/sci/ - Science & Math

Psst. Hey, kiddo. Come over here.

[math] \forall (x\in X). \neg \neg {\mathrm {LEM}} (P(x)) [/math]

where
[math] {\mathrm {LEM}} (A) := ( A \lor \neg A) [/math]

Since by a De Morgan's law, for any given [math]x\in X[/math], the statement [math] \neg \neg {\mathrm {LEM}} (P(x)) [/math] just says that [math] P(x) [/math] and its negation can't both be ruled out at the same time.

But consider now a more particular scenario.
The case where the [math] x [/math] are binary sequences and [math] P(x) [/math] is the claim that a sequence is forever constantly zero.

[math] X\ :=\ {\mathbb N}^{\mathbb N} [/math]

[math] P(x)\ :=\ \forall (n \in {\mathbb N}). x(n)=0 [/math]

If you prove something about an infinite object like a sequence, you can't possibly inspect all values, so much is clear anyhow. If you can prove something about an unending sequence, it's also a finite proof, so much is clear as well.
Given all the sequences and how complicated they can turn out to be, whether a sequence is the zero sequence surely can't be established for ALL the sequences.

[math] \neg \forall (x\in X). {\mathrm {LEM}} (P(x)) [/math]

I hope that helps.

Psst. Hey, kiddo. Come over here.

[math] \forall (x\in X). \neg \neg {\mathrm {LEM}} (P(x)) [/math]

where
[math] {\mathrm {LEM}} (A) := ( A \lor \neg A) [/math]

Since by a De Morgan's law, for any given [math]x\in X[/math], the statement [math] \neg \neg {\mathrm {LEM}} (P(x)) [/math] just says that [math] P(x) [/math] and its negation can't be true at the same time.

But consider now a more particular scenario.
The case where the [math] x [/math] are binary sequences and [math] P(x) [/math] is the claim that a sequence is forever constantly zero.

[math] X\ :=\ {\mathbb N}^{\mathbb N} [/math]

[math] P(x)\ :=\ \forall (n \in {\mathbb N}). x(n)=0 [/math]

If you prove something about an infinite object like a sequence, you can't possibly inspect all values, so much is clear anyhow. If you can prove something about an unending sequence, it's also a finite proof, so much is clear as well.
Given all the sequences and how complicated they can turn out to be, whether a sequence is the zero sequence surely can't be established for ALL the sequences.

[math] \neg \forall (x\in X). {\mathrm {LEM}} (P(x)) [/math]

I hope that helps.