Yes, the law of the excluded middle is an interesting one to exclude. You could also try and go without certain infinities: eg you could remove induction (and things equivalent to induction) from your toolset, and see how far you can go.
There's quite a bit of ideology / philosophy about excluding the law of the excluded middle. But even if you set these aside, it turns out that 'constructive logic' without the 'excluded middle' has enormous practical applications in eg computer science.
There's quite a bit of ideology / philosophy about excluding the law of the excluded middle. But even if you set these aside, it turns out that 'constructive logic' without the 'excluded middle' has enormous practical applications in eg computer science.