks either a least upper bound or a greatest lower bound. In fact, I almost gave up; I've been trying to do this for months.

However, I stumbled upon this page: http://thue.stanford.edu/bool.html

The author claims that in the boolean algebra on countably many generators (BACMG), "any set $X$ of variables has a least upper bound if and only if the set is finite."

This seems wrong to me: consider the set of basic formulas. If I am not mistaken, it has no upper bounds except the tautology, and thus the tautology must be its least upper bound.

Am I wrong, or is the site wrong? And if the site is wrong, is it possible to describe a subset of the BACMG which provably has no least upper bound or no greatest lower bound? If not can I prove that it is impossible to describe such a set?