In my previous post, I looked at Euler's infinite product over the primes derived from the harmonic series (the inverse natural numbers). Euler's infinite product over the primes is in fact an infinite product of inverse prime geometric power series. I consider this mathematical object the generating object of all inverse natural numbers. It kind represents the reverse of factorization. Instead of taking the inverse natural numbers and determining what prime factors a specific inverse natural number is composed of, we are instead generating/creating all inverse natural numbers from their generating object that supplies all of those prime factors, and this is basically just simple algebra, multiplicative distribution of the infinite product of inverse prime geometric power series.
But then I closed my prior post with this.
The above infinite paths, which can also be represented in the infinite multiplicative sequences model, never terminate to 1,1,1,1,1,... in their rightward progression. There is always another term in the rightward progression that is not a selected value of 1. The question is, do they really exist?
Things like the harmonic series are certainly fun. I would tend to think your infinitely factored set is related to the harmonic series set having infinite members (even though no member itself is infinite). Infinities are headache inducing and divergent infinite series means you can't even just check for approaching a limit but it actually is fun reading your work.
It has been 30 months since I received that response. I never replied back, as I was baffled. I couldn't understand how the infinitely factored (multiplicatively distributive) paths could not exist, how anyone could not see from both the visual trace path model and the infinite multiplicative sequences model that these infinitely factored inverse numbers existed. I also did not think the part of the response about infinities being divergent and difficult was even relevant to the question. The question is whether the infinitely factored numbers exist or not, not that they may be difficult to deal with if they do exist. That difficulty has nothing to do with whether they are extant or not. I had hoped someone would actually look at the logic and methods, and comment on the validity of the logic and methods. If the logic and methods are wrong, then why are they wrong?
I have come to the conclusion that not being able to see the infinitely factored numbers may come down to what I started this series of posts about to begin with, and that is the shortcuts and assumptions built into our mathematical education whose foundation is the counting numbers 1, 2, 3, 4, 5,...
More Than the Sum of Its Parts?
Euler's infinite product over the primes was derived from the harmonic series, the infinite summation series of all inverse natural numbers (all inverse counting numbers). How could the derived infinite product result have more numbers in it than Euler initially started with? It is counterintuitive. But that is indeed the assumption drilled into everyone from our mathematical education based on these counting numbers. The counting numbers are the foundation of everything we have today in mathematics and probably in society as a whole, all science, engineering, physics, etc. This foundation is drilled into us from when we are three years old and dominates our world view, with all of its assumptions and shortcuts. The counting numbers are basically the first principles in the hierarchy of mathematical education from day one. The question is, should they be the first principles that everything else is built on?
To me, the answer to my previous question is No. I think the infinite product of prime geometric power series should be the base, the foundation, the fundamental object, the first principles, and what arises from it becomes the second step.
If we examine the definition of the counting numbers or inverse counting numbers (naturals or inverse naturals), every element of each set has a finite value, meaning a finite set of prime factors. That is their definition. But if we take the infinite product of prime geometric power series as the foundation, it is plain as daylight that there are infinite multiplicative distributive paths that never have that infinite progression to the right where all values progress with an infinite signature of all ones, 1,1,1,1,1,... The only way to get rid of the infinite paths that always have another term to the right that is not a selected value of 1 is to impose a rule, an artificial constraint from the outside on the infinite product of prime geometric power series. We would have to create a rule that said, "No distinct/unique infinite multiplicative distributive path can have a signature where the infinite progression on the righthand side never becomes 1,1,1,1,1,... Now we can create that rule and inject it into our mathematical logic, but to me that is nonsense.
If we look at the infinite product of inverse prime geometric power series and use the infinite multiplicative sequences model as a proxy for the multiplicative distribution and we add that artificial new rule we just created, what would that mean?
Let's look at that.
In the infinite multiplicative sequences model, each term corresponds to a term in the infinite product of inverse prime geometric power series. There are infinite terms in the infinite product of inverse prime geometric power series, so we must have infinite terms in our infinite multiplicative sequences model. The values that are available to choose for each term of a multiplicative sequence are the corresponding values available from the corresponding term of the infinite product of inverse prime geometric power series; those values are either 1 (one) or an integer power of the corresponding prime number.
But that new rule we have just created limits us. It tells us that every distinct/unique combination of sequences we create must, at some point in its infinite rightward progression, somewhere there in that progression, we no longer have a choice. For a number to be finitely factored, every distinct/unique sequence you come up with, somewhere in that sequence, the choice of selecting either the number 1 (one) or a corresponding prime power for the next term, there is no choice, from that point rightward, you can only select the value 1 (one) for all of the rest of the infinite terms to the right. In effect, at some point you have no choice, your choice has already been determined for you, choice is no longer possible.
To me, this almost comes down to a philosophical argument. Do we have free will? Do we have choice? That is why I included 'Free Will and Choice' in the title of this post. Does Free Will and Choice exist?
To me, the answer is Yes. To me, I am unable to impose that artificial constraint on the infinite product of prime geometric power series. This is also why I think it is the 'infinite product of prime geometric power series' mathematical object that should be foundational, fundamental. If we make the 'infinite product of prime geometric power series' mathematical object foundational and we do not put any external artificial constraints on it, then what we see emerge from it are two sets of numbers. One where every member of the set has finite factors, and a second set where every member of the set has infinite factors.
Further observations of what I call this foundational mathematical object lead me to think that of all the numbers created by this foundational mathematical object, 1/2 are finite and 1/2 are infinite, and you cannot have one of the sets (finitely valued members) without the other set (infinitely valued members). In fact, both emerge bilaterally, dually, and simultaneously. The mathematical object responsible for creating members of both sets is a prime geometric power series term in the infinite product. If you remove, for instance, the inverse two power series from the infinite product, not only do all finitely factored members of the set of inverse natural numbers disappear from the result (all finitely factored numbers with a factor of 2 in them), but also all members of the parallel infinitely factored set also disappear (infinitely factored numbers with a factor of 2 in them). The two sets are bonded together, one cannot exist without the other existing.
So if these infinitely factored numbers do exist and I cannot see how they do not, then where are they in the literature? Well, basically, they are not in the literature. The closest thing in the literature is the formulation of the surreal numbers with the dyadics, birthdays, and day w, or the recent formulation of non-standard analysis and its notion of infinitesimals. But as I have mentioned before, this information (the parallel sets of infinitely factored numbers) could easily be derived 2300 years ago simply from Euclid's Elements or the Sieve of Eratosthenes, which were both available somewhere around the third century BCE.
I will not go into further information in this post on expanding this observation (the parallel infinitely factored sets arising from the infinite product of prime geometric power series), as proceeding further requires the acceptance that these infinitely factored numbers arise naturally just as the finitely factored numbers arise naturally from the infinite product of prime geometric power series, and until that Conundrum is decided upon, going further would be futile because all further expansion of the concept requires the acceptance of these infinitely factored numbers.
I will, however, briefly mention where my math paper goes that all this information comes from.
Accepting that the infinitely factored numbers exist, my paper continues and expands the same observations to the rational numbers and then to the real numbers via the introduction of a real valued variable into the exponents of every prime power series, similar to Euler's zeta function, then it continues to expand to two additional levels. In all, my paper proposes that there are six levels of numbers that are created by the infinite product of prime geometric power series.
- The inverse natural numbers
- The natural numbers
- The rational numbers
- The real numbers via an Euler like zeta function
- A first level where only infinitely factored numbers reside
- A second, even greater level where only infinitely factored numbers reside
Thus was born 'The Conundrum'.
Will this effort of mine lead anywhere or will it collapse?
Stay tuned...
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