Well, think about a system which would be rudimentary enough that you can list and prove everything that is true about it. Such systems exist. But as soon as you have structures able to contain natural numbers, this becomes impossible : there are true statements that can not be proven within your system. You need to enlarge your system with new axioms to prove those true statements.thomasxc said:im not seeing how math can end.
That's surely not what I meant to say.thomasxc said:mathematical progress could end if we found a set of basic...laws that could explain everything?
In math and music, 2^N refers to the total number of possible combinations when N different elements are combined. On the other hand, 2^n combinations refer to the number of unique combinations when n elements are chosen at a time. In simpler terms, 2^N is the total number of combinations, while 2^n is the number of combinations at a specific selection size.
2^n combinations are a subset of 2^N combinations. This means that 2^n combinations are a smaller group of combinations that make up the larger group of 2^N combinations. To get the total number of 2^n combinations from 2^N combinations, you can use the formula nCr (n choose r), where n is the total number of elements and r is the number of elements chosen at a time.
In math, 2^N and 2^n combinations are often used in probability and combinatorics to calculate the total number of possible outcomes in a given scenario. In music, 2^N combinations can represent the total number of different chord progressions or melodies that can be created using a set of chords or notes. 2^n combinations can represent the number of unique chord progressions or melodies that can be played in a specific song or section of a song.
Yes, the concept of 2^N and 2^n combinations can be applied in various fields such as computer science, genetics, and linguistics. In computer science, these combinations can be used in coding and data structures. In genetics, they can be used in analyzing DNA sequences. In linguistics, they can be used in understanding the structure and complexity of languages.
Yes, there are many real-life examples that illustrate the concept of 2^N and 2^n combinations. For instance, in a deck of cards, the total number of possible combinations is 52^N, while the number of unique combinations when choosing 5 cards at a time is 52^5. In genetics, the total number of possible genetic combinations from the parents is 2^N, while the number of unique combinations in their offspring is 2^n.