The explanation of primes, and their importance to number theory by definition must be both explainable, and justifiable with words as mathematics is an extension of language. That extension is by way of the principle that mathematics consists of ideas that are not only consistent, they are also communicable. Our proof that shows that an transfinite number of primes exists, does so by method of substitution. The principle that an entire idea can be expressible in a single symbol. It is quite possible to reverse substitute all of the mathematical symbols for the language that defines each symbol. Though the proof would become exceedingly long.
Basis for all numbers, and by consequence the basis of Number Theory, stems from the principle that all numbers are products of primes. This is the definition of Prime Factorization. What prime factorization does for us is put the composites in terms of primes. Further, the Sieve method eliminates composite numbers by consequence of their not meeting the definition of prime. That definition being any number that has exactly two factors, one, and itself. By this definition all negative numbers are not prime because they contain the additional factor of negative one. A composite number would be any number that contains more than two factors. Such as four, for example, in that Four has the factors of one, two, and four.
For primes this is obvious by the given definition, but less so for composites. Each composite number essentially is an unfactored product of primes. Knowing this further helps us when we reach beyond the primes we know into the realm of higher primes. This is quintessential when we are working with things like the Euclid algorithm or the Sieve algorithm. At this point eliminating numbers that are not prime is the only real method we know to find primes, which returns us the the “basis for all numbers” hinted at earlier.