*This law states that, for any proposition, either that proposition is true or its negation is.*

My argument are not perfect and might not be completely formal, i'm just a beginner. Stop doing that and just ask or help me to refine my question.

I have doubts and would love to iteratively understand it. Stop showing that you are an elite, its disgusting.

With the omission of the law of the excluded middle as an axiom, the remaining logical system has an existence property that classical logic does not have: whenever , often called a witness.

Thus the proof of the existence of a mathematical object is tied to the possibility of its construction.

Lets say we are inductively defining Natural Numbers.

In extensional equality you have to prove anything about it, so u'll need principle of mathematical induction while in case of Intentional equality u can't use it (Or there is no such principle, ).

This proof by contradiction is not constructively valid.

The constructive viewpoint involves a verificational interpretation of the existential quantifier, which is at odds with its classical interpretation. These include the program of intuitionism founded by Brouwer, the finitism of Hilbert and Bernays, the constructive recursive mathematics of Shanin and Markov, and Bishop's program of constructive analysis.

The chain of these numbers forms in itself an exceedingly useful instrument for the human mind; it presents an inexhaustible wealth of remarkable laws obtained by the introduction of the four fundamental operations of arithmetic.

Addition is the combination of any arbitrary repetitions of the above-mentioned simplest act into a single act; from it in a similar way arises multiplication.

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